If you want to vote by liking/disliking the video: “Agree with me” means 1/3 and “Disagree” means 1/2. Latest update (Nov 23, 2023): 217,332 agree with me, and 97,502 disagree with me.
I thought something similar at first too, but actually it is all carefully crafted to prevent this from being a valid answer. It is only when she is put "back to sleep" that she forgets, and what she forgets, is being woken up. So every time she is asked the question, she remembers the original explanation, the original time being put to sleep, and being woken up that time.
Whenever there's no consensus in probability puzzles like this one, it usually does boil down to subtle disagreements about what is actually being asked, not the answers themselves.
Yeah, "what was the probability that it came up heads?" vs "what is the probability that it came up heads?" can already make a difference to the answer. Only if you define questions properly can you answer them. I suppose that's why they were philosophy papers and not mathematics. In mathematics you need things to be defined unambiguously.
There is clearly a majority consensus on the entire thing with most people leaning towards the real world side instead of the fairytale book side. Why do you think they use a literal fairytale character to point this out? Math is 100% disconnected from reality. A concept. She's literally missing 25% of her ability to know what actually happened. She is at 75% comprehension of her reality since she can't tell the difference between waking up once or waking up twice. But the knowledge shown to her is letting her know, that she has two chances to respond on a tails flip, or once chance to respond on a heads flip. So she can take the chance of being right or wrong about a 50/50 chance twice in a row, or once. Her best chance of answering correctly on monday heads, monday tails, or tuesday tails is to realize that there is no tuesday heads and eliminate 25% of her ability to answer. Thus leaving 3 equal chance scenarios. Her real world probability is skewed by lack of information. Her fairytale probability is 1/2, because 1/2 is 1/2 and everyone knows 1/2 is 1/2.
If sleeping beauty was asked "What's the probability the coin came up heads?", I think she should say 1/2. If she was asked "What's the probability that you've been woken up as part of the outcome of a heads result?", I think she should say 1/3. I think the key thing with this question and the reason there isn't (and probably can't be) consensus comes down to how it's communicated and how we as individuals interpret what's being asked of us with the answer. If your goal is to reinforce your understanding about how the coin works, you are probably a halfer. If your goal is to be correct in answering the question from the perspective of sleeping beauty, you are probably a thirder.
But if sleeping beauty doesn't remember any times she's been woken up, every time is her first. So to her it's always 50/50. Any other wake-up (Tuesday) in _her_ existence never actually happened
I think there should be a distinction between asking "What is the probability A coin came up heads?" and "What is the probability THE coin came up heads?" The question is about THE coin, and given she is awake, the answer is the probability of her being awake.
Reminds me of the football coach who didn't want his quarterback to throw because two of three possible outcomes were bad. Interception and incomplete.😅
No. It’s not. If that was true, you would win any game every second round on average making only random choices, i.e., tossing a coin. Clearly, that’s absurd.
The whole thing can be solved by just acknowledging that there's two possible answers because of the unclear wording of the question rather than arguing over which is right while knowing that the question is ambiguous
the question was simple. What are the odds that the coin landed on heads. It is 1/2. He is confusing people by bringing up the irrelevant fact that she would be right more often if she guesses tails every time, but the question was not whether she would be correct more often.
No, I believe the wording is clear enough. You just have to be an impossible halfer... There is no rule that would be violated if she tied up her hair each time she wakes up. She can have this thought before the game starts. She will not forget her rule as she would also forget the game rules if that were the case. So if she awakens with hair down she can know it's Monday and ties it up and answers 50/50 odds, with certainty. When she awakens on Tuesday and her hair is up, she can be certain heads never came up and the odds are 0. So it's 50/50 that it's just 50/50 Monday and 50/50 that it's 50/50 Monday and impossible Tuesday. Either way, it's not 1/3. Now sure it could be but why force it; don't you want to be right? What if it's impossible to know what someone believes... Since you can only believe in what you do not know, how can someone else know if you cannot? If you knew you'd know it and not have to believe it ... Why does she believe it? We will never know...😢
There's a hidden lesson here about imbalanced classes in a dataset. Halfers are trying to model the distribution of the data generating function, while thirders are trying to minimize some loss function for the estimator.
@@orka6848 no, these are not two approaches to the same question, they are two different questions. Averaging them is kind of meaningless. Estimating the distribution is not the same as minimizing expected error.
funny, but no: the imbalance of the heads and tails here is only due to a deliberate mistake in sampling; because of a sampling error you record "tails" twice when a single "tails" event occurs, but only a single "heads" event is recorded for "heads" events. The dataset is seriously screwed up; when presented with a new "instance", the "thirder's classifier" will have its probability estimates wrong: it will be predicting "tails" with prob. 0.66 but it will only be "tails" with prob. 0.5.
0:01 I just love how the like button on my screen got highlighted with red color. I didn't know creaters could do this, or maybe this is a work of RUclips employee just for this video
The experimenters look on in horror as the coin rests upon its edge. They somberly pull the sheet over Sleeping Beauty's face. After an appropriate period of silence, Erwin asks, "You guys wanna put my cat in a box with an unstable nucleus, a hammer, and a vial of nerve gas?" "Not again, Erwin..."
I think the question is subtly mixing up the probability distribution of the coin toss with the probability distribution that the sleeping beauty was woken up with a certain coin toss. So it really comes down to what you think the question is asking for.
Yeah, one of the confusions is that "what's the probability that the coin came up heads" can mean different things. Halfers think it's a question about the behaviour of coins. Thirders think it's a question about your on-the-spot beliefs about past events.
@@wordsofcheresie936 No, thirders are answering the exact question asked. Sleeping Beauty wasn't asked "did the coin come up heads?" She was asked, "what are the *chances* that the coin came up heads?" In the soccer analogy @veritasium used, he talked about this difference without actually pointing it out. About ten billion humans have been born. So the odds of you being born as you is one in 10 billion. So when I ask if you are you, what is your response? If I ask what were the *chances* that you would be born exactly as you are, what is your answer? The questions are different and so the answers are as well.
The best way to explain it is the way he already did. Let's Make a Deal gives you 3 doors, with only one valid prize, heads. The other two have tails behind them. Then they take away a confirmed wrong door, giving your probability of choosing heads an increase. That's why you always switch the door you choose after the removal of a tails door. This method is simply presenting you with two possible doors but then adding a 3rd confirmed possible door. Your safest bet is to be realistic and realize that the original two doors always had a 1/3 chance of having heads no matter what door you chose. Changing doors still results in a 1/3 chance of choosing the heads door.
When I reached your poll, I didn't understand the controversy. If the question is "What is the probability that the coin WILL be heads?" the answer is 1/2. If the question is "What is the probability that the coin WAS heads?" is 1/3. These are two completely different questions. The first has to do with flipping a coin. The second is about what day it is.
But what if it was a 1% chance ( a 1 on a 100-sided die) to wake up 1 million times? Even if you were asked what the probability of the die being 1 WAS, it was still only 1%. Its unlikely that you were put to sleep a million times in the first place.
I've gone through this, and I think I've gotten to the conclusion that I'm a halver, but only on very specific conditions. I feel like two questions are being asked at the same time and each side chooses to focus on only one of them. Halvers are focusing on, sleeping beauty is woken up, she's asked what's the chance that it had come up heads. The answer is 50%, because it:s a fair coin and regardless of the day the answer is 50%. However, thirders are answering a DIFFERENT question, which is, every time sleeping beauty is woken up, what's the probability of her being right, should she always pick up heads. She's woken up everytime, is asked which one came every time, she picks head everytime, the chance of her being right is 33.3%, but it's not because of the coin, but because they're oversampling the wrong answer. Halvers are talking about the coin. Thirders are talking about sleeping beauty.
I really like how you worded this. And you're 100% percent correct. I personally believe that because of the way that the question was asked that it should be answered from sleeping beauty's perspective just as @rantingrodent416 stated, but the way you acknowledged both points of view without hating on either one I very much respect.
@@rantingrodent416 Well she has no way of telling if she was awaken or not, so her only guiding point would be her understanding of the fact that a coin has only two outcomes, so it would be 50%. If someone flips a coin and ask you what are the pobability of it being heads, with no previous context (as sleeping beauty didnt remember if she had been awaken) you would answer 50%, because there is no way for you to say how many times you have been asked that question.
The secret to this problem is that it is a trick question attempting to ask 2 different questions at the same time. Attaching probability to it just makes people think there is something more profound happening.
Yeah I agree, it's more about semantic than statistic. Derek just found a nice trick to get tons of likes and views with a question that is more intellectual masturbation than anything else.
@@En_theo exactly. And I love Derek and his content but this video just felt like a gotchya. And the worst part is I can't even express this to him by downvoting the video
I agree it's a trick question, but it's not two different questions. It's just one invalid question. The tail scenarios cannot be viewed as two separate outcomes: informationally they are identical to sleeping beauty, and therefore the same outcome. The question just arbitrarily labeled the tail scenarios as two outcomes, not with any kind logic compatible with reality, but with memory erasing magic.
@@lukatore123 - I think it was more like 2/5. Croatia's got a great team (maybe the best one per capita - amongst Uruguay and Portugal). Brazilian team, of course, had better individual quality, but Croatia had a very interesting collective game. Afterall, i think it was very well deserved
I think its the phrasing of the question that made this controversial. What if the question were " What is the chance you've been awakened due to a head coin toss?" Then to me its obvious, its one-third. Because sleeping beauty would be awakened more times due to a tail coin toss, even if she knew it is a fair coin. But if the question were " What is the chance the coin flip is a head " (With prior knowledge that she knew it is a fair one), it then would be 50-50.
What if the rules dictated that she would only be awakened and asked the question if the coin flip game up tails? Then, there would still be a 50-50 chance that the coin flip was heads. But given the information that she was being asked the question, she would know that the coin flip was not heads. The fact that she is being asked the question gives her additional information. What makes this "controversial" is that some people are unwilling to adjust their beliefs when given new information.
It's still just a matter of what's meant by the question. If a flip a coin, and you see that it's heads, and I ask you, what are that chances the coin landed heads, there are two answers depending on how you interpret my question. Either you answer 50% if you take my question as "what was the chance of what you've just seen occuring in general" or 100% if you interpret my question as "what is the chance that what you saw (the coin landed heads) is the actual state of the world (the coin landed heads)"
The probability that she guesses the side of the coin is ~1/6. ~1/2*1/3=1/6 But if you ask about the probability objectively, then of course ~0.707 It has no corelation to multiverse unless it exists (probability of Multiverse unknown)
I think this is more a problem with the question having multiple valid interpretations than it is an issue of the question having multiple valid answers. Halfers are focusing the question on the origin of the random event that causes a decision to be made at the start(i.e. the flipping of a coin). Thirders are focusing on the end result of the overall experiment (i.e. the number of ways sleeping beauty can be woken up). The tricky part in this whole scenario is that the question is presented as a single event with a single function to model it. However, from my perspective as a programmer, this scenario is better described as a chain or series of two functions. The first one generates a random 50-50 result (flipping the coin). That random result (heads vs tails) is that function's only output. Everyone can agree on the probability of each result for that function on its own. Now we take that outcome, and use it as the input for a separate function. This second function simply makes a decision on the number of times to wake sleeping beauty up. It becomes pretty obvious when looking at this function in isolation that its results are skewed towards the side that wakes her up more times. The second function essentially multiplies the likelihood of the input that would cause multiple wake-ups. Thus we arrive at the two interpretations of the original question and their different answers. Interpretation 1: How likely is the coin to come up heads? -> obviously 50%. Interpretation 2: How likely are you be woken up by the coin coming up heads vs tails? -> obviously 33%. Both are valid and so my personal stance on it is that the question is ill-formed by being ambiguous.
agree with this, but would say I'm a halfer in this instance because the exact question asked is 'what do you believe the probability of the coin being heads?' not 'what do you believe the probability of being woken up by the coin being heads?' subtle difference, but to one question I'm a halfer, the other a thirder.
@@superkeefo6951 This. That question sounds to me like question that would be asked in a hospital to check if my brain functions correctly like what's the date, who is current president etc. It made me 1/2er just because of semantics but I understood what he meant and in that context I'm 1/3er, so I don't know whether I should like or dislike
@@0NeeN0 but if you're saying there is context then you are essentially adding it and rephrasing the question given to you to be the second question. That's the point momo was making, the implied context makes you think you need to answer the second question. But really the question should be asked with that context or else it's 50/50
My reactions when I see a Veritasium video. Amazed by the title-> Understands the concept-> Trying to understand deeply-> Gets lost-> Forgets what was the video about-> Perplexed about the reality-->Video ends->Hits the like button.
"Good Morning, Beauty. What do you think was the probabilty for the coin to be heads?" "50%. Can I keep the coin?" "Sure." "What time is it?" "Well..." *opens Harry Potter's Gringotts vault*
4:30 but if you divide each section by the total number of tests, each section gets around one half of the probability. This is cause the two tails outcomes are inseparable. I think people need to stop separating the two tails outcomes because they’re honestly the same thing. If you get tails, both Monday and Tuesday will happen, not one of the other. So I don’t think that they have 1/4 or 1/3, but Monday and Tuesday both have a 1/2 chance.
Id say it’s rather a linguistic problem: It’s a 1/3 chance that if she is awake, it was Heads. It’s a 1/2 chance that it rolled Heads when she awakens at all.
It's a fairly complex situation, but I agree completely. If you jump to a conclusion you are ignoring the actual dilemma, which is how semmantics may affect our perceptions of the universe. There's no truly correct answer, only a correct answer given a chosen context. You wanna know the probability of heads vs tails? 1/2 You wanna know the probability of Sleeping Beauty correctly guessing if today is Tuesday? 1/3 etc Makes me think how much of actual science is affected by linguistic biases, I would guess most of it.
I disagree. It’s a 50 50 chance if when she’s awake it’s heads or not. It’s a 50 25 25 chance if she is waken when MH, MT, TT respectively, because it’s 50/50 whether it’s head or tails and then if tails 50/50 whether it’s Monday or Tuesday.
So I guess I think if she wants to say the actual probability, she would say 1/2, but she wants to be right more often, she would say 1/3. But does being right buy her anything? If no, I would say 1/2.
I've reasoned about this and I think it is correct to say 1/2. In my opinion 1/3 is simply wrong because it is not equally likely to be in any of the three cases. I'll copy here what I already said in other comments that are lost in the haystack. My opinion: When she is asked about the probability, the coin has already been flipped and its state is determined even if unknown to her. So here the word "probability" should be interpreted as her confidence that the coin landed heads. She is aware of the procedure and she knows that the coin is flipped one time at the beginning. Imagine she is asked the question immediately after the toss (of which she doesn't see the result) before being put to sleep. She would obviously answer 1/2. From now on there is no reason she should change her initial guess because the coin is tossed once for all and there is no subsequent event that could influence the output. It doesn't matter if it's the first or the millionth time she's being awakened: because she doesn't know what day it is she never gains new information and there's no reason she should update her initial guess. 1/3 is simply wrong because it assumes that the probability of being in one of the three cases is uniform while it is not. The probability is actually 1/2 of being Monday and it landed heads, 1/4 that is Monday and it landed tails and 1/4 that it is Tuesday and landed tails. The 1/3 argument moves from the wrong assumption that to the question "what day do you think it is today?" she should be 2/3 sure it is Monday. Actually she is instead 3/4 sure it is Monday to balance for the fact that there is no Tuesday/Heads combo. The probability it is Tuesday is in fact P(it landed tails) times P(it is Tuesday | it landed tails). I put video at 0.25x and he made a terrible error in his experiment. Look for yourself what he does. He simply writes a sign two times when the coin lands tails. He should have tossed the coin a second time to decide where to put ONE sign. If you do it right you get the expected 50-25-25 proportions. I wanna add something to make it more intuitive: in the case she is awakened 1 million times if it lands tails the probability that in any awakening that day is the first Monday is about 50% and not about 0%. Think of it this way: if she is asked "what day do you think it is today?" she is better off answering "The first Monday" because is much more likely to guess it landed heads and hence surely it is the first Monday than to guess it landed tails and then identify one of the million possible days.
The probability of the coin flip doesn't change with the way we want to measure it. If Sleeping Beauty was woken up a million times for a tails flip, it wouldn't make the coin flip any less likely to turn up heads. Being woken up two times instead of one doesn't make one outcome twice as likely as the other, as the thirder perspective implies. If we're asking about the probability of the coin flip alone, like the question in the video (1:04) very clearly is, then the answer cannot be anything other than 1/2. Now, if the question was anything like "For N times Sleeping Beauty was woken up, what is the probability of her being woken up because of a heads flip?", then it'd clearly be 1/3.
Let’s do a little thought experiment: I tell you: „I‘m about to flip a coin. If, and only if, the coin flips heads, I‘ll call you.“ The next day, I call you and say: „I flipped the coin now. What do you believe is the probability that the coin came up heads?“ What would be your answer?
I know it sounds counterintuitive,but the only correct answer for sleeping beauty is 1/3. When she wakes up, there are three possibilities: A: heads/monday, B: tails/monday, C: tails/tuesday. Obviously, A and B have the same probability, because it’s a fair coin flip, so if they would repeat the experiment every week, she would wake up every monday and the coin would have flipped each side 50% of the weeks. The probabilities of B and C must also be the same, because every week she wakes up on tails/monday, she also wakes up on tails/tuesday. So the probabilities of all three possible outcomes are the same. And the sum of the three possibilities must be 100%, because A, B and C are the only possible outcomes, and each time she wakes up, only one of them can be true. Thus, the probability of A: heads/monday is 1/3. P(A)+P(B)+P(C)=1 and P(A)=P(B)=P(C)=1/3
The dilemma is not "what is the correct answer", but "what is the question being asked?". If Sleeping Beauty is asked what is the probability the coin came up tails, her answer should be 1/2. If the question is "what was the result of the coin toss" and the challenge is to be right (significantly) more than 50% of the time, she should answer differently. In other words, the disagreement is not about what the answer should be, but about what the challenge was in the first place. The only sensible answer is therefore: Restate the question as to remove the ambiguity. Or 42. That works too. Same reason.
"what is the question being asked?" is not a dilemma. The question is clearly about "the probability that the coin came up Heads". Answer to that question is 50%. And I agree with you that those who answer 1/3 are answering the wrong question.
@@jonathanlavoie3115 what is the challenge being set, then. Is it to answer correctly on what the coin toss was, or something else? That's the dilemma here - not what is the correct answer, but what is being asked of her in the first place.
If the challenge was « guess the outcome and I give you 1$ » she would answer Tails, not because the probability is 2/3 but because the reward is twice. Just like I give you 1$ if you guess Heads right, and 2$ if you guess Tails right. You would answer Tails not because the probability is higher. It remains 50%. In the SB experiment, the question is the probability it came un Heads.
@@uRealReels Thank you. You're the first person who reply to me so kindly! A short anecdote about me: In my programming course there was an exam in probability and statistics. Three of the questions were about the same problem. In a basket containing 9 blue balls and 11 red balls, what are the probabilities of A) draw 2 blue balls. B) 2 red balls. C) 2 balls not the same color. Questions A and B are very easy. But for question C I knew that the teacher wanted us to use a complicated formula learned by heart. I didn't want to use this formula because 1- The formula is complicated and I'm lazy, 2- I don't like to use a ready-made formula that I don't fully understand and 3- I wasn't sure if the formula really applied to the situation. So, I solved question C by following this simple reasoning: Probability of 2 blue balls + probability of 2 red + probabilities of 2 different = 100%. Total must be 100% because there is no other possibility. As expected, the teacher's formula answer was not the same as my answer, and I had to argue to get the point, but he had no choice but to acknowledge that his formula didn't apply to the situation, and that my answer was correct. I argued my point in front of the review board, not because I needed the point (my average was already 98%) but because I like the truth. That's who I am...
Teo things are for sure: 1) The probability that the coin was tails is 1/2 2) The probability that sleeping beauty has a f*cked up sleep cycle at this point is 100%
I like how you state that the chance is a half as one of the two things that are 'sure', despite the dozens of scientific papers with discourse, this video, the other comments, and the whole nature of this debate. Guess you had the answer all along then.
To me it's the phrasing of the question asked that's important. If every time she's woken up, she's asked "do you think the coin came up heads or tails", she should always answer tails, because similar to the Monty hall problem, there will be more scenarios of her waking up and the outcome is tails. But the question isn't asking her what she thinks **the outcome** is, but instead it's asking her what she thinks **the probability** is. The probability of the coin toss is completely independent of how many times she wakes up, or even if she wakes up at all, and it is always 1/2. So even if she were to wake up and the actual outcome of the toss was tails, she is still correct by saying that **the probability** of the toss is 1/2.
My thoughts exactly! Was looking for this argument. What is the probability of coin came heads - 1/2, because that is the fact. What is the probability that we woke you because coin came heads - 1/3 and is very different question.
but she wasn't asked what is the probability a toss of a coin comes out heads. She was asked what is the probability the coin did come out heads. There is a big difference in asking about the probability of an event that has not occured vs the probability that a specific event has happened in the past so long as you gain knowledge when transition from that past point to the present. One view the point when asked what is the probability of A. Which is 50% What is the probability of A|B (A given B in statistics). The probability of A given I have information B modifies the probability of A having occurred. This is not an independent probability but a dependent one.
i agree with this because fundamentally she can't remember if she been woke up before (according to the experiment) so the fact that she is awake now can't be used to bias the answer dose 50/50 should be the right answer. correlation does not equal causation.
Honestly this seems more like a subtle semantic conflicts rather than a true paradox. I think it arises from a lack or rigorous clarity about the coin in question. Are we talking about the “probability” distribution of any single random coin flip or are we talking about what I will call the “outcome” distribution of a particular coin flip “result” which governs a deterministic finite state machine.
This problem is more of a word problem than a math problem. As i worked through it my understanding of the problem grew and as such my answer changed. The question "what is the probability the coin came up heads?" is two questions, depending on how you parse it. I think thirders and halfers are both correct and wrong, because they're answering different questions. One side is answering the probability of the coin turning up heads/tails when it was flipped. The other side is answering the probability of you being in a state where the coin came up heads vs tails. They're different problems with different solutions. What is the probability the coin came up heads? 50/50. What is the probability i will be right if i guess heads? 1/3rd.
"Came" is the keyword. It's past tense. When an event has already occurred, any information you can access regarding that event changes the probability that it occurred one way or another. What's the probability that the card I pulled out of the deck is the ace of spades? 1/52. But now you draw a card. It's not the ace of spades. Since you've removed that card from the list of possible cards I might have, the probability that the card I pulled at the beginning was the ace of spades is now 1/51. Since it's a past event, new information about it changes the probability. I draw another card. Now it's a 2/51 probability that I have the ace of spades. You draw another. One less possible card I could have, so now it's a 2/50 probability that I have the ace. And so on and so on until all the cards have been drawn and the probability becomes 1/1 whether I have the ace or not.
I do think at least for those fully understanding it that it's about how we value information. Thirders are incorporating the fact she lacks information. Halfers are assuming lacking information is irrelevant. For the sake of Halfers it's important we define the problem of her guess in one single instance based on the rules. There is inaruguably three states in the state space. She's awake on a Monday with tails, she's awake on Tuesday with tails and she's awake on Monday with heads. I actually think it's Halfers that have one extra step of justification. (unless you completely missed this is about the shared information that she lacks information.. it's not a matter of perspective). That extra step is say even though she knows there's three states in the state space there's ultimately only two that matter. The third being she doesn't know it's not Tuesday and heads so the question is like saying it's 50-50 on Monday or Tuesday.
The problem with doing the vote this way instead of a poll is that so many people are going to ignore the beginning and like the video because they like the video and not because they agree.
Knowing Derek, The like/dislike options is a study in of it's self. We'll get another video where the like is the wroner answer and then a later video examining the results.
I liked this question as a vote to the proposition that people expressing enjoying the video will have a massive distortive effect on any attempt at polling. (Edit: Wait don't use comments as polls! Dislikes just bury the poll itself!)
I am pretty sure he knows enough scientific methodology to know this liking/disliking thing is complete bs. It helps increase interaction so I guess it's a smart trick
That's hilarious. I dominated that class because of multiple degrees in philosophy. And went on to teach deductive, inductive, and probabilistic logic. And intro to inductive and probability logic class is pretty much proving the laws of statistics and much harder than any statistics class I ever took. Stats prof definitely hated me tho.
I think the real question is: does it matter what Sleeping Beauty thinks or how many times she’s right? The probability of the coin toss is still 50/50. EDIT: It seems (like most logic questions) that this is really a semantics issue. Is it: probability coin is heads based on it being flipped once, or based on which way the coin is facing up when she wakes up. So we’re not really learning any deeper truth to the world with this question, it’s just a matter of was our specific setup properly explained
Right. The important part is "she doesn't remember any times she's been woken up." So every Tuesday _her_ may we well have never happened. To her it's always the first wake up, which to her is 50/50.
But that wasn't the question. Thats the entire point (in my opinion) of this thought experiment: There are additional parameters at play (how often she is woken given a certain outcome) and given those parameters what are the odds? Put it another way: What if heads doesn't wake up? Then whenever she waked it will be 100% tails, even tho the coin has a 50/50 propability.
right? doesn't change that she'll wake up twice, it's not as if the coin is being flipped again everytime she wakes up. it's just that if one happens one set of events happen and if another happens a different set of events happen. no matter how frequent .
To me the answer is clearly 1/2 because the question is "what do you believe is the probability that the coin came up heads?". If the question is changed to "do you think the coin came up heads or tails?" and she'll lose 1 dollar if she's wrong, then assuming the purpose is to minimize the loss, then she should answer "tails". The last scenario above is the same as this scenario: * A fair coin will be flipped . Let's assume there are only 2 possibilities: the coin will come up heads or tails. * After it's flipped (you don't know the result), you're asked whether it came up heads or tails. * If it came up heads and you mistakenly answered tails, then you'll lose 1 dollar. * If it came up tails and you mistakenly answered heads, then you'll lose 2 dollars. In this scenario, any rational people who want to minimize their loss should of course answer "tails". Does it mean that in this scenario any rational people should also think that the probability of the coin came up heads is 1/3 (lower than the probability of the coin came up up tails) and that is why they should choose "tails"? Of course not. The probability stays the same for each (1/2). The rational people pick "tails" to minimize the loss, not because the probability of the coin came up tails is greater. I'll explain it again using the Brazil and Canada soccer match. Say that you agree that Brazil is a better team than Canada. Then you're asked who will win. The rule is that if you're wrong, you'll lose 1 dollar. In this case I'm sure you'll answer Brazil. Then say I change the rule as follow: * You'll lose 1 dollar if your answer is Canada and turns out Brazil won. * You'll lose 1 million dollars if your answer is Brazil and turns out Canada won. I'm pretty sure any rational people who want to minimize their loss will change their answer to "Canada". Does it mean they now think that Canada is a better team? Of course not. They still think that Brazil is a better team. They changed their answer to minimize the loss because the rule is changed, not because Canada suddenly becomes a better team.
Excellent analysis, however we do not say head or tail, we say HEADS or TAILS. It's so universal I have to assume a western English dialect is not your first language.
This is exactly how I was thinking about it, exactly what question is being asked matters. I’d be curious to know if this ambiguousness in phrasing occurs in other languages
Take away QM and there is no randomness in a coinflip. The coinflip outcome at any moment is predetermined from the beginning of time; however, since we have no information about the exact physical representation of an instance in time, we must make a statistical inference. This is where the value of 1/2 comes from. The sleeping beauty is asked what she personally believes the probability is. Since she is given more information about the coin, she is able to make a more informed response on the probability of this specific sample. If you saw the coin landing on tails, and I asked you "what do you believe the probability of this coin landing on tails is", you would say 100%. This is because you have the information needed to make a more informed response than 50/50. The same is true for the sleeping beauty.
My thought process for picking 1/2 is as follows: The coin is flipped only once. In the Tails scenario, both wakeups originate from a single coin toss. Since the coin is fair, the question if heads was up would be 50:50 for me. In my mind, there's no "third option" like shown on the paper (4:18), because whether its monday(tails) or tuesday(tails), it's still the same coin toss. If we sort by heads/tails instead of monday/tuesday, we have heads(monday) or tails(monday/tuesday). Now, if we rephrase the question as "What's the probability you were woken up because the coin landed on heads", then it's 1/3, because only 1 out of the three total wakeups originates from heads.
What if we change the problem, such that if the coin lands heads, she is never woken up. If the coin lands tails, she is woken and asked the question. In this situation, it's the same as if she can still see the coin on the table showing tails. The probability is 100% that the coin landed tails.
If she's asked EVERY time she woke up, then it'd be 1/3 because two times when asked, it had been tails. If she was asked only once, decided by the coin flipper, then it should be 1/2.
I don't think it matters to rephrase the question. If she had a record of how many times she guessed the face of the coin correctly through trial and error she would get to the probability being 1/3rd for heads. But this is only because she doesn't know if its Monday or Tuesday. So I agree with the first part of what you said. She you and I know the coin toss 50-50. But what's asked if is it's actually heads when she wakes up. This actually flips the assumptions around where it becomes obvious it should be 1/3. But I think people misunderstand what 1/2 would actually mean. It means that because she has no connected information between the time she wakes up the probability remains 1/2. So to believe 1/3 means you believe her inability to have information about waking up a second time is information.
@4:20 The reason there’s 1/3 of each in this scenario is precisely because the odds of the coin flip are 50/50. By virtue of the procedures you’ll always write down 2 tick marks for each tails. So each of the tails column will always be exactly the same. They are tied at the hip, so it doesn’t make sense to think of them as distinct probabilities. Then by virtue of probability between heads and tails, the number of times heads is flipped is the same number for tails. So you’ll always end up with 3 columns with the same number of tick marks. The third column is not a result of probability, but it was designed into the problem from the beginning.
If you really think about it, why should tails be weighted twice as much as heads (i.e. why should tails receive 2 tick marks and heads only receive 1 tick mark)? Then the coin will no longer be a "fair" coin since the results is weighted.
As he gets too at the end, they're really answering two different questions Halfers are answering the question what percentage of the time the coin was flipped did it land heads Thirders are answering the question what percentage of the times the question was askwed was heads the correct asnwer For the weighted coin flip at the end, 4/5 times the coin landed B, but 30/34 times the correct answer was C
Veritasium: it's a fair coin Sleeping beauty: is it fairer than me ? Veritasium: yes, we are living in a simulation Edit: wow thanks for the likes . Actually I was confused between the snow White and the sleeping beauty. Snow White is the fairest of them all. That's why she got killed
Prince Charming: [kisses Sleeping beauty] Researcher 1: Stop! you're wrecking the experiment! Researcher 2: Interesting, this proves we live in a disney simuatoin.
Derek: Okay so the game is about to start and you fall asleep... *ad starts to play and shows you a product that claims to help you sleep better* Me: Simulation theory sounds just about right.
The confusion arises from the same term “probability” being used for two different things: 1. the probability of getting heads when a coin is flipped (50%); 2. the probability of Sleeping Beauty in her confined situation guessing correctly if she believed that the coin had come out heads (in the past!). SB’s chances are, of course, skewed to tails. On Tuesday she may only guess when it had been tails. Had it been heads she would sleep and could not guess. In other words her guess entails her own dependence on the coin. Imagine you are lying on the operation table: The doctor tells you that you have a 50% chance of dying and never waking up from the narcosis. But what should you assume after you wake up? That the doctor comes and tells you: “Sorry, I goofed-you’re dead!” ??
exactly my answer. perspective and probability are two different things. and counting one event twice, as he did in the experiment when he got 1/3rd each does hurt people who do statistics.
In this case ‘probability’ refers to her ‘credence’ of the coin being heads, i.e. her subjective confidence all things considered that the coin was heads. That she would have slept through Monday on a tails flip is completely irrelevant as her credence of each scenario is not equal. It would only be rational to assign a higher probability to tails if her waking up eliminated some possibilities of heads which it doesn’t, so the chance of her being in a heads-world is exactly the same as being in a tails-world. Your doctor case is not analogous, since waking up eliminates all possibilities in which you die, so you gain information from the fact you wake up.
Sleeping Beauty actually does gain information from her awakening: It's no longer Sunday! It's either Mon- or Tuesday. On Sunday the coin is flipped: 50% chance for heads or tails. SB's Sunday credence is intact. Now she is awakened: Oops! Is it Monday? Is it Tuesday? She knows it not. Her presence depends on her past. Of course, her memory of the Sunday chance seems intact: 50% for heads or tails. But her memory of that past lies in the very presence which depends on it: a loop--not to be trusted! Ask her this question now: "What do you believe, my dear SB, is your chance of being awakened again?" Hm..., ...Monday 50%, ...Tuesday 0%. How to answer? She's no longer in a "Sunday mood". SB's credence has been compromised by the fact that her beautiful presence may have been tossed into a "Tuesdayish" tails-tails-nightmare already. The very bed she sleeps on has been gambled with. There's a chance she lies on a doomed bed (at least until the Prince of Mathematics appears on a white horse; allow me to cry for a while--but only with one eye). The SB-problem wants to not just entail the tossing of the coin but the tossing of the tossing itself. SB has already been tossed and turned in her bed (lousy sleep?) before she awakes. Her answer doesn't come from a 100% heads- or tails-world. Btw., to address another point in the video, I think, it doesn't matter if she'd be awakened 1.000.000 times with an original tails flip: It's Monday versus 999.999 days presenting Tuesday. The tossing has only been tossed once--not a million times.
For me, it is the wording of the question that tells me 1/2. "What is the probability" is a different question than "which outcome do you think happened".
Exactly. This is an independent event. Probability conditional on being awake though, I think that's different although I'm not sophisticated enough in probability to know how 😂
@@aubreydeangelo Not necessarily. The claim of them being independent is contentious among theories of probability. According to Bayesian probability theory, probabilities aren’t objective; instead, they reflect our degree of belief in X given Y information, so the totality of our information on the scenario actively affects the “probability” in the epistemological sense of an outcome. The existence of objective probabilities is tenuous at best; Quantum mechanics wave function collapse is a possible exception, be it contested. They are of course competing frequantist theories of probability however it being independent is not at all intuitive or obviously true.
Exactly, the question is ambiguous. There are 2 questions being conflated. - what is the probability that a fair coin came up as heads this week? - what is the probability that we woke you up because the coin came up heads?
@@nikhilweerakoon1793 There's no need to tap into some subjective probability nonsense. There are two probabilities at play. Implicitly, the question is stringing together two dependent probabilities: (1) a 50% chance of turning up as heads, and (2) 100% more likely to wake up due to Tails Let's use another example: I flip a coin. 50% chance it's heads and I wake you up. 50% chance it's tails and you die in your sleep. The next day you wake up. "What is the probability it landed on heads?" The probability is 100%. Because you have been woken up. The coin flip was a 50/50 chance, but the waking up was a 100/0 chance. Yes, flipping the coin _in general_ is a 50/50 shot at heads. But now that you have more information (the fact that you woke up), you need to factor that in. If you say "50% chance" because the independent coin flip had a 50% chance, you're just intentionally ignoring additional information in some kind of weird linguistic purism.
When he said "Don't hit the like / dislike button" , exactly at the same time my like and dislike icons in RUclips started "Glowing" .....what is that ? Magic?
On May 24, 1994 Canada and Brazil drew 1-1 at Commonwealth Stadium in Edmonton. I was at that game. It was an exhibition before the World Cup began later in the US which Brazil went on to win. (insert sad emoji of Roberto Baggio here)
5:26 - Reaching into a bag of one white marble and million black marbles is a fundamentally different exercise: the black marbles could be selected independently each other, but if Sleeping Beauty happens to have been woken up on day number 1000000, then she must also have been woken up on day number 999999, and the day before that, etc. Days two to one million are conditional on day one.
Yeah. It's more like reaching into a bag with one white marble and one black marble, but if you pull out the black marble you find out that it's actually a string of beads with 999999 other black marbles hanging out the bottom of the bag.
It’s like having 2 same size bags: one with 1 white marble and the other one with million black marbles. And putting your hand into a random bag to grab all the marbles. According to Derek, the chance of picking a black marble is million times higher. Well, you’ll have more black marbles in average, but the fact that you forget that all the black marbles are a result of a single outcome doesn’t make them being picked independently.
To me, similarly to how in your counting experiment you doubled up the tallies for Tails every time, it's 50% because the tails numbers are being arbitrarily inflated by double-counting. It's similar in my mind to if you said "toss a coin. If it lands on Heads count it as one, but if it lands on Tails, count it as if it happened twice". The coin is still 50% we're just counting it wrong (in my opinion, which isn't worth much).
Yep. Think about the marble example. Its not pulling one marble out of a bag of 1 million black marbles and one white marble. It should be stated - flip a fair coin then if its heads pull a marble from a bag of 1 white marble and if its tails pull a marble from a bag of 1 million black marbles. Whats the probability of a white marble? 50%
@@rakino4418 close. If it's tails, grab all the black marbles. Now you have a lot of black marbles and only one white marble. The chance to end up with that many black marbles is still 50%, but that analogy more closely resembles the mess that is this probability discussion :')
I think it boils down to being a language issue as each group interprets the question as a different problem (and it's probably wise to assume that there are more possible interpretations that lead to the same/similiar outcomes than these two). In the 1/2 case you are effectively asking "What do you believe is the probability of the initial experiment?", which is obviously the given chance, 1/2 in this case. The other interpretation would be something like "What do you believe is the state you have awakened in?", which would be 1/3 as there is one state for head and two for tails. Now you may argue that the interpretation here changes the rules, but an interpretation must be made in order to answer the question. I think it's pretty much a problem of different observers being implied by those questions. From the point of non-Sleeping Beauty observer, which Sleeping Beauty can imagine herself to be, the coin flips and sets off a series of definitive, deterministic events, thus the question boils down to the H/T flip. From the point of Sleeping Beauty herself, she guesses which state she has awakened into, which, given one H state and n T states, results in her waking up to T states n times as often. My intuitive interpretation is the one where only the initial flip plays a role, but I honestly think there is no definitive answer to this phrasing of the problem. It's the "what do YOU BELIEVE" part that causes this. Answering "What is the probability that the coin came up heads?" is always 1/2, because it refers to the initial experiment. The question asked in the video confuses you into evaluating the problem from the viewpoint of both Sleeping Beauty and Non-Sleeping Beauty observers by refering both to the initial flip and the viewpoint of Sleeping beauty.
I don't understand why taking the Beauty POV should equal to "try winning the odds". Because that's the interpretation for the "1/3" answer. I.e. "If you woke up, what's the probability that it happened because of head". Of course the answer is 1/3, but that's the point, why people think about "winning the odds" (similar to Monty Hall), when the question was about the probability of ending in two scenarios, where you wake up 1 time or 2 times. What I'm saying is that the duality is in the interpretation of the question and not being an examiner or beauty. In fack, the football example od the video is quite perfect for this. When you woke up you surely said "Brasil", because it had 80% of winning. Whil, if the experiment would have been "every time you guess, you win 1$, but if you don't guess you lose 10$". Of course it's better to say Canada, because they will ask about it 30 times, so even if Brasil wins, you only lose 10$ compared to 30$. Again, it all depends on the interpretation and type of experiment and not the pov.
I 100% agree and this is what I commented. The reason people see it differently is not because the answer is unclear, but because people inherently interpret the question differently.
That would be the direction of my "explanation". I feel we have not sufficiently transformed our imaginary scenario into hard math to even begin calculating a result.
Excellent analysis. Do you write articles/papers or make videos? I'd love to see more examples of your interpretations of similar ideas/themes. Any comment on the simulation theory part? I've tried reading Nick Bostrom but a lot of it is over my layman's mind.
The initial chance of heads is 1/2. When she wakes up, the chance for it to be Tuesday is 1/4, because there's a 1/2 chance the coin flip is tails, and only then a 1/2 chance that the current day IS Tuesday and not Monday. Same thing for waking the Monday after a tails flip. So the chance it landed heads is 1 - 1/4 (for tails Monday) - 1/4 (for Tuesday) giving us 1/2.
The right way to determine the possibility of the coin falling heads or tails is not by counting the coin sides or the virtual possible realities. The flip of the coin is a effect of a cause, that being the force of the hand of the coin flipper on the coin. Imagine that you give the coin to a robot that has the precision to flip the coin with the same force in the same position, he would certainly get one of the sides 100% of the time( if there is no other thing causing the coin to flip, like the wind for exemple).
It would be the same as thinking that if you droped a metal square on a flat table it would have 1/6th of a chance to every side to be hitting the table.
The key difference is that the event where she is asked about the coin happens twice with the same answer if it comes up as tails. The question is worded in a way that no matter what her perspective is, the answer is 1/2. However, if the question was instead: "If you guessed heads, how likely would you be to be right?" Then the answer would be 1/3 because she gets asked the question once if she's right, and twice if she's wrong.
In my opinion the question you seem to ask is: What is the probability that today isn’t Monday? The probability of a coin flip is always a coin flip as we say(50-50).. but the question refers to the coins state only in regard to what day it might be.. meaning that if it was H she’ll be asked once, while if it was T she’ll be asked twice, so it’s not really a philosophical problem, just that the question appears to be misleading
I agree, I put it this way: If you ask her what is the probability "a coin toss" *will* be heads or tails she should of course answer 1/2. You'd be referring to a random event that hasn't happened yet. But in her case the event has happened and is no longer a probability, she exists in a timeline created by its outcome. Really she's being asked, what is the probability you wound up in the timeline where a tails came up. She's more likely to be in the tails timeline by virtue that waking up in that timeline is statistically more probable.
The answer to your question (What is the probability that today isn't monday?) is actually 1/4. Since there is a 1/2 chance she is woken up only on monday and a 1/2 chance she is woken up on both days and then it is 1/2 chance for either day. The original problem is indeed flawed though. There are 2 ways to interpret this: 1. The experiment is only done one time, so the answer is 1/2 since the coinflip is either heads or tails. 2. The experiment is done an infinite number of times. In this case the answer is 1/3, since now you are woken up twice as much in case of tails. Even now we can see that the quenstion is badly phrased as the probability the coin came up heads is still 1/2. The probability she was woken up following heads did change: it is now 1/3.
@@lennertdevoldere7000 the coinflip probability being heads is 50%.. theres no debating that.. now the question refers to probability but then extends forward to requiring the "what day it maight be" reference as a parameter, which is irrelevant to probability. thats why im sayin its not a paradox but it only gets wierd with a misleading explanation. and the answer to my question is 1/3. yes theres 1/2 chance she gets waken up only monday and 1/2 waken up both days but in her world thats irrelevant.. like @neodonkey mentions above "she exists in a timeline created by its outcome". and to simplify matters lets say the coin is fair and we run this experiment 6 times.. so the outcome would be 3 times only monday and 3 more times both monday and tuesday, that makes 6 possible Mondays and 3 possible Tuesdays = 2/3 and 1/3..btw this is not my question, its the question implied in the explanation.
@@lennertdevoldere7000 no matter what the coins state is, the subject is goin to wake up on Monday, but theres 50% chance she also wakes up on Tuesday.. 100% Mondays, 50% Tuesdays which means 2-1 on favor of Monday which is 2/3 on a Monday possibility and 1/3 on a Teusday. Repetition only hepls visualize because you add up the outcomes on both coins states and just divide by the reps.. What your saying would be correct only if on heads she were to wake up twice on a Monday and on tails Monday-Teusday as is..
I am on the 1/2 side, here is my reasoning: The many wakings are not independent events (I think). If any one of them happens, you are certain that all of them will happen as well. Because of this, I think you can imagine all those sequential awakings as a single event. That removes the bias of "many possibilities in one branch" and you are left with single event in each branch. edit: This also makes sense to me for the argument of "no added information", because the sequential wakings are linked together.
That's the whole point..... The events are conditional from researchers perspective.... From sleeping Beauty's it's not....she doesn't know what number of time she has been woken
But any time you go to sleep, you either wake up and are asked about the coin flip, or wake up and are told the experiment is over. The fact that you're being asked about the coin flip *is* information, and it tells you that the experiment hasn't ended. Knowing that the experiment takes longer to end if you flipped tails, you use the information to update your probabilities.
@@elliotgengler3185 You could also take this scenario in a darker direction by slightly altering the experiment, whereby a 'heads' coin flip results in Beauty never waking up at all. If she is then awoken to be asked about the coin, she knows that the coin must have been tails.
Waking up on Tuesday always comes after the coin flips Tails. There is no other reason for this event. If we want to correctly relate the question to the situation it's incorrect to put more variables into the equation such as "there is one more day in which she may be awake". The question has to be only related to the thrown of the coin which possibility is 1/2.
But then you would disregard the information she already has, and information is important in mathematics. If she gets woken up, theres a 1/3 chance that its tuesday.
@@gorgit That's not the question. First, she has no information other than the rules of the game. The question is "what does she believe the probability is that the coin landed on heads?" When she gets up, she isn't given anything other than the question. 50%... problem solved.
@@JohnJJSchmidt she is given knowledge that one of the days she could be awoken is Tuesday, but she doesn't have the ability to tell what day it is when she is awake. Therefore she has the knowledge that it could either be Monday after a heads, Monday after tails or Tuesday after tails. Even though the question she is asked doesn't change the probability the pre existing knowledge she has does.
@@jaredhahn7970 The probability of what? Again. The question is what does SHE THINK the prob the coin landed heads, not the prob of it being a certain day. If she says anything other than 50% with a blank mind, she is letting the pre-existing knowledge of the rules interfere with really basic reasoning.
If we phrase it as “what is the probability that she woke up in a scenario that the coin was heads?”, it’s 1/3; but yeah, given that the question is, “which way did the coin come up?”, it’s still just 1/2.
This is what happened to me during my school exams. I would forget what I studied the previous day. The teacher would think the probability I studied for the test was 1/10 from the marks.
The problem here was your approach to study. That last-minute "cramming" is easily forgotten, especially in a stressful situation. Repetition and practice over a much longer period, or reflective self-study to the point where you reach genuine understanding, was the way to make sure you passed those tests. But I'm completely aware that very few kids at school would heed that advice, including my younger self!
A lot of the other scenarios were not equivalent to the Sleeping Beauty scenario. They were more like asking Sleeping Beauty "Do you think it's Monday?" That's an entirely different question from "What are the chances the coin landed on heads?"
This is a brilliant comment. Contrasts very well the difference that is muddled in the "what was the probability" question. The answer to, "what is the probability the coin was heads?" is objectively 50%. The answer to the question, "what is the probability that the coin was heads AND that your answer is correct?" is 1/3.
@@bbanks42 What? So for the first question your answer doesnt need to be correct for it be... correct? The answer for "what is the probability of the coin to flip heads" is indeed 50%. But thats not the question, the question is "what is the probability the coin FLIPPED heads" with the given that if you are being asked that question you woke up. Similarly if someone flips a coin and it results in a Tail, it would be correct to say the probability of flipping tails in the past is still 50%, but wouldnt be correct to say the probability the coin that was flipped was tails was 50%, because you are clearly already seeing the result, and its 100%. Imagine if the SB only woke up if the coin was Tails and was asked "what is the probability the coin FLIPPED heads?" , it would be ridiculous for her to say the chance is 50% after being asked that question, because she knows she wouldnt be asked that question if it was Heads.
Makes no sense. The setup is wrong. You cant make her forget that she woke up yesterday .... and then ask a logical realistic question. If she cannot remember yesterday, then the asking person might have forgotten who they are altogether, or whom to ask. Like : you are my banker, with a brilliant mind, and you can recall all of my bank statements from memory. But i always forget my adress, my name, my job and which bank to go to. Now you want to ask me a meaningful logical game theory question on how to save money better ? Makes no sense.
....also the coin is a half half deal. The monday or monday-tuesday thing is a scam. Please try Mon and then Mon Tue with a coin. Heads comes up .... or both sides come up. Great.
Another variation that should provide the same answer: 1) Wake SB both days. 2) Have her write down on a notecard the probability for the event "Today is Monday and the coin landed on Heads." Below that, write the three other probabilities for Monday&Tails, Tuesday&Heads, and Tuesday&Tails, 3) Lock the notecard in a safe that can only be opened by her voice. 4) Give her the amnesia drug. Include the sleep drug ONLY IF it is Tuesday, after Heads. 5) Once amnesia has taken effect, if she is awake, have her open the safe. 6) Ask her, with the full knowledge of these steps and the notecard, for the probability that the coin landed Heads. The correct probabilities, in step 2, are all 1/4. In step 6, she can cross out the probability for Tuesday&Heads. The remaining three need to be renormalized do they sum to 1. You do this by dividing each by their current sum, 3/4. This makes each 1/3, But step 6 is identical to the video's problem. With the full knowledge of the procedures, she didn't need to go through steps 2 thru 5, she knows those probabilities applied at the time just before she was wakened. The answer is 1/3. The error made by Halfers is not recognizing that Tuesday&Heads is just as much a possible outcome of the experiment as the other combinations. What causes them to make this error, is pretending that her being awake is a prerequisite for it to exist.
I think the confusion arises because of the question "What is the probability she should assign to the event?" It's a separate issue of the probability she should assign versus what prediction she should make due to her being asked the question more frequently given a certain outcome.
This is what confused me about the question. The probability the coin flip was heads is 1/2; but the probability that she was woken up by a "heads" coin is 1/3.
Yep. It's a whole genre, these silly faux-dilemmas. They try to pass-off a multi-step process as a one-and-done. In the case of SB and similar DT puzzles, it's, as you say, more than one prediction, serially. In this guy's argument, he even replaces the fair coin with an extremely unfair coin, if you will, and STILL courts the controversy. Infantilizing, if you ask me.
The number is still being transmitted with the page, but not being displayed. If you want to see the numbers again, you can use a browser extension such as "Return RUclips Dislike". This is why I'm able to tell you that at the moment, this video has 114k likes and 61k dislikes.
@@mattgies The RUclips API doesn't include the Dislike Dataset no more. The Extension just looks at every Extension user which likes or dislikes the video and extrapolate the data to get a rough estimation of the like to dislike ratio
@@anonymouscommentator Youtubе Vanced on mobile does include Return RUclips Dislike as a toggleable setting, so it is still possible to use. However, that estimation is still a factor.
Firmly in the 1/2er camp. Irrespective of how many times she was woken up, the coin flip was a singular event. That event had a 50/50 chance of either result.
You are absolutely correct, the problem is that its a trick question, change the question from "what is the probability that the coin came up heads" to "what is the probability that we woke you up when the coin came up heads". Now the second question was not literally asked, but it was presupposed by the fact that she woke up being part of the experiment and with the information she was told prior to the experiment. With the second question, its 1/3, it has to be. But the first question tricks people into believing its just about a coin toss, but it is not. It is about her waking up in this experiment involving coin tosses, and because she gets to wake up twice on tails, it weighed in favor of tails.
@@emperorpicard4901 why are you saying he is right then? The question is clearly "what is the state of the coin NOW". Not what was the probability that it would land this way. And it will be heads 1/3 of the times it is asked.
The chance the coin flip will be heads or tails is a 50/50, but the chance it was tails from her perspective, given she's woken up, will be higher, as being woken up in a world where tails was true is more likely given it happened to her multiple times
The question is not about the physical probability represented by the coin toss. It is about the epistemic probability that SB should assign to the outcome -- in other words, her credence. Bayesian inference dictates she assign 1/3 to heads and 2/3 to tails. She knows it is either Monday or Tuesday, but not which. She knows "I am awake on this particular day". For the hypothesis "The coin came up tails", the conditional probability of "I am awake on this particular day" is 100%. For the alternative hypothesis, the conditional probability of "I am awake on this particular day" is 50%. That gives a likelihood ratio of 2:1, which you multiply by the prior odds for tails of 1:1, to get odds of 2:1 for tails (Bayes's rule in odds form). That translates to probability of heads = 1/3.
@@rsm3t What I'm not following there is what value there is in treating the wake-ups from the 2nd outcome as unique/why the day she wakes up matters to the question being asked.
The REAL problem is that there's an implied reward: if she's asked the "probability" of heads, then it's 1/2. If she's being rewarded for GUESSING whether the coin was heads or tails, she should always answer tails, because she'll get rewarded twice in that scenario (vs once if the coin flip was heads).
@@hisuianarcanine9379 Nah, the question was clearly "What is the probability that the coin came up heads", that fits perfectly the first case of OP and it's unambiguously 1/2
Exactly what I think! It's unclear which question is being asked from this video, and we have to be very specific when asking the question. Like you said, if the question is "what is the probability that the coin landed on heads", the answer is always 0.5. Sure, sleeping beauty will be wrong multiple times if the coin landed on tails, but that's not relevant to the question being asked here. The fact that the coin is two-sided does NOT change, and the sleeping beauty's knowledge is identical every single time. It's an entirely different question if sleeping beauty is trying to 'win' as many times as she can, then the best answer is quite obviously tails. If she is woken up N times when tails is thrown, and once when heads is thrown, she will get the correct answer N times out of N+1 guesses, on average when repeating the problem.
@@angivaretv4475 "The probability that a fair coin comes up heads" is undoubtedly 1/2, but "the probability that you are waking up on a heads monday" is 1/3.
@@myeloon It would be a waste of everyone's time to go into the exposition of her being waken up monday monday/tuesday if all we are going to ask is that given a random fair coin that is absolutely irrelevant to her situation. Of course the question is "given that you were just woken up, what is the probability that heads came up". If that is not the intention, I hope the people who developed this problem and six degrees of separation from themnever wakes up again.
Bob watches Alice flip a coin. The coin comes up heads. Alice: "What do you believe is the probability that the coin came up heads?" Bob: "What do you mean? That the coin *has* come up heads? That the coin *would've* come up heads? That the coin comes up heads in general?" Alice: "Pick one." This isn't a math problem. It's linguistics.
Considering how important it is to write formulae correctly in math, it does feel like not just a linguistics problem but one that's intentionally vague so as to make two different outcomes seem equally obvious.
A fun problem where the two answers are actually answering two different questions! The skill is not figuring out which is right, but understanding how the two questions are subtly different. Good thinking exercise and excellent video as usual.
There is no question to which the correct answer is 1/3. The whole thirder perspective is flawed because it treats the possible "states" as equally likely and independent, but they are not independent.
@@nekekaminger The question would be, ‘What is the probability you were woken up by a flip of heads?’ I think the answer to the sleeping beauty question is 1/2 though because like the original comment said they are answering two different questions.
@@reubensavage2067 She's always woken up, otherwise you couldn't ask her. Prepending the question with the pseudo-condition of her being woken up doesn't actually change anything because it always happens. The question is fully equivalent for "What's the chance heads came up?" which is clearly 50%. I see what you are trying to do. You view each waking up event as an independent event and try to assign a probability to that event (just like Derek proposes in the video), but that approach is flawed since they are not independent. Monday Tails and Tuesday Tails cannot happen without the their also happening. Imagine you have a somewhat unusual coin that instead of heads has one dot on one side and instead of tails it has two dots on the other side. Each dot represents a waking up "event". After the toss pick one of the dots you see (which is either just one, in which case the choice is simple, or two, in which case you just randomly pick one, since SB can't remember being woken up before, the order does not matter) and ask yourself "What is the chance I see this particular dot because the coin came up with the single dotted side?". If two dots were up you answer the same question for the other dot. The experiment is exactly equivalent (if you don't agree, please explain). Do you still think the answer is 1/3?
@@nekekaminger The part that I disagree with is that I'd argue they are independent events. She could be woken up on Monday Heads and be asked the question, or woken up on Monday Tails and be asked the question, or be woken up on Tuesday Tails and be asked a question. As others have stated, it really comes down to which question is asked of her. If she's asked "What do you believe is the probability that the coin came up heads?", then she should answer 1/2. Because the coin either came up heads, or it came up tails. It doesn't matter which day she woke up; the coin was either heads, or tails. If the question is "What do you believe is the probability that you were awoken on heads?", then she should answer 1/3. Because as I mentioned in my first paragraph, if she's asked this question on Monday Heads, she would be right. If she's asked on Monday Tails, she would be wrong. If she's asked on Tuesday Tails, then she would again be wrong. So it's a 1/3 chance of her being right about the 2nd question.
@@ThrowAway-hy5sp you haven’t tackled his point that these events are not independent. Monday tails and Tuesday tails are essentially the same event. For the example where she wakes up “a million times” it’s 1/2 chance that she’ll wake up a million times or 1/2 chance she wakes up once. Either way if she wakes up on the thousandth Tuesday and is asked “what’s the chance that you will wake up another thousand or so days”, its 1/2 as is “what’s the chance you only wake up today on the monday”. There’s not “more chance” of waking up in the millionth day like it’s compared to being in a simulation. It would be like saying the chance of you living in reality is 1/2, and the chance of you living in any of the millions of situations is also 1/2. 1/3 would be the answer to “what’s the probability today Is Tuesday” regarding the original question.
1/3 seems obvious because the frame of reference for her evaluation is different than ours. If we think from her frame of reference, then it must be 1/3, from ours, 1/2. Since the question was asked to her in her frame and her circumstances (which offer some certainty whereas ours offer only probability), then it must be evaluated from that frame. 1/3 is the obvious answer. I would also note that the agree-to-disagree ratio is nearly 2/3 to 1/3.
@@jogadorjncbut that’s not how it works in this case right? He did mention that she will forget that she has been woken. To her every time she wakes, she would feel that this was her first time, no matter what day it was. Right?
@@The-Meaning-of-Life-is-42 She gains information by waking up, even without memory. Imagine if the setup was to not wake her up if it landed heads. By the fact that she woke up she'd be able to tell with certainty that it landed tails even if the coin was balanced.
@@jogadorjncbut isn’t the information gained irrelevant to “what’s the probability the coin came to head?” Let’s rather than waking her up 2 days, wake her up instead for 50 days? Wouldn’t that means by the vid logic, you’re saying the chance the coin came up head is near 0%?
The question is the trap as you explained in the video: "What do you believe is the probability that the coin came up heads?" You would have to disagree and ask them to clarify if they mean: "What are the chances of a fair coin flipping heads or tails?" OR "What are the chances you are in either stage of waking up in the experiment". Edit: i hate these kinds of "math problems" since they are almost always about the question being asked in a stupid/inaccurate/unfair way to the situation at hand and then people just going "what if we actually try to answer the unfair question seriously". Then it inevitably ends up with the same conlusion as the first paragraph where the authors assume one of X interpretations of the question and continue to calculate and answer that. But in that case you could have just asked the correct question from the start in the problem. This is why I always tell my friends to think about what they are saying, if it can be understood in mulitple ways it wont help you get your point across. Write so that your intention can only be interpreted in one clear way.
If the question for the sleeping beauty was to tell if the coin flipped heads or tails. She is woken up three times,two times if it's tails and just one time if it's heads. If she says tails all three times, she'd be correct 2/3 times. If she says heads, she'll be correct 1/3 times. In conclusion, the probability of her getting the answer correct if the outcome is heads is 1/3. Whereas,the probability of the coin flipping to either heads or tails is 1/2. She would be right 1/3 times,but then answer is 1/2 as per the question.
@@architlal8594 the question doesn't ask her to predict whether the coin was heads or tails. it asks her what the probability is that it was heads. so her response wouldn't be 'heads' or 'tails'. the trick with this problem is that people are fooled into thinking that monday (tails) and tuesday (tails) are independent events. but they aren't. they're actually the same event. the reason you get the 1/3 distribution is that she gets woken up twice on a tails. and therefore gets asked twice from the same coin flip.
I feel like that arguing with people about politics and society all the time. In the absence of an obvious answer on a lot of those issues - unless you are very well informed on them, which a lot of people aren't, often times people just try to roll you with fallacies like that. I believe usually even unknowingly so and thinking "they got you". But it's very tough to effectively counter that, especially in the moment, because, as this video shows, unraveling such fallacies can be very hard. Often much harder than coming up with them.
Actually a lot people that understoodd this question as "did I flip a heads or tails?" respond with "I dont know its 50/50." This isnt some word game this is a sort of paradox. The people that disliked this video isnt arguing that a coinflip is always 50/50.
Okay, I actually gave it some thought before listening to the proposed answers, and initially I also answered 1/3, but then I read the comments, did some more thinking, and switched to 1/2. First, as others said, the wake-ups are not independent, and the question comes down to "which branch do you think you're in", and the probability of that is 50%. Second, even in the situation with 1 vs a million awakenings, you can't consider being in each of them equally likely. Finding yourself in the branch with a million awakenings is dependent on the actual coin toss (50%), and the probability of finding yourself in a specific position in the second branch must be calculated using Bayes' theorem. 50% it is.
A branching reality is the best way to describe it. You don't care about the probabilities in one branch, only the probability of ending up in one branch or the other.
I think you misunderstood the question. If you wake up a million times more often if it hits tails, then it is a million times more likely that when you wake up, it was tails.
That's why I'm set on 50% too. There's only one coin flip that happens at the start so it's gotta be 50% because whether it's Monday or Tuesday doesn't matter. The question is "What do you believe the probability is that the coin came up heads?", and the probability of a coin flip coming up heads doesn't change when you ask someone "What's the probability of a coin flip coming up heads?" two days in a row. There are three possible states of them asking the question (Monday + Heads, Monday + Tails, Tuesday + Tails), but the question doesn't ask anything about what day it is or if she thinks she's been woken up once before, so the day of the week is irrelevant and only the coin flip matters. But the researchers are asking Sleeping Beauty, who's from a fairy tale originally written in the 14th century, so I think her answer would actually be "What's a probability?"
@@pelleas9091...but if she's woken up on a Tuesday, there is a zero percent chance it landed on heads, so how many times she's woken up does matter, making it 1/3 overall.
Your ad was perfectly timed after you had us “go to sleep” 😂 As soon as the screen went black, it cut to Patrick Stewart’s smiling face telling the camera, “Hello, I’m Patrick Stewart”
For my part, I think fundamentally the question is malformed, and that's why we have such issues with it. There are two possible meanings of the question, and commensurately two possible ways of looking at the data. A: "What do you believe is the probability of the coin landing as heads?" B: "What do you believe is the probability, given you are awake now, that the coin actually was heads on Sunday?" The ways of looking at the data (if we treat it as sampling whether Sleeping Beauty thinks the coin actually did land heads/tails): 1. In each _trial_ of the SBP, which answer will be most consistently correct? 2. For each _awakening_ of Sleeping Beauty herself, which answer will be most consistently correct? If we base our statistics around the per-awakening result, then 1/3 is correct, and indeed it should be 1/(n+1), where n is the number of times you awaken Sleeping Beauty if you flip tails. If we base our statistics around the per-trial result, then 50% is correct. The former is true because, when we look at the percent likelihood *on any given awakening* that Sleeping Beauty was awakened on a trial that flipped Heads, that of course must fall to zero as the number of Tails-awakenings tends to infinity--the vast majority of awakenings are Tails-awakenings. That the latter is true is a bit more complicated, but can be expressed as follows. Perform the same test, but simply ask SB whether she actually DOES believe the coin flipped heads, yes or no. Tally up the answers. If the coin _did_ flip heads, then she will either be right once or wrong once. If the coin _did not_ flip heads, then she will either be right N times (for the N awakenings), or she will be wrong N times. All told, there are 2N+2 possibilities, and out of them, (N+1)/(2N+2) = 50% are correct. Hence, it depends on whether you examine the data from a per-awakening basis or a per-trial basis. The question is malformed, ambiguous, and that is why it leads to an alleged "paradox."
Of course. The statement as 1:53 is simply false. These are two different questions, each yielding a different probability distribution and thus different answers.
It's two levels of abstraction using the same symbol so the English confuses the math. Lets do the same exercise but change the coin to marbles when we present the new abstraction instead of hiding it behind the same name (coin flip) Flip a coin every time it turns HEAD place a RED marble in a bag Every time its face up Tails put two BLUE marbles in the bag Now if we ask the question "What are the odds of the coin" well its 50/50 What are the odds of pulling a red marble out of the bag? Well 1/3 Paradoxes are cool, this isn't one, just a poorly worded question
Yeah this was my thought too. However, it does ask "...IS the probably that the coin CAME UP..." So it is not asking you how often the coin did anything. It is asking how often you will wake up because of tails as opposed to heads, therefore it must indeed be 1/3rd.
@@identifiesas65.wheresmyche95 There is no probability for events that have already occurred. Hence, the question if phrased that way is about whether you *believe* it did or not, and that belief is where the probability component enters the picture. Whether Bayesian or frequentist, you'll be thinking about two things: "What is my belief that a fair coin would have already been heads or tails?" (naturally, ½), or, "What is my belief that this awakening is a heads awakening?" (naturally, ⅓). If we are clear about which question we are asking, the problem goes away. Edit: I think it's actually really useful to treat this as one would the Monty Hall problem. There, it becomes a lot more clear what's going on if you presume a hundred doors, or a thousand, or the like. If you pick one door out of a thousand, and Monty opens *every single other door except one,* would you switch? It seems pretty clear you should. You only had a 0.1% chance to pick the right door at first. Monty has now eliminated every other door *except one.* The odds are enormously in favor of that other door. It just happens to be hardest to intuitively see that when you have the smallest possible number of doors (3, in this case.) We see the same thing with the SBP. We only have three awakenings (well, one vs two). What happens if we make it one vs 999? Further, what if we add some expected value to the answer? Consider: Sleeping Beauty wins $1000 if she correctly picks Heads, and $1 if she correctly picks Tails. The expected value now depends on how you view the question! If we structure things on a *per trial* basis, then half the time the coin is heads, and half the time it is tails (before any awakenings have occurred), this is agreed by all parties. Hence, *per trial,* the expected value is $1000 if she guesses heads correctly, and $999 if she guesses tails correctly. Since each is equally likely *before* any awakenings have occurred, she should choose heads every time; she will net more money, albeit slowly. If, however, we award her the exact same prize for any correct guess on each awakening (e.g. "if you pick a side of the coin and are correct, you win $1"), then she should 100% always choose tails, because she can win $1 on half of trials, or $999 on half of trials. The preponderance of *awakenings* is on tails paths. Someone asserting that the probability must be ⅓ is claiming that, for _all_ experimental setups, the higher expected value for these Sleeping Beauty prizes must be from picking tails. This is not true. By offering prizes based only on the coin's facing, *not* on the number of times Beauty awakens, we can clearly see the difference between the two approaches.
@@identifiesas65.wheresmyche95 - Does she even know about the multiple awakenings? It's not made clear in this video (and it's my only familiarity with the problem).
I feel really bad leaving a Veritasium video on "dislike", I even almost liked the video as a habit after I was done watching it. which also leaves me to believe the like dislike ratio is not to be trusted that much.
Yeah I feel like the number of likes is mostly a combination of people using the like button to actually "like" the video and others who only listened to the first part. The comments seem pretty consistently to agree that it's 50/50.
I think he designed it so that he'll receive more likes than dislikes since statistically, more people will choose the 1/3 option. I'm pretty sure he thought through it.
The question should be asked like "what is the probability that sleeping beauty is awakened on monday and that of heads."(It's a conditional probability). We will obtain 1/2 as answer if the question is "what is the probability that sleeping beauty is awakened on only ones a week or twice in a week.OR even more simpler by stating heads or tails. The probability of a coin flip is always "1/2" untill a conditional probability is involved. EXAMPLE:From two differently colored bags containing two different colored identical balls.probability to draw a one of the colured ball from any of the bag (or) from a specific bag.Then the answer varies.
For me it becomes less paradoxical when I think of the question as rephrased as "How likely is it that Heads is responsible for you waking up this particular time?"
"what's the probability it came up heads" contains a hidden knowledge assumption, that being with the knowledge that it just woke you up or prior to that being someone's state. It's a partly a matter of how you understand the meaning of the question, but the best answer is based on all knowledge, which is that you just woke up. Imagine she's never woken up for heads. She's woken up, what was the probability that it was heads? In an abstract sense 50:50, but based on the full knowledge of evidence, it's zero
The problem here is that to define probability, you need a clear indication of what counts as a trial. If, in a series of these experiments, a trial is marked by each time the coin is flipped, the probability is 1/2. If a trial is marked by Sleeping Beauty waking up, then the probability is 1/3. For any practical application, Sleeping Beauty would be advised to mark trials by waking up, as if she had to bet on the result of the coin, her breakeven point would be the same as betting with open information on a coin that came up heads 1/3 of the time. In a scenario with no applications where she only cared about being "correct," either method of marking trials could be chosen, but having a specific goal for her allows the problem to be concrete enough to be optimized.
even if the trial is marked by each time sleeping beauty wakes up the probability is still 1/2 because her wake is dependent on the coin flip in the first place
@@coolzmasterz the only way 1/3 can be a bit logical is when you have to bet with money on the answer but dosent it make more sense to bet 100% on tails , i mean if your wrong you only lose once but if your right you win a lot
To me it is just how the question is phrased. I'd say it this way: -the probablility of the coin came up heads was 50% -the probability of me being awaken by the coin on heads, is 33%
Sleeping Beauty not remembering her previous wake-ups is akin to a coin not remembering its previous flips. The # of times she is woken up can be thought of as the payout. In this case, betting tails would yield a greater payout, but it does not mean tails will come up more often.
"but it does not mean tails will come up more often." - that's exactly the point though. When do you "count" a flip? Only when it's actually flipped, or when the sleeping beauty is awakened? From the beauty's perspective she cannot know if she's awaken for the first time or the second time, so she is forced to count the two separately.
@@ShaharHarshuv Are we in agreement that this occurs in a single coin flip? The scenario is if that flip goes heads, [outcome], if it goes tails [outcome]. The coin flip is 50%, that has to be accepted. Meaning the Monday Wake ups both have an equal 50% possibility. If it’s Monday Tails, there is a 100% chance that you will wake up Tuesday, as that is the basis of the scenario. This 100% rate of Tuesday wake ups within the 50% rate of Monday Tails wake ups, when multiplied, gives a 50% chance for Tuesday Tails wake ups.
It's a matter of perspective: The odds of her flipping heads is 1/2.. but the odds of her waking up on Monday is 1/3. But the result of waking up on Monday can only happen if she flips heads. So you must assume the probability of the consequences rather than just the result of the coin toss. Therefor it is 1/3 and there is no debate about it; because she is waking up more often than the coin is tossed, so that changes the likelihood of the standard 50/50 chance. The fact that waking her up is a qualifier, completely changes the standard parameters of a 50/50 coin toss. It's easier to think of it as Monday is heads/ Monday is tails and Tuesday is NO TOSS. So every time she is woken up, she has to think, is it a heads or tails Monday, or is it no toss Tuesday. So she has 3 results to consider. The answer is 1/3, because there are 3 possibilities. But what is very interesting: From the perspective of the "coin toss man"-- he gets to go directly from toss to toss, so even though tails has multiple consequences, it does not effect his odds one way or another. So from the perspective of the coin toss man, it is 1/2 chance. But that is not the question. The question is from HER perspective of having to wake up and answer, when she has to wake up more times than the coin is tossed; so from her perspective, it's 1/3 chance, because of the "waking up" prerequisite.
@@ShaharHarshuv You count the flip when it's actually flipped. The amount of times she is awakened, (2-infinity), is all triggered by that one flip of tails.
globally tails will not come up more often, but subjectively for here it will come up more often. Is she asked about the global upcomming or the subjective upcomming?
This is a great lesson in clarifying the problem. As addressed in the video toward the end, are we asking the state of the coin after the flip or after waking. Be careful with the empirical study. By definition Wake on Tuesday = Wake on Monday following the event “coin lands on tails.” On E(Tails) you will talley in both columns E(Mo|Tails) and E(Tu|Tails). This will give the appearance of E(Heads) = 1/3 and E(Tails) = 2/3, but you’re really tallying E(Day|State of Coin). Thirders are answering the question, given that Sleeping Beauty was woken up, what is the state of the coin? Halfers are answering the question, Sleeping Beauty was woken up. What is the state of the coin?
But sleeping beauty wakes up all the time / half the time on heads monday / 1/4 time wakes up tails swith monday , 1/4 time tails on tuesday , She does NOT know which day it is , so this information is irrelevant . i think Thirders try to calculate the probability of she guess right ! we do not ask that
I'm answering the question 'How many times was the coin tossed ?'....which is clearly only once and is I think the correct question to ask. How many times Sleeping Beauty is woken up is an irrelevant red herring.
I think the _really_ clever thing here is that Derek has carefully orchestrated a video to generate a high "like" _and_ "dislike" count. That kind of controversy will be irresistible to the almighty algorithm 😎
@@theeraphatsunthornwit6266 Pretty much all social media is optimised for controversy or moral outrage, because that's what drives the most interaction. I don't honestly have any idea about the YT algorithm, but we can be pretty sure it'll rank videos with a widely split opinion above a video that has just a high number of likes, especially if there are lots of comments too.
I'm in the half camp. Here's why: In the sample there are three days in play but only two coin flips. Heads is good for one day and tails gets you two. There are only two "events" resulting from the coin flip: one day or two days so the chances of any waking choice in this scenario is always 50/50. Additionally, the question would be posed to Sleeping Beauty who has not recollection of time passing except for the few moments she is awake. There are no "2 days running" for her; only one day at a time, each day resetting the coin flip probability.
I see a major difference in the Monty Hall problem and this example: in MH, the contestant sees everything throughout the process, and indeed, the probability goes fro 33% to 67%. In Sleeping Beauty, there is only a single flip of the coin for each trial with a 50% probability of coming up heads. If it comes up heads, she will be asked the question once, if tails, twice. Because she knows how the experiment is set up and there was a single coin flip, she knows the probability of it being heads is 50%. That is the answer she will give *every* time because the whole set up is based on *one* coin flip. The experimenter is doing something different. They are keeping a tally of what she says each time, so obviously , if the coin flip comes up tails, she will give that same answer on two occasions, rather than one. The fact that two tally marks were recorded doesn't change the nature of the flip itself, which is what the question is about. Recording tally marks isn't gathering anything useful because her answer will always be the same. In Derek's coin flipping example, he was recording the results of multiple flips, but tallying the tails results twice. That is artificially doubling the tails tally and has nothing to do with the probability of the outcome of the coin flip nor to the answer SB will give. NOW, if the question was changed to "what day is it today" then the answer will be different because of three possible outcomes (waking up on a given day) instead of two (results of a coin flip).
50/50 seems obvious to me. Which makes me think I'm not understanding something. But your description fits. The question now is... why are people picking 1/3rd?
That's because they ARE two different problems. I don't think he meant that they were the same. If he did, he is wrong because MH has an actual, non-ambiguous answer whereas this [currently] does not. Edit: you're also misunderstanding the question. He's not recording each tails flip twice more than he should; he's doing that because if it IS tails she will wake up twice but will have no way of knowing which day it is, or that she has woken up before or will wake up again (to put that another way: she doesn't know if it was heads or tails). They are recording two tally marks because she woke up twice: monday AND tuesday. So because she has no memory of the first time (if it is tuesday) and has no way of knowing there will be a second time (if it is monday), and 2/3 scenarios are tails scenarios, 1/3 is a totally valid answer. She doesn't know if it's Monday or Tuesday, but she _can_ know that if it WAS tails, she will wake up twice while heads she will wake up once. That very well may affect the answer.. we just don't know at this juncture in time. "what day is today?" is actually not an interesting question at all, and you can find the answer quite easily: given that she is being asked what day it is (and she is never asked on tuesday if it was heads), then it's 2/3 likely to be Monday, and 1/3 likely to be Tuesday. Idk what you're missing, but if you're analogizing this to the Monty Hall problem then that might be doing more harm than good!
yeah, I agreed with this etirely, if heads you pour 1 liter in a tub, and tails you pour two in another tub, the second tub will be twce as full about,, but not twice as likely to be chosen, it's still fifty fifty. I thought out of all people, veritasum would be a halfer, and I'm really perplexed that it''s this big of an imbalance between the agreeers and the disagreeers.
The dilemma becomes much easier if you realize that "what are the odds that something WILL happen" and "what are the odds that something DID happen" are two completely different concepts. One is prediction, one is deduction. And really, "the odds that something happened in the past" isn't all that meaningful. It already happened, or not. There is no probability associated with it. The question to Sleeping Beauty could be phrased as odds, if you can relate it to the future: "Given all that you know right now, what are the odds that you will be correct if you conclude that the coin landed on heads?" This is a much easier question to digest, the answer is based 100% upon deduction, and zero percent upon the toss of a coin. There is quite an attractive quantum rabbit-hole here, but it is Super Bowl Sunday, and I have no time! :-)
I think it depends on what exactly is at stake. In the Brazil example, if you have to bet $1 on who the winner is when you wake up, you take Canada because there's a chance you'll get woken up 30x in a row and lose a dollar each time. On the other hand, if your life was at stake for getting the answer wrong, you take Brazil.
@@billybob-uz6wz that only works because you can only lose your life once. If instead it was shortened (or prolonged) by a year each time you should still answer Canada.
@@AbsoluteHuman Indeed. More broadly it depends on the consequence of being right/wrong and if it compounds. In your scenario if Brazil loses and you pick Canada, you lose 1 year of your life and that's the worst case scenario for picking Canada--but 20% of the time you gain 30 years. If you pick Brazil the best you can get is 1 more year, but you lose 30 years 20% of the time. Basically, without knowing the parameters and end goal, there's no real solution here.
This! This is the answer. By casting the question in terms of "probability" the questioner is trying to confuse the discussion. There are no probabilities about the past. Asking someone something they don't know is similarly wrong-headed. They can use deduction to maximize the payout on a bet they make with you over the question, but that's not knowing AND not probability.
There's no relevance to additional instances because the multiple experiences exists within a 50% probability across the possibility of heads or either tails. You're being given irrelevant information, then taking that information and treating it as a factor in how a coin has been flipped. For example, on the tails side, you would be woken up ten thousand times and you're currently the seven hundredth time to answer the question, your response will always be that the coin was only ever flipped once.
The part where you said that if you choose 1/2 you are more often correct about the outcome of the coin tosses, but if you choose 1/3 you answer the question correctly more often made a lot of sense at first, but then I realized it's not the same question at all. The football/soccer question is asking which outcome actually happened, but the Sleeping Beauty question is asking what the likelihood is of a specific outcome happening. Sleeping Beauty may be woken up twice when the coin landed tails, but both times it was still a result of the same initial coin toss, which had a 1/2 probability of landing on either side. If she had been asked whether or not it did land on heads, then the football/soccer analogy would apply.
The thing is that in the soccer question even if Canada won and it's Tuesday you still don't remember previously being awakened nor do you know what day it is. So you should (and probably would) still guess Brazil won.
Exactly, the question didn't ask what the probability of the outcome was for that particular day, it just asked what the probability was if it landing on heads or tails.
It's a matter of perspective: The odds of her flipping heads is 1/2.. but the odds of her waking up on Monday is 1/3. But the result of waking up on Monday can only happen if she flips heads. So you must assume the probability of the consequences rather than just the result of the coin toss. Therefor it is 1/3 and there is no debate about it; because she is waking up more often than the coin is tossed, so that changes the likelihood of the standard 50/50 chance. The fact that waking her up is a qualifier, completely changes the standard parameters of a 50/50 coin toss. It's easier to think of it as Monday is heads/ Monday is tails and Tuesday is NO TOSS. So every time she is woken up, she has to think, is it a heads or tails Monday, or is it no toss Tuesday. So she has 3 results to consider. The answer is 1/3, because there are 3 possibilities. But what is very interesting: From the perspective of the "coin toss man"-- he gets to go directly from toss to toss, so even though tails has multiple consequences, it does not effect his odds one way or another. So from the perspective of the coin toss man, it is 1/2 chance. But that is not the question. The question is from HER perspective of having to wake up and answer, when she has to wake up more times than the coin is tossed; so from her perspective, it's 1/3 chance, because of the "waking up" prerequisite.
@@Sirithil The comparison with the soccer game was extremely misleading in my opinion, because it was a completely different question. The initial question was about the _probability,_ but for the soccer match he asked about the _outcome._
@@joshpollnitz1618 Each time she wakes up it's 50/50 chance she either exists in the heads sequence or the tails sequence. The sequence is one complete package. It doesn't matter how long the sequence is, it's still one package. The reason being that the conditional statement is the fair coin which only has two outcomes at 50% chance each.
You are confusing the probability of the coin coming up heads _before_ it was flipped with the probability that it _did_ comes up heads, after it was flipped and the observer has some amount of information pertaining to what the outcome was. These are two very different things. The answer to the first is always 50%, but the answer to the second is not necessarily 50%, depending on what information the observer has observed. This is the fundamental principle behind Bayesian probability.
@@therainman7777 Sigh. Leftism is a religion upon which the idea that an individual's perceptions can dictate reality is the central object of divine worship, thus the solipsistic imposition of the self's mind onto the world makes the Leftist a god in his own eyes. Pay attention to me. There is no secondary set of probabilities. There exists no such thing as metaphysical probability. That's a perceptual lie that the illogical parts of the human brain concoct in order to alleviate the pain and stress caused by the sheer gravity of reality pressing on you from all sides forcing you to accept that you cannot escape objective truth. There is no such thing as "probability after the fact" nor is there any such thing as a "choose your own adventure" kind of probability. Probability is determined by material scientific forces that can be measured and predicted to a certain level of accuracy. In this case though we have from the outset a theoretical "perfect" object in the form of the fair coin. We know from the get go what the probabilities are because those probabilities are deterministic and they are determined by a physical object whose attributes are already established -- the coin is an ideal one. According to the parameters set by the use of such an ideal instrument we are bound to make the conclusion IMPOSED by the parameters set by the ideal instrument. You cannot escape the physical imposition of physical laws even when those laws are purely theoretical because those laws still follow the rules of logic. The coin is set to have a 50/50 chance to land either heads or tails therefore those are THE ONLY outcomes you can measure in the experiment. Nothing else besides those two outcomes exist. Period. End of story. --
6:10 - relates to my argument why the Universe exists at all: the probability of there being something is infinitely higher than there being nothing because there's infinite variations of somethingness as opposed to only one possible state of nothigness. 😁
That's a tricky one, really twisted my brain at first, but after a while I came to conclusion there is no paradox :) The core of the problem is that we compare two completely different values - on one hand we have a toin coss probability which is we know to be 50% for both heads and tails, but the second measurement is basically "how many times the princess was right" and it actually has nothing to do with coin toss probability, it only depends on how many times she was asked... So if we ask her (aka wake her up) 1 time for heads and 1 time for tails, it would be 50% guess for right answer, and same holds for any amount of questions as long as its the same for heads and tails. But in that ptoblem we ask her 2 times more questions for tails than for heads, so she can safely say "its tails" every time and she will be right 66% of times and it has nothing to do with toin coss chance which still remains at 50%. Now if we take that example form one of the articles of asking her a million times for tails - everything will still hold up, the heads/tails is 50% as always but she can always say "tails" and be right 99.999% chance because of the amount of questions she gets when its tails. So, yeah, don't thank me thouthands of matematicians all over the world, just remember this post when next Noble prize in math is being issued!
No. That's not the question she is asked. They ask her what she believes the probability is. They do not ask her which side the coin landed on. Her answer is "1/2" or "1/3", not "heads" or "tails".
The difficult thing with this problem is defining the question you want to answer. If Beaty has to make a guess for heads or tails when she awakes and she always guesses heads, what is the chance she is right? Is it about how many times she answers right(1/3 of the time) or about how many weeks she is right(1/2 of the weeks)?
Will she answer right 1/3 of the time though? Not when it turned out to be heads! That's the problem. It is N=1. In that case the so called 2/3 of potential actually didn't turn out to be 2/3
@@jogadorjnc That's not spelling it out, that's intentionally obfuscating it. Probability of correctly guessing heads is _never_ less than 1/2; only the probability of getting _more right answers_ by guessing heads declines. Until _that_ question is what's actually asked, this is just another cheap semantic deception.
@@a-walpatches6460 The answer becomes 1/3 assuming you know that you get woken up twice a week on tails, and only once a week on heads. I think that's where the lack of clarity comes into the mix. If you know that on heads you only get woken up on mondays, that means that if you guessed heads 3 times in a row you would likely be wrong 2 out of 3 times, because you get woken up twice as often on tails. Or no, I think you are right after all. I think my maths is right, but it answers a different question from the one being asked. "The probability that the coin came up heads" is simply 1 in 2, and will always be. The other question might be phrased as "What do you think was the result of the coin toss", or "Based on your limitd information, do you find it more likely that heads or tails was rolled?"
@@holysecret2 Same answer for those questions, 50/50. It doesn't matter how many times you get woken up, each time you're woken up is the same from your perspective, 50/50.
@@a-walpatches6460 Nope 50/50 only makes sense if you're asked what is the probability the coin *will* be heads or tails, but that is the wrong question since the event already happened, you now live in the timeline created by the outcome and you're more likely to be awake in that timeline. So she's really answering what are the chances it *was* heads or tails and information about that outcome has leaked via the fact she's more likely to be in the second timeline by virtue of being awake.
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If you want to vote by liking/disliking the video: “Agree with me” means 1/3 and “Disagree” means 1/2.
Latest update (Nov 23, 2023): 217,332 agree with me, and 97,502 disagree with me.
ok
👍
First
I disagree with u.
😅
"What coin? What are you talking about? Where am I? Who are you?"
I thought something similar at first too, but actually it is all carefully crafted to prevent this from being a valid answer. It is only when she is put "back to sleep" that she forgets, and what she forgets, is being woken up. So every time she is asked the question, she remembers the original explanation, the original time being put to sleep, and being woken up that time.
That would logically mean X, but I don't like X, so it doesn't mean X.
Great science right there, chief 👍
Wait, why am I naked?
Professor Farnworth response lol
You'd like my music
As a Canadian, I would be quite happy for a 20% chance of winning against Brazil
fax
Amen
In the particular universe that produces Canadian dominance of Brazil in soccer...pigs can fly. Pig guano everywhere.
As a Brazilian, I’m happy there’s at least one scenario where we are more likely to win against Canada
@@diegototti we are also more likely to win against canada in a war. they would apologize for being involved in a fight and raise the white flag
Whenever there's no consensus in probability puzzles like this one, it usually does boil down to subtle disagreements about what is actually being asked, not the answers themselves.
Yeah, it just seems like semantics. I depends on whose perspective you are using
Semantics, asking the wrong question, wrong definition, etc.
That's what makes Monty Hall problem so great - it's not about words, it's about the actual concept itself.
Yeah, "what was the probability that it came up heads?" vs "what is the probability that it came up heads?" can already make a difference to the answer. Only if you define questions properly can you answer them. I suppose that's why they were philosophy papers and not mathematics. In mathematics you need things to be defined unambiguously.
There is clearly a majority consensus on the entire thing with most people leaning towards the real world side instead of the fairytale book side. Why do you think they use a literal fairytale character to point this out? Math is 100% disconnected from reality. A concept. She's literally missing 25% of her ability to know what actually happened. She is at 75% comprehension of her reality since she can't tell the difference between waking up once or waking up twice. But the knowledge shown to her is letting her know, that she has two chances to respond on a tails flip, or once chance to respond on a heads flip. So she can take the chance of being right or wrong about a 50/50 chance twice in a row, or once. Her best chance of answering correctly on monday heads, monday tails, or tuesday tails is to realize that there is no tuesday heads and eliminate 25% of her ability to answer. Thus leaving 3 equal chance scenarios. Her real world probability is skewed by lack of information. Her fairytale probability is 1/2, because 1/2 is 1/2 and everyone knows 1/2 is 1/2.
4:52 id say "put me back to sleep"
😂
Another one: why do I keep gettung woken up
but remeber he said in this experiment every time you wake up you have no knowledge if this is your first time or this has happened before
If sleeping beauty was asked "What's the probability the coin came up heads?", I think she should say 1/2. If she was asked "What's the probability that you've been woken up as part of the outcome of a heads result?", I think she should say 1/3. I think the key thing with this question and the reason there isn't (and probably can't be) consensus comes down to how it's communicated and how we as individuals interpret what's being asked of us with the answer. If your goal is to reinforce your understanding about how the coin works, you are probably a halfer. If your goal is to be correct in answering the question from the perspective of sleeping beauty, you are probably a thirder.
Agree.
I like the way you explained this. His statement “Something changed” was important because it matters that an event occurred between observations.
This is the correct answer. It depends on how the question is interpreted.
But if sleeping beauty doesn't remember any times she's been woken up, every time is her first. So to her it's always 50/50. Any other wake-up (Tuesday) in _her_ existence never actually happened
I think there should be a distinction between asking "What is the probability A coin came up heads?" and "What is the probability THE coin came up heads?" The question is about THE coin, and given she is awake, the answer is the probability of her being awake.
I'm a simple man. The probability of everything is always 50-50. It either happens, or it doesn't.
exactly
Reminds me of the football coach who didn't want his quarterback to throw because two of three possible outcomes were bad. Interception and incomplete.😅
@Glenn Clark ahh the coach is wrong. It's still 50-50. The pass either reaches the teammate or it doesn't. 🤣
Average mulla thought process
No. It’s not. If that was true, you would win any game every second round on average making only random choices, i.e., tossing a coin. Clearly, that’s absurd.
Lessons learned: never let a researcher put you to sleep and never pay them in cash
That would logically mean X, but I don't like X, so it doesn't mean X.
Great science right there, chief 👍
the researcher just wanted to kiss sleeping beauty multiple times
@@AuGAlaNstud
The whole thing can be solved by just acknowledging that there's two possible answers because of the unclear wording of the question rather than arguing over which is right while knowing that the question is ambiguous
Exactly. It’s just a matter of clarity. If you allow for more scenarios in your question, then there are more possibilities.
I love how this comment has four likes but is sandwiched in-between comments with 1k and 1.4k.
the question was simple. What are the odds that the coin landed on heads. It is 1/2. He is confusing people by bringing up the irrelevant fact that she would be right more often if she guesses tails every time, but the question was not whether she would be correct more often.
@itoibo4208 that's part of the setup, the question is "how should she answer?"
No, I believe the wording is clear enough. You just have to be an impossible halfer... There is no rule that would be violated if she tied up her hair each time she wakes up. She can have this thought before the game starts. She will not forget her rule as she would also forget the game rules if that were the case. So if she awakens with hair down she can know it's Monday and ties it up and answers 50/50 odds, with certainty. When she awakens on Tuesday and her hair is up, she can be certain heads never came up and the odds are 0. So it's 50/50 that it's just 50/50 Monday and 50/50 that it's 50/50 Monday and impossible Tuesday. Either way, it's not 1/3. Now sure it could be but why force it; don't you want to be right? What if it's impossible to know what someone believes... Since you can only believe in what you do not know, how can someone else know if you cannot? If you knew you'd know it and not have to believe it ... Why does she believe it? We will never know...😢
Derik: vote with like and dislike button
RUclips: makes dislikes invisible
Fr 🙄
71% of viewers liked the video. I just checked :)
@@poseidonsmafia1160 Wow how did u know that??
@@nezukoyaegerr
There are Chrome Addons that let you see dislikes. I use "Return RUclips Dislike".
@@poseidonsmafia1160 it doesn't let you see actual dislikes though, it only shows those by people who also have that same extension
“Do not hit the like button” 87 people instantly ignored him
Now 2,479...
How are you able to see number of dislikes?
@@Varma17 the title is the amount of likes (agree) to dislike (disagree)
@@Varma17 I think the title updates periodically, the number of people disagree
@@Varma17 there are browser plug-ins to show dislikes again. the one I use is "return youtube dislikes". cheers
There's a hidden lesson here about imbalanced classes in a dataset. Halfers are trying to model the distribution of the data generating function, while thirders are trying to minimize some loss function for the estimator.
Then take them both to the consideration and calculate the average. That would be the real solution to this dilemma.
@@orka6848 no, these are not two approaches to the same question, they are two different questions. Averaging them is kind of meaningless.
Estimating the distribution is not the same as minimizing expected error.
@@johnmorrell3187 I think you hit the nail on the head - those who agree with him are answering a different question than those who do not.
funny, but no: the imbalance of the heads and tails here is only due to a deliberate mistake in sampling; because of a sampling error you record "tails" twice when a single "tails" event occurs, but only a single "heads" event is recorded for "heads" events. The dataset is seriously screwed up; when presented with a new "instance", the "thirder's classifier" will have its probability estimates wrong: it will be predicting "tails" with prob. 0.66 but it will only be "tails" with prob. 0.5.
@@orka6848 here we have the engeneer
0:01 I just love how the like button on my screen got highlighted with red color. I didn't know creaters could do this, or maybe this is a work of RUclips employee just for this video
The experimenters look on in horror as the coin rests upon its edge. They somberly pull the sheet over Sleeping Beauty's face. After an appropriate period of silence, Erwin asks, "You guys wanna put my cat in a box with an unstable nucleus, a hammer, and a vial of nerve gas?"
"Not again, Erwin..."
Split the difference!
@@ChrisContin They divvied up the hadrons amongst themselves and Erwin got a new cat.
Ahh I dont have enough neurons in my brain to understand this, someone please do the honors.
@@bluzter it is a reference to Shrodinger’s Cat
Wait, hidden result, hidden protocol. Then they wake her up -1/12 times and Poland wins the Super Bowl
"Waking up on Monday with head" gets me every time.
Best way to start a Monday
That's why I pick heads everytime
Some people prefer waking up with tail.
By Veritasium? I'd only want it to be from Sleeping Beauty. If not, I'd pass
bruh
Veritasium uploaded: 0 People Agree With Me, 0 Disagree
0.3 is probability of one side of the coin vs 0.707 probability of the other side
Well no sh*t
@@aucklandnewzealand2023huh? Where does that come from?
@@aucklandnewzealand2023more like 0.33 and 0.66
That ad timing at 7:03 was genius, I don't know if the ad displayed for everyone but that was absolutely perfect
I think the question is subtly mixing up the probability distribution of the coin toss with the probability distribution that the sleeping beauty was woken up with a certain coin toss. So it really comes down to what you think the question is asking for.
Yeah, one of the confusions is that "what's the probability that the coin came up heads" can mean different things. Halfers think it's a question about the behaviour of coins. Thirders think it's a question about your on-the-spot beliefs about past events.
@@AzrgExplorers I agree. Thirders actually think that the question is, "what are the chances that you were woken up once before?"
yup, like nearly all things, the readers interpretation is what truly matters... and yet the world doesnt care
@@wordsofcheresie936 No, thirders are answering the exact question asked. Sleeping Beauty wasn't asked "did the coin come up heads?" She was asked, "what are the *chances* that the coin came up heads?" In the soccer analogy @veritasium used, he talked about this difference without actually pointing it out.
About ten billion humans have been born. So the odds of you being born as you is one in 10 billion. So when I ask if you are you, what is your response? If I ask what were the *chances* that you would be born exactly as you are, what is your answer?
The questions are different and so the answers are as well.
The best way to explain it is the way he already did. Let's Make a Deal gives you 3 doors, with only one valid prize, heads. The other two have tails behind them. Then they take away a confirmed wrong door, giving your probability of choosing heads an increase. That's why you always switch the door you choose after the removal of a tails door.
This method is simply presenting you with two possible doors but then adding a 3rd confirmed possible door. Your safest bet is to be realistic and realize that the original two doors always had a 1/3 chance of having heads no matter what door you chose. Changing doors still results in a 1/3 chance of choosing the heads door.
When I reached your poll, I didn't understand the controversy. If the question is "What is the probability that the coin WILL be heads?" the answer is 1/2. If the question is "What is the probability that the coin WAS heads?" is 1/3. These are two completely different questions. The first has to do with flipping a coin. The second is about what day it is.
Thank you
Exactly. Seems fairly obvious
But what if it was a 1% chance ( a 1 on a 100-sided die) to wake up 1 million times? Even if you were asked what the probability of the die being 1 WAS, it was still only 1%. Its unlikely that you were put to sleep a million times in the first place.
@@alexs1277 Not clear what variation you are describing here. But If 99% to wake once, and 1% to wake 10^6 times, chance die WAS 1/100 is 99.990%. No?
Changing the tense of the question has no impact. Again just consult the soccer game analogy. It’s obvious
I've gone through this, and I think I've gotten to the conclusion that I'm a halver, but only on very specific conditions. I feel like two questions are being asked at the same time and each side chooses to focus on only one of them. Halvers are focusing on, sleeping beauty is woken up, she's asked what's the chance that it had come up heads. The answer is 50%, because it:s a fair coin and regardless of the day the answer is 50%.
However, thirders are answering a DIFFERENT question, which is, every time sleeping beauty is woken up, what's the probability of her being right, should she always pick up heads. She's woken up everytime, is asked which one came every time, she picks head everytime, the chance of her being right is 33.3%, but it's not because of the coin, but because they're oversampling the wrong answer.
Halvers are talking about the coin. Thirders are talking about sleeping beauty.
The formulation of the question directly tells you to consider it from sleeping beauty's perspective.
In other words, if we repeat the experiment every week for the rest of eternity, is she trying to be right most on *days* or right on most *weeks* ?
I really like how you worded this. And you're 100% percent correct. I personally believe that because of the way that the question was asked that it should be answered from sleeping beauty's perspective just as @rantingrodent416 stated, but the way you acknowledged both points of view without hating on either one I very much respect.
Flip heads, put one green bean in the bowl. Flip tails, put two red beans in the bowl. You pick a bean, what are the odds it is green?
@@rantingrodent416 Well she has no way of telling if she was awaken or not, so her only guiding point would be her understanding of the fact that a coin has only two outcomes, so it would be 50%. If someone flips a coin and ask you what are the pobability of it being heads, with no previous context (as sleeping beauty didnt remember if she had been awaken) you would answer 50%, because there is no way for you to say how many times you have been asked that question.
Thanks, this was the looming existential dread I was missing this year, thanks Veritasium!
The secret to this problem is that it is a trick question attempting to ask 2 different questions at the same time. Attaching probability to it just makes people think there is something more profound happening.
Yeah I agree, it's more about semantic than statistic. Derek just found a nice trick to get tons of likes and views with a question that is more intellectual masturbation than anything else.
@@En_theo exactly. And I love Derek and his content but this video just felt like a gotchya. And the worst part is I can't even express this to him by downvoting the video
Maybe it's a social experiment on how much influence his opinion has
Exactly
I agree it's a trick question, but it's not two different questions. It's just one invalid question. The tail scenarios cannot be viewed as two separate outcomes: informationally they are identical to sleeping beauty, and therefore the same outcome. The question just arbitrarily labeled the tail scenarios as two outcomes, not with any kind logic compatible with reality, but with memory erasing magic.
As a Canadian, I'm really thankful you gave Canada one in five odds of winning against Brazil 😂
As a Brazilian I'm thankful for 4 out of 5... Canadian team is getting better and better (Brazilian team have been a lot better).
And there's a 100% chance of another balloon flying over Canada will be shot down by an F22. :D :D
As a Croatian, we beat you both, even though Brazil was better but unlucky against us. It was that 1/5 win for us 🙂
Good luck to Brazil!
@@lukatore123 - I think it was more like 2/5. Croatia's got a great team (maybe the best one per capita - amongst Uruguay and Portugal). Brazilian team, of course, had better individual quality, but Croatia had a very interesting collective game.
Afterall, i think it was very well deserved
@@BillAnt 🤫🤫🤫
I think its the phrasing of the question that made this controversial. What if the question were " What is the chance you've been awakened due to a head coin toss?" Then to me its obvious, its one-third. Because sleeping beauty would be awakened more times due to a tail coin toss, even if she knew it is a fair coin. But if the question were " What is the chance the coin flip is a head " (With prior knowledge that she knew it is a fair one), it then would be 50-50.
Facts I don’t get how the root problem is that complex or controversial lol
@@kuribohoverlord2432 cause you are a genius mate, congratulations
What if the rules dictated that she would only be awakened and asked the question if the coin flip game up tails? Then, there would still be a 50-50 chance that the coin flip was heads. But given the information that she was being asked the question, she would know that the coin flip was not heads.
The fact that she is being asked the question gives her additional information.
What makes this "controversial" is that some people are unwilling to adjust their beliefs when given new information.
It's still just a matter of what's meant by the question. If a flip a coin, and you see that it's heads, and I ask you, what are that chances the coin landed heads, there are two answers depending on how you interpret my question. Either you answer 50% if you take my question as "what was the chance of what you've just seen occuring in general" or 100% if you interpret my question as "what is the chance that what you saw (the coin landed heads) is the actual state of the world (the coin landed heads)"
Thank you. You phrased it beautifully
Thanks!
My guy just asked a sleeping beauty problem and just left me on a thought about multiverses. I love this channel.
The probability that she guesses the side of the coin is ~1/6. ~1/2*1/3=1/6
But if you ask about the probability objectively, then of course ~0.707
It has no corelation to multiverse unless it exists (probability of Multiverse unknown)
@@aucklandnewzealand2023 stop dude you're talking to an anime pfp
@@aucklandnewzealand2023 Honestly, that isn't just as justifying seeing that she could have done any other operation
@@roddraft3466 the general consensus is that your pfp doesn't affect your comment
fr lolll
i hate how probabilities twist my minds, but i love how philosophical questions leave me questioning my existence-
I think this is more a problem with the question having multiple valid interpretations than it is an issue of the question having multiple valid answers.
Halfers are focusing the question on the origin of the random event that causes a decision to be made at the start(i.e. the flipping of a coin). Thirders are focusing on the end result of the overall experiment (i.e. the number of ways sleeping beauty can be woken up). The tricky part in this whole scenario is that the question is presented as a single event with a single function to model it. However, from my perspective as a programmer, this scenario is better described as a chain or series of two functions. The first one generates a random 50-50 result (flipping the coin). That random result (heads vs tails) is that function's only output. Everyone can agree on the probability of each result for that function on its own. Now we take that outcome, and use it as the input for a separate function. This second function simply makes a decision on the number of times to wake sleeping beauty up. It becomes pretty obvious when looking at this function in isolation that its results are skewed towards the side that wakes her up more times. The second function essentially multiplies the likelihood of the input that would cause multiple wake-ups. Thus we arrive at the two interpretations of the original question and their different answers.
Interpretation 1: How likely is the coin to come up heads? -> obviously 50%. Interpretation 2: How likely are you be woken up by the coin coming up heads vs tails? -> obviously 33%. Both are valid and so my personal stance on it is that the question is ill-formed by being ambiguous.
agree with this, but would say I'm a halfer in this instance because the exact question asked is 'what do you believe the probability of the coin being heads?' not 'what do you believe the probability of being woken up by the coin being heads?' subtle difference, but to one question I'm a halfer, the other a thirder.
Danm
@@superkeefo6951 This. That question sounds to me like question that would be asked in a hospital to check if my brain functions correctly like what's the date, who is current president etc. It made me 1/2er just because of semantics but I understood what he meant and in that context I'm 1/3er, so I don't know whether I should like or dislike
@@0NeeN0 but if you're saying there is context then you are essentially adding it and rephrasing the question given to you to be the second question. That's the point momo was making, the implied context makes you think you need to answer the second question. But really the question should be asked with that context or else it's 50/50
This! 100% this! The problem is that the language being used isn't precise enough.
My reactions when I see a Veritasium video. Amazed by the title-> Understands the concept-> Trying to understand deeply-> Gets lost-> Forgets what was the video about-> Perplexed about the reality-->Video ends->Hits the like button.
aandd you became a thirder
Same
That’s just his fans
real
"Good Morning, Beauty. What do you think was the probabilty for the coin to be heads?"
"50%. Can I keep the coin?"
"Sure."
"What time is it?"
"Well..."
*opens Harry Potter's Gringotts vault*
4:30 but if you divide each section by the total number of tests, each section gets around one half of the probability. This is cause the two tails outcomes are inseparable. I think people need to stop separating the two tails outcomes because they’re honestly the same thing. If you get tails, both Monday and Tuesday will happen, not one of the other. So I don’t think that they have 1/4 or 1/3, but Monday and Tuesday both have a 1/2 chance.
Id say it’s rather a linguistic problem: It’s a 1/3 chance that if she is awake, it was Heads. It’s a 1/2 chance that it rolled Heads when she awakens at all.
Not it’s still 1/3 when she awaken because she awakens twice if it’s tails
It's a fairly complex situation, but I agree completely. If you jump to a conclusion you are ignoring the actual dilemma, which is how semmantics may affect our perceptions of the universe. There's no truly correct answer, only a correct answer given a chosen context.
You wanna know the probability of heads vs tails? 1/2
You wanna know the probability of Sleeping Beauty correctly guessing if today is Tuesday? 1/3
etc
Makes me think how much of actual science is affected by linguistic biases, I would guess most of it.
It’s always the language that is the issue in these kind of paradoxes. Write this problem using only math and suddenly there is no paradoxes
I disagree. It’s a 50 50 chance if when she’s awake it’s heads or not. It’s a 50 25 25 chance if she is waken when MH, MT, TT respectively, because it’s 50/50 whether it’s head or tails and then if tails 50/50 whether it’s Monday or Tuesday.
Wittgenstein is proven right yet again
So I guess I think if she wants to say the actual probability, she would say 1/2, but she wants to be right more often, she would say 1/3. But does being right buy her anything? If no, I would say 1/2.
I've reasoned about this and I think it is correct to say 1/2. In my opinion 1/3 is simply wrong because it is not equally likely to be in any of the three cases. I'll copy here what I already said in other comments that are lost in the haystack.
My opinion: When she is asked about the probability, the coin has already been flipped and its state is determined even if unknown to her. So here the word "probability" should be interpreted as her confidence that the coin landed heads. She is aware of the procedure and she knows that the coin is flipped one time at the beginning. Imagine she is asked the question immediately after the toss (of which she doesn't see the result) before being put to sleep. She would obviously answer 1/2. From now on there is no reason she should change her initial guess because the coin is tossed once for all and there is no subsequent event that could influence the output. It doesn't matter if it's the first or the millionth time she's being awakened: because she doesn't know what day it is she never gains new information and there's no reason she should update her initial guess.
1/3 is simply wrong because it assumes that the probability of being in one of the three cases is uniform while it is not. The probability is actually 1/2 of being Monday and it landed heads, 1/4 that is Monday and it landed tails and 1/4 that it is Tuesday and landed tails.
The 1/3 argument moves from the wrong assumption that to the question "what day do you think it is today?" she should be 2/3 sure it is Monday. Actually she is instead 3/4 sure it is Monday to balance for the fact that there is no Tuesday/Heads combo. The probability it is Tuesday is in fact P(it landed tails) times P(it is Tuesday | it landed tails).
I put video at 0.25x and he made a terrible error in his experiment. Look for yourself what he does. He simply writes a sign two times when the coin lands tails. He should have tossed the coin a second time to decide where to put ONE sign. If you do it right you get the expected 50-25-25 proportions.
I wanna add something to make it more intuitive: in the case she is awakened 1 million times if it lands tails the probability that in any awakening that day is the first Monday is about 50% and not about 0%. Think of it this way: if she is asked "what day do you think it is today?" she is better off answering "The first Monday" because is much more likely to guess it landed heads and hence surely it is the first Monday than to guess it landed tails and then identify one of the million possible days.
Shocking take
The probability of the coin flip doesn't change with the way we want to measure it. If Sleeping Beauty was woken up a million times for a tails flip, it wouldn't make the coin flip any less likely to turn up heads. Being woken up two times instead of one doesn't make one outcome twice as likely as the other, as the thirder perspective implies. If we're asking about the probability of the coin flip alone, like the question in the video (1:04) very clearly is, then the answer cannot be anything other than 1/2.
Now, if the question was anything like "For N times Sleeping Beauty was woken up, what is the probability of her being woken up because of a heads flip?", then it'd clearly be 1/3.
Let’s do a little thought experiment: I tell you: „I‘m about to flip a coin. If, and only if, the coin flips heads, I‘ll call you.“ The next day, I call you and say: „I flipped the coin now. What do you believe is the probability that the coin came up heads?“
What would be your answer?
I know it sounds counterintuitive,but the only correct answer for sleeping beauty is 1/3.
When she wakes up, there are three possibilities: A: heads/monday, B: tails/monday, C: tails/tuesday.
Obviously, A and B have the same probability, because it’s a fair coin flip, so if they would repeat the experiment every week, she would wake up every monday and the coin would have flipped each side 50% of the weeks.
The probabilities of B and C must also be the same, because every week she wakes up on tails/monday, she also wakes up on tails/tuesday.
So the probabilities of all three possible outcomes are the same.
And the sum of the three possibilities must be 100%, because A, B and C are the only possible outcomes, and each time she wakes up, only one of them can be true.
Thus, the probability of A: heads/monday is 1/3.
P(A)+P(B)+P(C)=1 and P(A)=P(B)=P(C)=1/3
I think this scenario highlights, more than anything, that it’s odd to phrase a question with multiple answers with a yes or no prompt.
Maybe that was the real point the originator was trying to make but people just totally missed it and now here we are
My first reaction was that the problem is too contrived to be interesting.
actually, the probability of it being monady or tuesday is 33 percent, but the odds of it being tails is 50 percent
@@Sad_cat_studio no the odds of it being Monday is 66%
That would logically mean X, but I don't like X, so it doesn't mean X.
Great science right there, chief 👍
Plot twist, the coin didn't land on heads or tails, it landed on the side. Now she never wakes up.
She does with true love's kiss, but how many dudes who weren't her true love kissed her?
@@RoderickEtheria Ain't she 118 years old when a prince wakes her up?
The dilemma is not "what is the correct answer", but "what is the question being asked?". If Sleeping Beauty is asked what is the probability the coin came up tails, her answer should be 1/2. If the question is "what was the result of the coin toss" and the challenge is to be right (significantly) more than 50% of the time, she should answer differently.
In other words, the disagreement is not about what the answer should be, but about what the challenge was in the first place. The only sensible answer is therefore: Restate the question as to remove the ambiguity.
Or 42. That works too. Same reason.
"what is the question being asked?" is not a dilemma. The question is clearly about "the probability that the coin came up Heads". Answer to that question is 50%. And I agree with you that those who answer 1/3 are answering the wrong question.
that is so perfect an answer. how did you make it so easy,, in that, what is your background?
@@jonathanlavoie3115 what is the challenge being set, then. Is it to answer correctly on what the coin toss was, or something else?
That's the dilemma here - not what is the correct answer, but what is being asked of her in the first place.
If the challenge was « guess the outcome and I give you 1$ » she would answer Tails, not because the probability is 2/3 but because the reward is twice. Just like I give you 1$ if you guess Heads right, and 2$ if you guess Tails right. You would answer Tails not because the probability is higher. It remains 50%. In the SB experiment, the question is the probability it came un Heads.
@@uRealReels Thank you. You're the first person who reply to me so kindly!
A short anecdote about me:
In my programming course there was an exam in probability and statistics. Three of the questions were about the same problem. In a basket containing 9 blue balls and 11 red balls, what are the probabilities of A) draw 2 blue balls. B) 2 red balls. C) 2 balls not the same color.
Questions A and B are very easy. But for question C I knew that the teacher wanted us to use a complicated formula learned by heart. I didn't want to use this formula because 1- The formula is complicated and I'm lazy, 2- I don't like to use a ready-made formula that I don't fully understand and 3- I wasn't sure if the formula really applied to the situation.
So, I solved question C by following this simple reasoning: Probability of 2 blue balls + probability of 2 red + probabilities of 2 different = 100%. Total must be 100% because there is no other possibility. As expected, the teacher's formula answer was not the same as my answer, and I had to argue to get the point, but he had no choice but to acknowledge that his formula didn't apply to the situation, and that my answer was correct.
I argued my point in front of the review board, not because I needed the point (my average was already 98%) but because I like the truth. That's who I am...
Teo things are for sure:
1) The probability that the coin was tails is 1/2
2) The probability that sleeping beauty has a f*cked up sleep cycle at this point is 100%
Underrated comment lol
I like how you state that the chance is a half as one of the two things that are 'sure', despite the dozens of scientific papers with discourse, this video, the other comments, and the whole nature of this debate. Guess you had the answer all along then.
2/3
You are incorrect about #1.
The probability that the coin was tails is either 0% or 100%, depending on its result.
@@feha92 thats actually true no joke, since he specified "was tails" and anything that happened in the past either happened or didn't happen
To me it's the phrasing of the question asked that's important. If every time she's woken up, she's asked "do you think the coin came up heads or tails", she should always answer tails, because similar to the Monty hall problem, there will be more scenarios of her waking up and the outcome is tails.
But the question isn't asking her what she thinks **the outcome** is, but instead it's asking her what she thinks **the probability** is. The probability of the coin toss is completely independent of how many times she wakes up, or even if she wakes up at all, and it is always 1/2. So even if she were to wake up and the actual outcome of the toss was tails, she is still correct by saying that **the probability** of the toss is 1/2.
EXACTLY, probability? heads, obviously, what you think the result for this run was? tails, obviously
My thoughts exactly! Was looking for this argument.
What is the probability of coin came heads - 1/2, because that is the fact.
What is the probability that we woke you because coin came heads - 1/3 and is very different question.
What I was about to type.
but she wasn't asked what is the probability a toss of a coin comes out heads. She was asked what is the probability the coin did come out heads.
There is a big difference in asking about the probability of an event that has not occured vs the probability that a specific event has happened in the past so long as you gain knowledge when transition from that past point to the present.
One view the point when asked what is the probability of A. Which is 50%
What is the probability of A|B (A given B in statistics).
The probability of A given I have information B modifies the probability of A having occurred.
This is not an independent probability but a dependent one.
i agree with this because fundamentally she can't remember if she been woke up before (according to the experiment) so the fact that she is awake now can't be used to bias the answer dose 50/50 should be the right answer. correlation does not equal causation.
Honestly this seems more like a subtle semantic conflicts rather than a true paradox. I think it arises from a lack or rigorous clarity about the coin in question. Are we talking about the “probability” distribution of any single random coin flip or are we talking about what I will call the “outcome” distribution of a particular coin flip “result” which governs a deterministic finite state machine.
This problem is more of a word problem than a math problem. As i worked through it my understanding of the problem grew and as such my answer changed. The question "what is the probability the coin came up heads?" is two questions, depending on how you parse it.
I think thirders and halfers are both correct and wrong, because they're answering different questions.
One side is answering the probability of the coin turning up heads/tails when it was flipped. The other side is answering the probability of you being in a state where the coin came up heads vs tails. They're different problems with different solutions.
What is the probability the coin came up heads? 50/50. What is the probability i will be right if i guess heads? 1/3rd.
Agreed. It is the perspective.
"Came" is the keyword. It's past tense. When an event has already occurred, any information you can access regarding that event changes the probability that it occurred one way or another. What's the probability that the card I pulled out of the deck is the ace of spades? 1/52. But now you draw a card. It's not the ace of spades. Since you've removed that card from the list of possible cards I might have, the probability that the card I pulled at the beginning was the ace of spades is now 1/51.
Since it's a past event, new information about it changes the probability.
I draw another card. Now it's a 2/51 probability that I have the ace of spades. You draw another. One less possible card I could have, so now it's a 2/50 probability that I have the ace. And so on and so on until all the cards have been drawn and the probability becomes 1/1 whether I have the ace or not.
I do think at least for those fully understanding it that it's about how we value information. Thirders are incorporating the fact she lacks information. Halfers are assuming lacking information is irrelevant. For the sake of Halfers it's important we define the problem of her guess in one single instance based on the rules. There is inaruguably three states in the state space. She's awake on a Monday with tails, she's awake on Tuesday with tails and she's awake on Monday with heads. I actually think it's Halfers that have one extra step of justification. (unless you completely missed this is about the shared information that she lacks information.. it's not a matter of perspective). That extra step is say even though she knows there's three states in the state space there's ultimately only two that matter. The third being she doesn't know it's not Tuesday and heads so the question is like saying it's 50-50 on Monday or Tuesday.
It's pretty clear that he asked the first question, it's explicitly written on the screen. So thirders are just wrong.
I agree. And as a halfer I have to point that the question is "What is the probability that the coin was heads"
The problem with doing the vote this way instead of a poll is that so many people are going to ignore the beginning and like the video because they like the video and not because they agree.
Knowing Derek, The like/dislike options is a study in of it's self. We'll get another video where the like is the wroner answer and then a later video examining the results.
@@BrandonFrancey That makes a lot of sense. I'd bet that is the actual purpose of this video.
I liked this question as a vote to the proposition that people expressing enjoying the video will have a massive distortive effect on any attempt at polling.
(Edit: Wait don't use comments as polls! Dislikes just bury the poll itself!)
I have liked your comment because I agree with it.
I am pretty sure he knows enough scientific methodology to know this liking/disliking thing is complete bs.
It helps increase interaction so I guess it's a smart trick
I tried to apply philosophy to probability in my Probabilities class in college and almost failed the course. So, you know what my vote is.
That's hilarious. I dominated that class because of multiple degrees in philosophy. And went on to teach deductive, inductive, and probabilistic logic. And intro to inductive and probability logic class is pretty much proving the laws of statistics and much harder than any statistics class I ever took. Stats prof definitely hated me tho.
Philosophy begins where science ends
Or is it the other way round
@@nyjsackexchange
Mathematicians and physicists were philosophers at one time.
@@aglawe1 Science is born of philosophy
The scientific method begins with a question
@@nyjsackexchange
So it is an iterative process, philosophers ask questions and scientists try to answer them.
That was a rollercoaster. I went from a confident 50/50 to a confident 1/3 and ended with an uncertain 50/50.
I think the real question is: does it matter what Sleeping Beauty thinks or how many times she’s right? The probability of the coin toss is still 50/50.
EDIT: It seems (like most logic questions) that this is really a semantics issue. Is it: probability coin is heads based on it being flipped once, or based on which way the coin is facing up when she wakes up. So we’re not really learning any deeper truth to the world with this question, it’s just a matter of was our specific setup properly explained
Right. The important part is "she doesn't remember any times she's been woken up." So every Tuesday _her_ may we well have never happened. To her it's always the first wake up, which to her is 50/50.
But that wasn't the question. Thats the entire point (in my opinion) of this thought experiment: There are additional parameters at play (how often she is woken given a certain outcome) and given those parameters what are the odds? Put it another way: What if heads doesn't wake up? Then whenever she waked it will be 100% tails, even tho the coin has a 50/50 propability.
She is either correct or incorrect she is answering with a probability.
right? doesn't change that she'll wake up twice, it's not as if the coin is being flipped again everytime she wakes up. it's just that if one happens one set of events happen and if another happens a different set of events happen. no matter how frequent .
Exactly, the extra steps to validate a non function was just mental gymnastics, but after listening to it once more a coin flip is a coin flip aka ½
To me the answer is clearly 1/2 because the question is "what do you believe is the probability that the coin came up heads?".
If the question is changed to "do you think the coin came up heads or tails?" and she'll lose 1 dollar if she's wrong, then assuming the purpose is to minimize the loss, then she should answer "tails".
The last scenario above is the same as this scenario:
* A fair coin will be flipped . Let's assume there are only 2 possibilities: the coin will come up heads or tails.
* After it's flipped (you don't know the result), you're asked whether it came up heads or tails.
* If it came up heads and you mistakenly answered tails, then you'll lose 1 dollar.
* If it came up tails and you mistakenly answered heads, then you'll lose 2 dollars.
In this scenario, any rational people who want to minimize their loss should of course answer "tails".
Does it mean that in this scenario any rational people should also think that the probability of the coin came up heads is 1/3 (lower than the probability of the coin came up up tails) and that is why they should choose "tails"?
Of course not. The probability stays the same for each (1/2). The rational people pick "tails" to minimize the loss, not because the probability of the coin came up tails is greater.
I'll explain it again using the Brazil and Canada soccer match.
Say that you agree that Brazil is a better team than Canada.
Then you're asked who will win. The rule is that if you're wrong, you'll lose 1 dollar.
In this case I'm sure you'll answer Brazil.
Then say I change the rule as follow:
* You'll lose 1 dollar if your answer is Canada and turns out Brazil won.
* You'll lose 1 million dollars if your answer is Brazil and turns out Canada won.
I'm pretty sure any rational people who want to minimize their loss will change their answer to "Canada".
Does it mean they now think that Canada is a better team?
Of course not. They still think that Brazil is a better team. They changed their answer to minimize the loss because the rule is changed, not because Canada suddenly becomes a better team.
Excellent analysis, however we do not say head or tail, we say HEADS or TAILS. It's so universal I have to assume a western English dialect is not your first language.
This is exactly how I was thinking about it, exactly what question is being asked matters. I’d be curious to know if this ambiguousness in phrasing occurs in other languages
@@Artcore103 Thanks for the correction. I've edited my comment.
@@Artcore103 please leave sir, thank you.
Take away QM and there is no randomness in a coinflip. The coinflip outcome at any moment is predetermined from the beginning of time; however, since we have no information about the exact physical representation of an instance in time, we must make a statistical inference. This is where the value of 1/2 comes from.
The sleeping beauty is asked what she personally believes the probability is. Since she is given more information about the coin, she is able to make a more informed response on the probability of this specific sample.
If you saw the coin landing on tails, and I asked you "what do you believe the probability of this coin landing on tails is", you would say 100%. This is because you have the information needed to make a more informed response than 50/50. The same is true for the sleeping beauty.
My thought process for picking 1/2 is as follows:
The coin is flipped only once.
In the Tails scenario, both wakeups originate from a single coin toss. Since the coin is fair, the question if heads was up would be 50:50 for me.
In my mind, there's no "third option" like shown on the paper (4:18), because whether its monday(tails) or tuesday(tails), it's still the same coin toss. If we sort by heads/tails instead of monday/tuesday, we have heads(monday) or tails(monday/tuesday).
Now, if we rephrase the question as "What's the probability you were woken up because the coin landed on heads", then it's 1/3, because only 1 out of the three total wakeups originates from heads.
What if we change the problem, such that if the coin lands heads, she is never woken up. If the coin lands tails, she is woken and asked the question. In this situation, it's the same as if she can still see the coin on the table showing tails. The probability is 100% that the coin landed tails.
Bruh that's the same question
If she's asked EVERY time she woke up, then it'd be 1/3 because two times when asked, it had been tails. If she was asked only once, decided by the coin flipper, then it should be 1/2.
I don't think it matters to rephrase the question. If she had a record of how many times she guessed the face of the coin correctly through trial and error she would get to the probability being 1/3rd for heads. But this is only because she doesn't know if its Monday or Tuesday. So I agree with the first part of what you said. She you and I know the coin toss 50-50. But what's asked if is it's actually heads when she wakes up. This actually flips the assumptions around where it becomes obvious it should be 1/3. But I think people misunderstand what 1/2 would actually mean. It means that because she has no connected information between the time she wakes up the probability remains 1/2. So to believe 1/3 means you believe her inability to have information about waking up a second time is information.
@@Furiends wording matters. P(head) and P(head|awake) are two different question, the video seems to be asking the former
It hurts to dislike a Veritasium video but there we go
@4:20 The reason there’s 1/3 of each in this scenario is precisely because the odds of the coin flip are 50/50. By virtue of the procedures you’ll always write down 2 tick marks for each tails. So each of the tails column will always be exactly the same. They are tied at the hip, so it doesn’t make sense to think of them as distinct probabilities. Then by virtue of probability between heads and tails, the number of times heads is flipped is the same number for tails. So you’ll always end up with 3 columns with the same number of tick marks. The third column is not a result of probability, but it was designed into the problem from the beginning.
If you really think about it, why should tails be weighted twice as much as heads (i.e. why should tails receive 2 tick marks and heads only receive 1 tick mark)? Then the coin will no longer be a "fair" coin since the results is weighted.
As he gets too at the end, they're really answering two different questions
Halfers are answering the question what percentage of the time the coin was flipped did it land heads
Thirders are answering the question what percentage of the times the question was askwed was heads the correct asnwer
For the weighted coin flip at the end, 4/5 times the coin landed B, but 30/34 times the correct answer was C
Perfect
Ye but it's not the coin toss itself so the question is a trick (and wrong)
@@Trepur349 thank you. Very good summary and re-explanation
Veritasium: it's a fair coin
Sleeping beauty: is it fairer than me ?
Veritasium: yes, we are living in a simulation
Edit: wow thanks for the likes .
Actually I was confused between the snow White and the sleeping beauty. Snow White is the fairest of them all. That's why she got killed
Veritasum: now, will you eat the red apple or the blue apple
Prince Charming: [kisses Sleeping beauty]
Researcher 1: Stop! you're wrecking the experiment!
Researcher 2: Interesting, this proves we live in a disney simuatoin.
@@blankregistration7301 Or do we? 🤔 * suspenseful veritasium music *
Derek: Okay so the game is about to start and you fall asleep...
*ad starts to play and shows you a product that claims to help you sleep better*
Me: Simulation theory sounds just about right.
Targeted ads. It means stop wasting your life on youtube and go to sleep. Rofl.
Had an ad for a Canadian University for the football match lol
Same , I mean not the same ad but perfectly timed
“Do not hit the like button”
RUclips gaslighting me to do it:
The confusion arises from the same term “probability” being used for two different things: 1. the probability of getting heads when a coin is flipped (50%); 2. the probability of Sleeping Beauty in her confined situation guessing correctly if she believed that the coin had come out heads (in the past!). SB’s chances are, of course, skewed to tails. On Tuesday she may only guess when it had been tails. Had it been heads she would sleep and could not guess. In other words her guess entails her own dependence on the coin. Imagine you are lying on the operation table: The doctor tells you that you have a 50% chance of dying and never waking up from the narcosis. But what should you assume after you wake up? That the doctor comes and tells you: “Sorry, I goofed-you’re dead!” ??
Good analysis!
exactly my answer. perspective and probability are two different things. and counting one event twice, as he did in the experiment when he got 1/3rd each does hurt people who do statistics.
In this case ‘probability’ refers to her ‘credence’ of the coin being heads, i.e. her subjective confidence all things considered that the coin was heads.
That she would have slept through Monday on a tails flip is completely irrelevant as her credence of each scenario is not equal. It would only be rational to assign a higher probability to tails if her waking up eliminated some possibilities of heads which it doesn’t, so the chance of her being in a heads-world is exactly the same as being in a tails-world.
Your doctor case is not analogous, since waking up eliminates all possibilities in which you die, so you gain information from the fact you wake up.
Sleeping Beauty actually does gain information from her awakening: It's no longer Sunday! It's either Mon- or Tuesday. On Sunday the coin is flipped: 50% chance for heads or tails. SB's Sunday credence is intact. Now she is awakened: Oops! Is it Monday? Is it Tuesday? She knows it not. Her presence depends on her past. Of course, her memory of the Sunday chance seems intact: 50% for heads or tails. But her memory of that past lies in the very presence which depends on it: a loop--not to be trusted! Ask her this question now: "What do you believe, my dear SB, is your chance of being awakened again?" Hm..., ...Monday 50%, ...Tuesday 0%. How to answer? She's no longer in a "Sunday mood". SB's credence has been compromised by the fact that her beautiful presence may have been tossed into a "Tuesdayish" tails-tails-nightmare already. The very bed she sleeps on has been gambled with. There's a chance she lies on a doomed bed (at least until the Prince of Mathematics appears on a white horse; allow me to cry for a while--but only with one eye).
The SB-problem wants to not just entail the tossing of the coin but the tossing of the tossing itself. SB has already been tossed and turned in her bed (lousy sleep?) before she awakes. Her answer doesn't come from a 100% heads- or tails-world.
Btw., to address another point in the video, I think, it doesn't matter if she'd be awakened 1.000.000 times with an original tails flip: It's Monday versus 999.999 days presenting Tuesday. The tossing has only been tossed once--not a million times.
Are the odds skewed for tails tough? When it's heads she won't actually wake up on tuesday.
It's a variation on Pascal's wager.
For me, it is the wording of the question that tells me 1/2. "What is the probability" is a different question than "which outcome do you think happened".
Exactly. This is an independent event. Probability conditional on being awake though, I think that's different although I'm not sophisticated enough in probability to know how 😂
@@aubreydeangelo Not necessarily. The claim of them being independent is contentious among theories of probability. According to Bayesian probability theory, probabilities aren’t objective; instead, they reflect our degree of belief in X given Y information, so the totality of our information on the scenario actively affects the “probability” in the epistemological sense of an outcome. The existence of objective probabilities is tenuous at best; Quantum mechanics wave function collapse is a possible exception, be it contested. They are of course competing frequantist theories of probability however it being independent is not at all intuitive or obviously true.
Exactly, the question is ambiguous. There are 2 questions being conflated.
- what is the probability that a fair coin came up as heads this week?
- what is the probability that we woke you up because the coin came up heads?
@@nicksmith9521 thank you! None of the research papers would be necessary if the question was specified.
@@nikhilweerakoon1793 There's no need to tap into some subjective probability nonsense. There are two probabilities at play.
Implicitly, the question is stringing together two dependent probabilities: (1) a 50% chance of turning up as heads, and (2) 100% more likely to wake up due to Tails
Let's use another example: I flip a coin. 50% chance it's heads and I wake you up. 50% chance it's tails and you die in your sleep. The next day you wake up. "What is the probability it landed on heads?"
The probability is 100%. Because you have been woken up. The coin flip was a 50/50 chance, but the waking up was a 100/0 chance.
Yes, flipping the coin _in general_ is a 50/50 shot at heads. But now that you have more information (the fact that you woke up), you need to factor that in. If you say "50% chance" because the independent coin flip had a 50% chance, you're just intentionally ignoring additional information in some kind of weird linguistic purism.
When he said "Don't hit the like / dislike button" , exactly at the same time my like and dislike icons in RUclips started "Glowing" .....what is that ? Magic?
AI when it hear like it glows
On May 24, 1994 Canada and Brazil drew 1-1 at Commonwealth Stadium in Edmonton. I was at that game. It was an exhibition before the World Cup began later in the US which Brazil went on to win. (insert sad emoji of Roberto Baggio here)
5:26 - Reaching into a bag of one white marble and million black marbles is a fundamentally different exercise: the black marbles could be selected independently each other, but if Sleeping Beauty happens to have been woken up on day number 1000000, then she must also have been woken up on day number 999999, and the day before that, etc. Days two to one million are conditional on day one.
Yeah. It's more like reaching into a bag with one white marble and one black marble, but if you pull out the black marble you find out that it's actually a string of beads with 999999 other black marbles hanging out the bottom of the bag.
Yeahhh now I am getting it after reading this comment
Savagely well understood.
@@johnoldroyd94 you as well.
It’s like having 2 same size bags: one with 1 white marble and the other one with million black marbles. And putting your hand into a random bag to grab all the marbles. According to Derek, the chance of picking a black marble is million times higher.
Well, you’ll have more black marbles in average, but the fact that you forget that all the black marbles are a result of a single outcome doesn’t make them being picked independently.
To me, similarly to how in your counting experiment you doubled up the tallies for Tails every time, it's 50% because the tails numbers are being arbitrarily inflated by double-counting.
It's similar in my mind to if you said "toss a coin. If it lands on Heads count it as one, but if it lands on Tails, count it as if it happened twice". The coin is still 50% we're just counting it wrong (in my opinion, which isn't worth much).
👍
+
It's this bloody simple. Counting tails twice doesn't change the chance.
Yep. Think about the marble example. Its not pulling one marble out of a bag of 1 million black marbles and one white marble.
It should be stated - flip a fair coin then if its heads pull a marble from a bag of 1 white marble and if its tails pull a marble from a bag of 1 million black marbles. Whats the probability of a white marble? 50%
@@rakino4418 close. If it's tails, grab all the black marbles. Now you have a lot of black marbles and only one white marble.
The chance to end up with that many black marbles is still 50%, but that analogy more closely resembles the mess that is this probability discussion :')
Exactly, what that test is saying is that 1 heads = 1 heads, while 1 tails = 2 tails. The latter is then saying 1 = 2, which cannot be.
I think it boils down to being a language issue as each group interprets the question as a different problem (and it's probably wise to assume that there are more possible interpretations that lead to the same/similiar outcomes than these two). In the 1/2 case you are effectively asking "What do you believe is the probability of the initial experiment?", which is obviously the given chance, 1/2 in this case. The other interpretation would be something like "What do you believe is the state you have awakened in?", which would be 1/3 as there is one state for head and two for tails. Now you may argue that the interpretation here changes the rules, but an interpretation must be made in order to answer the question.
I think it's pretty much a problem of different observers being implied by those questions. From the point of non-Sleeping Beauty observer, which Sleeping Beauty can imagine herself to be, the coin flips and sets off a series of definitive, deterministic events, thus the question boils down to the H/T flip. From the point of Sleeping Beauty herself, she guesses which state she has awakened into, which, given one H state and n T states, results in her waking up to T states n times as often.
My intuitive interpretation is the one where only the initial flip plays a role, but I honestly think there is no definitive answer to this phrasing of the problem. It's the "what do YOU BELIEVE" part that causes this. Answering "What is the probability that the coin came up heads?" is always 1/2, because it refers to the initial experiment. The question asked in the video confuses you into evaluating the problem from the viewpoint of both Sleeping Beauty and Non-Sleeping Beauty observers by refering both to the initial flip and the viewpoint of Sleeping beauty.
I don't understand why taking the Beauty POV should equal to "try winning the odds".
Because that's the interpretation for the "1/3" answer. I.e. "If you woke up, what's the probability that it happened because of head". Of course the answer is 1/3, but that's the point, why people think about "winning the odds" (similar to Monty Hall), when the question was about the probability of ending in two scenarios, where you wake up 1 time or 2 times.
What I'm saying is that the duality is in the interpretation of the question and not being an examiner or beauty.
In fack, the football example od the video is quite perfect for this. When you woke up you surely said "Brasil", because it had 80% of winning.
Whil, if the experiment would have been "every time you guess, you win 1$, but if you don't guess you lose 10$".
Of course it's better to say Canada, because they will ask about it 30 times, so even if Brasil wins, you only lose 10$ compared to 30$.
Again, it all depends on the interpretation and type of experiment and not the pov.
I 100% agree and this is what I commented. The reason people see it differently is not because the answer is unclear, but because people inherently interpret the question differently.
That would be the direction of my "explanation".
I feel we have not sufficiently transformed our imaginary scenario into hard math to even begin calculating a result.
Excellent analysis. Do you write articles/papers or make videos? I'd love to see more examples of your interpretations of similar ideas/themes. Any comment on the simulation theory part? I've tried reading Nick Bostrom but a lot of it is over my layman's mind.
The initial chance of heads is 1/2.
When she wakes up, the chance for it to be Tuesday is 1/4, because there's a 1/2 chance the coin flip is tails, and only then a 1/2 chance that the current day IS Tuesday and not Monday.
Same thing for waking the Monday after a tails flip.
So the chance it landed heads is 1 - 1/4 (for tails Monday) - 1/4 (for Tuesday) giving us 1/2.
The right way to determine the possibility of the coin falling heads or tails is not by counting the coin sides or the virtual possible realities. The flip of the coin is a effect of a cause, that being the force of the hand of the coin flipper on the coin.
Imagine that you give the coin to a robot that has the precision to flip the coin with the same force in the same position, he would certainly get one of the sides 100% of the time( if there is no other thing causing the coin to flip, like the wind for exemple).
It would be the same as thinking that if you droped a metal square on a flat table it would have 1/6th of a chance to every side to be hitting the table.
The key difference is that the event where she is asked about the coin happens twice with the same answer if it comes up as tails. The question is worded in a way that no matter what her perspective is, the answer is 1/2. However, if the question was instead: "If you guessed heads, how likely would you be to be right?" Then the answer would be 1/3 because she gets asked the question once if she's right, and twice if she's wrong.
Well said, Bismuth.
Well said Bismuth. Funny to see you here instead of a speedrunning video!
but both monday and tuesday tails should count as 1 guess, so it would be 1/2
Did you speed run the answers too?
This is what I came to. This 1/3 business seems to be lateral thinking about how the question was asked.
In my opinion the question you seem to ask is: What is the probability that today isn’t Monday? The probability of a coin flip is always a coin flip as we say(50-50).. but the question refers to the coins state only in regard to what day it might be.. meaning that if it was H she’ll be asked once, while if it was T she’ll be asked twice, so it’s not really a philosophical problem, just that the question appears to be misleading
I agree, I put it this way: If you ask her what is the probability "a coin toss" *will* be heads or tails she should of course answer 1/2. You'd be referring to a random event that hasn't happened yet. But in her case the event has happened and is no longer a probability, she exists in a timeline created by its outcome. Really she's being asked, what is the probability you wound up in the timeline where a tails came up. She's more likely to be in the tails timeline by virtue that waking up in that timeline is statistically more probable.
The answer to your question (What is the probability that today isn't monday?) is actually 1/4. Since there is a 1/2 chance she is woken up only on monday and a 1/2 chance she is woken up on both days and then it is 1/2 chance for either day.
The original problem is indeed flawed though. There are 2 ways to interpret this:
1. The experiment is only done one time, so the answer is 1/2 since the coinflip is either heads or tails.
2. The experiment is done an infinite number of times. In this case the answer is 1/3, since now you are woken up twice as much in case of tails. Even now we can see that the quenstion is badly phrased as the probability the coin came up heads is still 1/2. The probability she was woken up following heads did change: it is now 1/3.
@@lennertdevoldere7000 the coinflip probability being heads is 50%.. theres no debating that.. now the question refers to probability but then extends forward to requiring the "what day it maight be" reference as a parameter, which is irrelevant to probability. thats why im sayin its not a paradox but it only gets wierd with a misleading explanation. and the answer to my question is 1/3. yes theres 1/2 chance she gets waken up only monday and 1/2 waken up both days but in her world thats irrelevant.. like @neodonkey mentions above "she exists in a timeline created by its outcome". and to simplify matters lets say the coin is fair and we run this experiment 6 times.. so the outcome would be 3 times only monday and 3 more times both monday and tuesday, that makes 6 possible Mondays and 3 possible Tuesdays = 2/3 and 1/3..btw this is not my question, its the question implied in the explanation.
@@ΚάλληΤόσκα yes, but this again implies the experiment is repeated. In that case it will be 1/3. The outcome changes if it is only done one time.
@@lennertdevoldere7000 no matter what the coins state is, the subject is goin to wake up on Monday, but theres 50% chance she also wakes up on Tuesday.. 100% Mondays, 50% Tuesdays which means 2-1 on favor of Monday which is 2/3 on a Monday possibility and 1/3 on a Teusday. Repetition only hepls visualize because you add up the outcomes on both coins states and just divide by the reps.. What your saying would be correct only if on heads she were to wake up twice on a Monday and on tails Monday-Teusday as is..
I am on the 1/2 side, here is my reasoning:
The many wakings are not independent events (I think). If any one of them happens, you are certain that all of them will happen as well. Because of this, I think you can imagine all those sequential awakings as a single event. That removes the bias of "many possibilities in one branch" and you are left with single event in each branch.
edit: This also makes sense to me for the argument of "no added information", because the sequential wakings are linked together.
true.
dislike guys. we have to rise.
That's the whole point..... The events are conditional from researchers perspective.... From sleeping Beauty's it's not....she doesn't know what number of time she has been woken
But any time you go to sleep, you either wake up and are asked about the coin flip, or wake up and are told the experiment is over. The fact that you're being asked about the coin flip *is* information, and it tells you that the experiment hasn't ended. Knowing that the experiment takes longer to end if you flipped tails, you use the information to update your probabilities.
Well said!
@@elliotgengler3185 You could also take this scenario in a darker direction by slightly altering the experiment, whereby a 'heads' coin flip results in Beauty never waking up at all. If she is then awoken to be asked about the coin, she knows that the coin must have been tails.
How can I click dislike on a video that made me confused within the first couple of seconds, but I carried on watching anyway?
Waking up on Tuesday always comes after the coin flips Tails. There is no other reason for this event. If we want to correctly relate the question to the situation it's incorrect to put more variables into the equation such as "there is one more day in which she may be awake". The question has to be only related to the thrown of the coin which possibility is 1/2.
But then you would disregard the information she already has, and information is important in mathematics.
If she gets woken up, theres a 1/3 chance that its tuesday.
@@gorgit That's not the question. First, she has no information other than the rules of the game. The question is "what does she believe the probability is that the coin landed on heads?" When she gets up, she isn't given anything other than the question. 50%... problem solved.
@@JohnJJSchmidt she is given knowledge that one of the days she could be awoken is Tuesday, but she doesn't have the ability to tell what day it is when she is awake. Therefore she has the knowledge that it could either be Monday after a heads, Monday after tails or Tuesday after tails. Even though the question she is asked doesn't change the probability the pre existing knowledge she has does.
@@jaredhahn7970 The probability of what? Again. The question is what does SHE THINK the prob the coin landed heads, not the prob of it being a certain day. If she says anything other than 50% with a blank mind, she is letting the pre-existing knowledge of the rules interfere with really basic reasoning.
If we phrase it as “what is the probability that she woke up in a scenario that the coin was heads?”, it’s 1/3; but yeah, given that the question is, “which way did the coin come up?”, it’s still just 1/2.
This is what happened to me during my school exams. I would forget what I studied the previous day. The teacher would think the probability I studied for the test was 1/10 from the marks.
This! This right here.
The problem here was your approach to study. That last-minute "cramming" is easily forgotten, especially in a stressful situation. Repetition and practice over a much longer period, or reflective self-study to the point where you reach genuine understanding, was the way to make sure you passed those tests. But I'm completely aware that very few kids at school would heed that advice, including my younger self!
A lot of the other scenarios were not equivalent to the Sleeping Beauty scenario. They were more like asking Sleeping Beauty "Do you think it's Monday?" That's an entirely different question from "What are the chances the coin landed on heads?"
This is a brilliant comment. Contrasts very well the difference that is muddled in the "what was the probability" question. The answer to, "what is the probability the coin was heads?" is objectively 50%. The answer to the question, "what is the probability that the coin was heads AND that your answer is correct?" is 1/3.
@@bbanks42 What? So for the first question your answer doesnt need to be correct for it be... correct?
The answer for "what is the probability of the coin to flip heads" is indeed 50%. But thats not the question, the question is "what is the probability the coin FLIPPED heads" with the given that if you are being asked that question you woke up. Similarly if someone flips a coin and it results in a Tail, it would be correct to say the probability of flipping tails in the past is still 50%, but wouldnt be correct to say the probability the coin that was flipped was tails was 50%, because you are clearly already seeing the result, and its 100%.
Imagine if the SB only woke up if the coin was Tails and was asked "what is the probability the coin FLIPPED heads?" , it would be ridiculous for her to say the chance is 50% after being asked that question, because she knows she wouldnt be asked that question if it was Heads.
Makes no sense. The setup is wrong. You cant make her forget that she woke up yesterday .... and then ask a logical realistic question.
If she cannot remember yesterday, then the asking person might have forgotten who they are altogether, or whom to ask.
Like : you are my banker, with a brilliant mind, and you can recall all of my bank statements from memory. But i always forget my adress, my name, my job and which bank to go to.
Now you want to ask me a meaningful logical game theory question on how to save money better ?
Makes no sense.
....also the coin is a half half deal. The monday or monday-tuesday thing is a scam. Please try Mon and then Mon Tue with a coin.
Heads comes up .... or both sides come up.
Great.
@@crockmans1386 go watch cartoon
Another variation that should provide the same answer:
1) Wake SB both days.
2) Have her write down on a notecard the probability for the event "Today is Monday and the coin landed on Heads." Below that, write the three other probabilities for Monday&Tails, Tuesday&Heads, and Tuesday&Tails,
3) Lock the notecard in a safe that can only be opened by her voice.
4) Give her the amnesia drug. Include the sleep drug ONLY IF it is Tuesday, after Heads.
5) Once amnesia has taken effect, if she is awake, have her open the safe.
6) Ask her, with the full knowledge of these steps and the notecard, for the probability that the coin landed Heads.
The correct probabilities, in step 2, are all 1/4.
In step 6, she can cross out the probability for Tuesday&Heads. The remaining three need to be renormalized do they sum to 1. You do this by dividing each by their current sum, 3/4. This makes each 1/3,
But step 6 is identical to the video's problem. With the full knowledge of the procedures, she didn't need to go through steps 2 thru 5, she knows those probabilities applied at the time just before she was wakened. The answer is 1/3.
The error made by Halfers is not recognizing that Tuesday&Heads is just as much a possible outcome of the experiment as the other combinations. What causes them to make this error, is pretending that her being awake is a prerequisite for it to exist.
I think the confusion arises because of the question "What is the probability she should assign to the event?" It's a separate issue of the probability she should assign versus what prediction she should make due to her being asked the question more frequently given a certain outcome.
Exactly, I feel like this is convoluted for the sake of being convoluted cos the question is confusing. Interesting nonetheless
This is what confused me about the question.
The probability the coin flip was heads is 1/2; but the probability that she was woken up by a "heads" coin is 1/3.
Took me two watches to catch that. I even simulated the second “unasked” question, confirmed the 1/3 outcome, but that wasn’t what she was asked.
Yeah. Depending on the exact framing of the question, either answer is valid.
Yep. It's a whole genre, these silly faux-dilemmas. They try to pass-off a multi-step process as a one-and-done. In the case of SB and similar DT puzzles, it's, as you say, more than one prediction, serially. In this guy's argument, he even replaces the fair coin with an extremely unfair coin, if you will, and STILL courts the controversy. Infantilizing, if you ask me.
such an awesome "feature" that youtube not only removed the surveys in videos, but also the dislike count 😍😍
The number is still being transmitted with the page, but not being displayed. If you want to see the numbers again, you can use a browser extension such as "Return RUclips Dislike". This is why I'm able to tell you that at the moment, this video has 114k likes and 61k dislikes.
@@mattgies unfortunately that does not work for mobile. and that extension is not accurate, it is only a rough estimate :/
@@mattgies The RUclips API doesn't include the Dislike Dataset no more. The Extension just looks at every Extension user which likes or dislikes the video and extrapolate the data to get a rough estimation of the like to dislike ratio
@@dnoldGames Oh wow! I had no idea
@@anonymouscommentator Youtubе Vanced on mobile does include Return RUclips Dislike as a toggleable setting, so it is still possible to use. However, that estimation is still a factor.
Firmly in the 1/2er camp. Irrespective of how many times she was woken up, the coin flip was a singular event. That event had a 50/50 chance of either result.
You are absolutely correct, the problem is that its a trick question, change the question from "what is the probability that the coin came up heads" to "what is the probability that we woke you up when the coin came up heads". Now the second question was not literally asked, but it was presupposed by the fact that she woke up being part of the experiment and with the information she was told prior to the experiment. With the second question, its 1/3, it has to be.
But the first question tricks people into believing its just about a coin toss, but it is not. It is about her waking up in this experiment involving coin tosses, and because she gets to wake up twice on tails, it weighed in favor of tails.
@@emperorpicard4901 why are you saying he is right then? The question is clearly "what is the state of the coin NOW". Not what was the probability that it would land this way. And it will be heads 1/3 of the times it is asked.
The chance the coin flip will be heads or tails is a 50/50, but the chance it was tails from her perspective, given she's woken up, will be higher, as being woken up in a world where tails was true is more likely given it happened to her multiple times
The question is not about the physical probability represented by the coin toss. It is about the epistemic probability that SB should assign to the outcome -- in other words, her credence. Bayesian inference dictates she assign 1/3 to heads and 2/3 to tails. She knows it is either Monday or Tuesday, but not which. She knows "I am awake on this particular day". For the hypothesis "The coin came up tails", the conditional probability of "I am awake on this particular day" is 100%. For the alternative hypothesis, the conditional probability of "I am awake on this particular day" is 50%. That gives a likelihood ratio of 2:1, which you multiply by the prior odds for tails of 1:1, to get odds of 2:1 for tails (Bayes's rule in odds form). That translates to probability of heads = 1/3.
@@rsm3t What I'm not following there is what value there is in treating the wake-ups from the 2nd outcome as unique/why the day she wakes up matters to the question being asked.
she would say its heads, as even if its a tail, she would have forgotten that she had woken up prior.
The REAL problem is that there's an implied reward: if she's asked the "probability" of heads, then it's 1/2. If she's being rewarded for GUESSING whether the coin was heads or tails, she should always answer tails, because she'll get rewarded twice in that scenario (vs once if the coin flip was heads).
This is exactly it. I feel like this problem wasn’t really posed thoroughly and that causes confusion
@@hisuianarcanine9379 Nah, the question was clearly "What is the probability that the coin came up heads", that fits perfectly the first case of OP and it's unambiguously 1/2
Exactly what I think! It's unclear which question is being asked from this video, and we have to be very specific when asking the question.
Like you said, if the question is "what is the probability that the coin landed on heads", the answer is always 0.5. Sure, sleeping beauty will be wrong multiple times if the coin landed on tails, but that's not relevant to the question being asked here. The fact that the coin is two-sided does NOT change, and the sleeping beauty's knowledge is identical every single time.
It's an entirely different question if sleeping beauty is trying to 'win' as many times as she can, then the best answer is quite obviously tails. If she is woken up N times when tails is thrown, and once when heads is thrown, she will get the correct answer N times out of N+1 guesses, on average when repeating the problem.
@@angivaretv4475 "The probability that a fair coin comes up heads" is undoubtedly 1/2, but "the probability that you are waking up on a heads monday" is 1/3.
@@myeloon It would be a waste of everyone's time to go into the exposition of her being waken up monday monday/tuesday if all we are going to ask is that given a random fair coin that is absolutely irrelevant to her situation. Of course the question is "given that you were just woken up, what is the probability that heads came up". If that is not the intention, I hope the people who developed this problem and six degrees of separation from themnever wakes up again.
Bob watches Alice flip a coin.
The coin comes up heads.
Alice: "What do you believe is the probability that the coin came up heads?"
Bob: "What do you mean? That the coin *has* come up heads? That the coin *would've* come up heads? That the coin comes up heads in general?"
Alice: "Pick one."
This isn't a math problem. It's linguistics.
Considering how important it is to write formulae correctly in math, it does feel like not just a linguistics problem but one that's intentionally vague so as to make two different outcomes seem equally obvious.
Good point. Once the coin has already landed, it's no longer 50/50, it's either 100% or 0% heads.
Specifically, it's semantics
Yes and no. This is just a rephrasing of the Monty Hall problem.
@@kirbwarriork3371 Yep it seems like a lot of information is left out just to create a clash between two different answers to two different questions.
A fun problem where the two answers are actually answering two different questions! The skill is not figuring out which is right, but understanding how the two questions are subtly different. Good thinking exercise and excellent video as usual.
There is no question to which the correct answer is 1/3. The whole thirder perspective is flawed because it treats the possible "states" as equally likely and independent, but they are not independent.
@@nekekaminger The question would be, ‘What is the probability you were woken up by a flip of heads?’
I think the answer to the sleeping beauty question is 1/2 though because like the original comment said they are answering two different questions.
@@reubensavage2067 She's always woken up, otherwise you couldn't ask her. Prepending the question with the pseudo-condition of her being woken up doesn't actually change anything because it always happens. The question is fully equivalent for "What's the chance heads came up?" which is clearly 50%.
I see what you are trying to do. You view each waking up event as an independent event and try to assign a probability to that event (just like Derek proposes in the video), but that approach is flawed since they are not independent. Monday Tails and Tuesday Tails cannot happen without the their also happening.
Imagine you have a somewhat unusual coin that instead of heads has one dot on one side and instead of tails it has two dots on the other side. Each dot represents a waking up "event". After the toss pick one of the dots you see (which is either just one, in which case the choice is simple, or two, in which case you just randomly pick one, since SB can't remember being woken up before, the order does not matter) and ask yourself "What is the chance I see this particular dot because the coin came up with the single dotted side?". If two dots were up you answer the same question for the other dot. The experiment is exactly equivalent (if you don't agree, please explain). Do you still think the answer is 1/3?
@@nekekaminger The part that I disagree with is that I'd argue they are independent events. She could be woken up on Monday Heads and be asked the question, or woken up on Monday Tails and be asked the question, or be woken up on Tuesday Tails and be asked a question.
As others have stated, it really comes down to which question is asked of her.
If she's asked "What do you believe is the probability that the coin came up heads?", then she should answer 1/2. Because the coin either came up heads, or it came up tails. It doesn't matter which day she woke up; the coin was either heads, or tails.
If the question is "What do you believe is the probability that you were awoken on heads?", then she should answer 1/3. Because as I mentioned in my first paragraph, if she's asked this question on Monday Heads, she would be right. If she's asked on Monday Tails, she would be wrong. If she's asked on Tuesday Tails, then she would again be wrong. So it's a 1/3 chance of her being right about the 2nd question.
@@ThrowAway-hy5sp you haven’t tackled his point that these events are not independent. Monday tails and Tuesday tails are essentially the same event. For the example where she wakes up “a million times” it’s 1/2 chance that she’ll wake up a million times or 1/2 chance she wakes up once. Either way if she wakes up on the thousandth Tuesday and is asked “what’s the chance that you will wake up another thousand or so days”, its 1/2 as is “what’s the chance you only wake up today on the monday”. There’s not “more chance” of waking up in the millionth day like it’s compared to being in a simulation. It would be like saying the chance of you living in reality is 1/2, and the chance of you living in any of the millions of situations is also 1/2. 1/3 would be the answer to “what’s the probability today Is Tuesday” regarding the original question.
1/3 seems obvious because the frame of reference for her evaluation is different than ours. If we think from her frame of reference, then it must be 1/3, from ours, 1/2. Since the question was asked to her in her frame and her circumstances (which offer some certainty whereas ours offer only probability), then it must be evaluated from that frame. 1/3 is the obvious answer.
I would also note that the agree-to-disagree ratio is nearly 2/3 to 1/3.
The probability doesn’t change, she just answers twice in one scenario.
Which means the probability does change, from her perspective. She gained information by the fact that she was woken up.
@@jogadorjncbut that’s not how it works in this case right? He did mention that she will forget that she has been woken. To her every time she wakes, she would feel that this was her first time, no matter what day it was. Right?
@@The-Meaning-of-Life-is-42 She gains information by waking up, even without memory.
Imagine if the setup was to not wake her up if it landed heads.
By the fact that she woke up she'd be able to tell with certainty that it landed tails even if the coin was balanced.
@@jogadorjnc that’s the probability of her guessing right, not the probability of the coin toss itself
@@jogadorjncbut isn’t the information gained irrelevant to “what’s the probability the coin came to head?” Let’s rather than waking her up 2 days, wake her up instead for 50 days? Wouldn’t that means by the vid logic, you’re saying the chance the coin came up head is near 0%?
The question is the trap as you explained in the video: "What do you believe is the probability that the coin came up heads?"
You would have to disagree and ask them to clarify if they mean: "What are the chances of a fair coin flipping heads or tails?" OR "What are the chances you are in either stage of waking up in the experiment".
Edit: i hate these kinds of "math problems" since they are almost always about the question being asked in a stupid/inaccurate/unfair way to the situation at hand and then people just going "what if we actually try to answer the unfair question seriously". Then it inevitably ends up with the same conlusion as the first paragraph where the authors assume one of X interpretations of the question and continue to calculate and answer that. But in that case you could have just asked the correct question from the start in the problem. This is why I always tell my friends to think about what they are saying, if it can be understood in mulitple ways it wont help you get your point across. Write so that your intention can only be interpreted in one clear way.
i also feel like they just missing the obvious lol
If the question for the sleeping beauty was to tell if the coin flipped heads or tails. She is woken up three times,two times if it's tails and just one time if it's heads. If she says tails all three times, she'd be correct 2/3 times. If she says heads, she'll be correct 1/3 times. In conclusion, the probability of her getting the answer correct if the outcome is heads is 1/3. Whereas,the probability of the coin flipping to either heads or tails is 1/2. She would be right 1/3 times,but then answer is 1/2 as per the question.
@@architlal8594 the question doesn't ask her to predict whether the coin was heads or tails. it asks her what the probability is that it was heads.
so her response wouldn't be 'heads' or 'tails'.
the trick with this problem is that people are fooled into thinking that monday (tails) and tuesday (tails) are independent events. but they aren't. they're actually the same event. the reason you get the 1/3 distribution is that she gets woken up twice on a tails. and therefore gets asked twice from the same coin flip.
I feel like that arguing with people about politics and society all the time. In the absence of an obvious answer on a lot of those issues - unless you are very well informed on them, which a lot of people aren't, often times people just try to roll you with fallacies like that. I believe usually even unknowingly so and thinking "they got you". But it's very tough to effectively counter that, especially in the moment, because, as this video shows, unraveling such fallacies can be very hard. Often much harder than coming up with them.
Actually a lot people that understoodd this question as "did I flip a heads or tails?" respond with "I dont know its 50/50." This isnt some word game this is a sort of paradox. The people that disliked this video isnt arguing that a coinflip is always 50/50.
Okay, I actually gave it some thought before listening to the proposed answers, and initially I also answered 1/3, but then I read the comments, did some more thinking, and switched to 1/2.
First, as others said, the wake-ups are not independent, and the question comes down to "which branch do you think you're in", and the probability of that is 50%.
Second, even in the situation with 1 vs a million awakenings, you can't consider being in each of them equally likely. Finding yourself in the branch with a million awakenings is dependent on the actual coin toss (50%), and the probability of finding yourself in a specific position in the second branch must be calculated using Bayes' theorem. 50% it is.
I agree that it's 1/2. Even if it was 1 vs infinity I would say 50%
That makes the most sense to me as well
A branching reality is the best way to describe it. You don't care about the probabilities in one branch, only the probability of ending up in one branch or the other.
I think you misunderstood the question. If you wake up a million times more often if it hits tails, then it is a million times more likely that when you wake up, it was tails.
@@johnnysilverhand1733 1:04 how the question isn't what the likelihood of a coin coming up heads or tails ??
Where, in the initial phrasing of the question, does it say anything about a second coin flip? It’s 50/50 because the coin is only flipped once.
That's why I'm set on 50% too. There's only one coin flip that happens at the start so it's gotta be 50% because whether it's Monday or Tuesday doesn't matter. The question is "What do you believe the probability is that the coin came up heads?", and the probability of a coin flip coming up heads doesn't change when you ask someone "What's the probability of a coin flip coming up heads?" two days in a row. There are three possible states of them asking the question (Monday + Heads, Monday + Tails, Tuesday + Tails), but the question doesn't ask anything about what day it is or if she thinks she's been woken up once before, so the day of the week is irrelevant and only the coin flip matters.
But the researchers are asking Sleeping Beauty, who's from a fairy tale originally written in the 14th century, so I think her answer would actually be "What's a probability?"
@@pelleas9091...but if she's woken up on a Tuesday, there is a zero percent chance it landed on heads, so how many times she's woken up does matter, making it 1/3 overall.
Your ad was perfectly timed after you had us “go to sleep” 😂
As soon as the screen went black, it cut to Patrick Stewart’s smiling face telling the camera, “Hello, I’m Patrick Stewart”
It will scares me if one day this happens in real life
what is this man saying? is he from another universe?
I got a bounty commercial lmao and I also posted a comment about it before I saw yours😂
@@mgkelley2609 😂❤️
Well that's a nice dream!
For my part, I think fundamentally the question is malformed, and that's why we have such issues with it. There are two possible meanings of the question, and commensurately two possible ways of looking at the data.
A: "What do you believe is the probability of the coin landing as heads?"
B: "What do you believe is the probability, given you are awake now, that the coin actually was heads on Sunday?"
The ways of looking at the data (if we treat it as sampling whether Sleeping Beauty thinks the coin actually did land heads/tails):
1. In each _trial_ of the SBP, which answer will be most consistently correct?
2. For each _awakening_ of Sleeping Beauty herself, which answer will be most consistently correct?
If we base our statistics around the per-awakening result, then 1/3 is correct, and indeed it should be 1/(n+1), where n is the number of times you awaken Sleeping Beauty if you flip tails. If we base our statistics around the per-trial result, then 50% is correct. The former is true because, when we look at the percent likelihood *on any given awakening* that Sleeping Beauty was awakened on a trial that flipped Heads, that of course must fall to zero as the number of Tails-awakenings tends to infinity--the vast majority of awakenings are Tails-awakenings.
That the latter is true is a bit more complicated, but can be expressed as follows. Perform the same test, but simply ask SB whether she actually DOES believe the coin flipped heads, yes or no. Tally up the answers. If the coin _did_ flip heads, then she will either be right once or wrong once. If the coin _did not_ flip heads, then she will either be right N times (for the N awakenings), or she will be wrong N times. All told, there are 2N+2 possibilities, and out of them, (N+1)/(2N+2) = 50% are correct.
Hence, it depends on whether you examine the data from a per-awakening basis or a per-trial basis. The question is malformed, ambiguous, and that is why it leads to an alleged "paradox."
Of course. The statement as 1:53 is simply false. These are two different questions, each yielding a different probability distribution and thus different answers.
It's two levels of abstraction using the same symbol so the English confuses the math.
Lets do the same exercise but change the coin to marbles when we present the new abstraction instead of hiding it behind the same name (coin flip)
Flip a coin every time it turns HEAD place a RED marble in a bag
Every time its face up Tails put two BLUE marbles in the bag
Now if we ask the question "What are the odds of the coin" well its 50/50
What are the odds of pulling a red marble out of the bag? Well 1/3
Paradoxes are cool, this isn't one, just a poorly worded question
Yeah this was my thought too. However, it does ask "...IS the probably that the coin CAME UP..." So it is not asking you how often the coin did anything. It is asking how often you will wake up because of tails as opposed to heads, therefore it must indeed be 1/3rd.
@@identifiesas65.wheresmyche95 There is no probability for events that have already occurred. Hence, the question if phrased that way is about whether you *believe* it did or not, and that belief is where the probability component enters the picture. Whether Bayesian or frequentist, you'll be thinking about two things: "What is my belief that a fair coin would have already been heads or tails?" (naturally, ½), or, "What is my belief that this awakening is a heads awakening?" (naturally, ⅓).
If we are clear about which question we are asking, the problem goes away.
Edit: I think it's actually really useful to treat this as one would the Monty Hall problem. There, it becomes a lot more clear what's going on if you presume a hundred doors, or a thousand, or the like. If you pick one door out of a thousand, and Monty opens *every single other door except one,* would you switch? It seems pretty clear you should. You only had a 0.1% chance to pick the right door at first. Monty has now eliminated every other door *except one.* The odds are enormously in favor of that other door. It just happens to be hardest to intuitively see that when you have the smallest possible number of doors (3, in this case.)
We see the same thing with the SBP. We only have three awakenings (well, one vs two). What happens if we make it one vs 999? Further, what if we add some expected value to the answer?
Consider: Sleeping Beauty wins $1000 if she correctly picks Heads, and $1 if she correctly picks Tails. The expected value now depends on how you view the question! If we structure things on a *per trial* basis, then half the time the coin is heads, and half the time it is tails (before any awakenings have occurred), this is agreed by all parties. Hence, *per trial,* the expected value is $1000 if she guesses heads correctly, and $999 if she guesses tails correctly. Since each is equally likely *before* any awakenings have occurred, she should choose heads every time; she will net more money, albeit slowly.
If, however, we award her the exact same prize for any correct guess on each awakening (e.g. "if you pick a side of the coin and are correct, you win $1"), then she should 100% always choose tails, because she can win $1 on half of trials, or $999 on half of trials. The preponderance of *awakenings* is on tails paths.
Someone asserting that the probability must be ⅓ is claiming that, for _all_ experimental setups, the higher expected value for these Sleeping Beauty prizes must be from picking tails. This is not true. By offering prizes based only on the coin's facing, *not* on the number of times Beauty awakens, we can clearly see the difference between the two approaches.
@@identifiesas65.wheresmyche95 - Does she even know about the multiple awakenings? It's not made clear in this video (and it's my only familiarity with the problem).
I feel really bad leaving a Veritasium video on "dislike", I even almost liked the video as a habit after I was done watching it.
which also leaves me to believe the like dislike ratio is not to be trusted that much.
Yeah I feel like the number of likes is mostly a combination of people using the like button to actually "like" the video and others who only listened to the first part. The comments seem pretty consistently to agree that it's 50/50.
The ratio in the title would also cause a lot of people’s opinions to change due to peer pressure.
I think he designed it so that he'll receive more likes than dislikes since statistically, more people will choose the 1/3 option. I'm pretty sure he thought through it.
The question should be asked like "what is the probability that sleeping beauty is awakened on monday and that of heads."(It's a conditional probability).
We will obtain 1/2 as answer if the question is "what is the probability that sleeping beauty is awakened on only ones a week or twice in a week.OR even more simpler by stating heads or tails.
The probability of a coin flip is always "1/2" untill a conditional probability is involved.
EXAMPLE:From two differently colored bags containing two different colored identical balls.probability to draw a one of the colured ball from any of the bag (or) from a specific bag.Then the answer varies.
For me it becomes less paradoxical when I think of the question as rephrased as "How likely is it that Heads is responsible for you waking up this particular time?"
that is so much better of a question, kudos
Exactly dude, like it's simply not possible for it to exist. 1/2 is simply the answer.
"what's the probability it came up heads" contains a hidden knowledge assumption, that being with the knowledge that it just woke you up or prior to that being someone's state. It's a partly a matter of how you understand the meaning of the question, but the best answer is based on all knowledge, which is that you just woke up. Imagine she's never woken up for heads. She's woken up, what was the probability that it was heads? In an abstract sense 50:50, but based on the full knowledge of evidence, it's zero
Yes, indeed, the answer to the question in the video is 1/2 and the answer to your question is 1/3 (the 'intended' answer)
Your question is simply different from the original
The problem here is that to define probability, you need a clear indication of what counts as a trial. If, in a series of these experiments, a trial is marked by each time the coin is flipped, the probability is 1/2. If a trial is marked by Sleeping Beauty waking up, then the probability is 1/3. For any practical application, Sleeping Beauty would be advised to mark trials by waking up, as if she had to bet on the result of the coin, her breakeven point would be the same as betting with open information on a coin that came up heads 1/3 of the time. In a scenario with no applications where she only cared about being "correct," either method of marking trials could be chosen, but having a specific goal for her allows the problem to be concrete enough to be optimized.
Id like to see someone argue this, pretty spot on
even if the trial is marked by each time sleeping beauty wakes up the probability is still 1/2 because her wake is dependent on the coin flip in the first place
Completely agree. A fraction only means anything when the whole is defined in any way.
@@drbrain4483 can you elaborate on this please
@@coolzmasterz the only way 1/3 can be a bit logical is when you have to bet with money on the answer but dosent it make more sense to bet 100% on tails , i mean if your wrong you only lose once but if your right you win a lot
i love how just when i feel certain that im right he makes me question it
Only a Sith is based.
@@kaizokujimbei143 no obi-wan! no!!!!
hello there
Let me make you question it again, the answer is 60% tails and 40% heads for 2 flips :)
I sense great turmoil and hesitancy in you
To me it is just how the question is phrased. I'd say it this way:
-the probablility of the coin came up heads was 50%
-the probability of me being awaken by the coin on heads, is 33%
Sleeping Beauty not remembering her previous wake-ups is akin to a coin not remembering its previous flips. The # of times she is woken up can be thought of as the payout. In this case, betting tails would yield a greater payout, but it does not mean tails will come up more often.
"but it does not mean tails will come up more often." - that's exactly the point though. When do you "count" a flip? Only when it's actually flipped, or when the sleeping beauty is awakened? From the beauty's perspective she cannot know if she's awaken for the first time or the second time, so she is forced to count the two separately.
@@ShaharHarshuv Are we in agreement that this occurs in a single coin flip?
The scenario is if that flip goes heads, [outcome], if it goes tails [outcome]. The coin flip is 50%, that has to be accepted. Meaning the Monday Wake ups both have an equal 50% possibility. If it’s Monday Tails, there is a 100% chance that you will wake up Tuesday, as that is the basis of the scenario. This 100% rate of Tuesday wake ups within the 50% rate of Monday Tails wake ups, when multiplied, gives a 50% chance for Tuesday Tails wake ups.
It's a matter of perspective:
The odds of her flipping heads is 1/2.. but the odds of her waking up on Monday is 1/3. But the result of waking up on Monday can only happen if she flips heads. So you must assume the probability of the consequences rather than just the result of the coin toss. Therefor it is 1/3 and there is no debate about it; because she is waking up more often than the coin is tossed, so that changes the likelihood of the standard 50/50 chance. The fact that waking her up is a qualifier, completely changes the standard parameters of a 50/50 coin toss.
It's easier to think of it as Monday is heads/ Monday is tails and Tuesday is NO TOSS. So every time she is woken up, she has to think, is it a heads or tails Monday, or is it no toss Tuesday. So she has 3 results to consider. The answer is 1/3, because there are 3 possibilities.
But what is very interesting: From the perspective of the "coin toss man"-- he gets to go directly from toss to toss, so even though tails has multiple consequences, it does not effect his odds one way or another. So from the perspective of the coin toss man, it is 1/2 chance. But that is not the question. The question is from HER perspective of having to wake up and answer, when she has to wake up more times than the coin is tossed; so from her perspective, it's 1/3 chance, because of the "waking up" prerequisite.
@@ShaharHarshuv You count the flip when it's actually flipped. The amount of times she is awakened, (2-infinity), is all triggered by that one flip of tails.
globally tails will not come up more often, but subjectively for here it will come up more often. Is she asked about the global upcomming or the subjective upcomming?
This is a great lesson in clarifying the problem. As addressed in the video toward the end, are we asking the state of the coin after the flip or after waking. Be careful with the empirical study. By definition Wake on Tuesday = Wake on Monday following the event “coin lands on tails.” On E(Tails) you will talley in both columns E(Mo|Tails) and E(Tu|Tails). This will give the appearance of E(Heads) = 1/3 and E(Tails) = 2/3, but you’re really tallying E(Day|State of Coin). Thirders are answering the question, given that Sleeping Beauty was woken up, what is the state of the coin? Halfers are answering the question, Sleeping Beauty was woken up. What is the state of the coin?
But sleeping beauty wakes up all the time / half the time on heads monday / 1/4 time wakes up tails swith monday , 1/4 time tails on tuesday , She does NOT know which day it is , so this information is irrelevant . i think Thirders try to calculate the probability of she guess right ! we do not ask that
The problem is it’s a bad question as it wasn’t clear what is being asked
I'm answering the question 'How many times was the coin tossed ?'....which is clearly only once and is I think the correct question to ask. How many times Sleeping Beauty is woken up is an irrelevant red herring.
@@Agnostic_Mind can you explain why you think the coin is more likely to be heads if it's monday?
@@peterstanbury3833 The coin is only tossed once, he never says the coin gets tossed again after she wakes up.
I think the _really_ clever thing here is that Derek has carefully orchestrated a video to generate a high "like" _and_ "dislike" count. That kind of controversy will be irresistible to the almighty algorithm 😎
Stolen from Tom Scott. Doesn't bother me, but it is.
RUclips algo like that? It might get changed soon if it can be abused.
@@theeraphatsunthornwit6266 Pretty much all social media is optimised for controversy or moral outrage, because that's what drives the most interaction. I don't honestly have any idea about the YT algorithm, but we can be pretty sure it'll rank videos with a widely split opinion above a video that has just a high number of likes, especially if there are lots of comments too.
@@bingbongthegong what video did Tom Scott do that abused the like count?
@@bingbongthegongThat would be the like and dislike number update
The ideo of ratios was not stolen though
I'm in the half camp. Here's why: In the sample there are three days in play but only two coin flips. Heads is good for one day and tails gets you two. There are only two "events" resulting from the coin flip: one day or two days so the chances of any waking choice in this scenario is always 50/50. Additionally, the question would be posed to Sleeping Beauty who has not recollection of time passing except for the few moments she is awake. There are no "2 days running" for her; only one day at a time, each day resetting the coin flip probability.
this is just the existential crisis i needed at 5pm on a saturday. thanks so much, vertiasium 🤣
There is now a 50/25/25 chance of you falling asleep :V
I can put your mind at ease....Just think about the coin landing *ON EDGE*
Don't worry, I got the same one later.
I see a major difference in the Monty Hall problem and this example: in MH, the contestant sees everything throughout the process, and indeed, the probability goes fro 33% to 67%. In Sleeping Beauty, there is only a single flip of the coin for each trial with a 50% probability of coming up heads. If it comes up heads, she will be asked the question once, if tails, twice. Because she knows how the experiment is set up and there was a single coin flip, she knows the probability of it being heads is 50%. That is the answer she will give *every* time because the whole set up is based on *one* coin flip. The experimenter is doing something different. They are keeping a tally of what she says each time, so obviously , if the coin flip comes up tails, she will give that same answer on two occasions, rather than one. The fact that two tally marks were recorded doesn't change the nature of the flip itself, which is what the question is about. Recording tally marks isn't gathering anything useful because her answer will always be the same. In Derek's coin flipping example, he was recording the results of multiple flips, but tallying the tails results twice. That is artificially doubling the tails tally and has nothing to do with the probability of the outcome of the coin flip nor to the answer SB will give. NOW, if the question was changed to "what day is it today" then the answer will be different because of three possible outcomes (waking up on a given day) instead of two (results of a coin flip).
100% agree with this. The amount of times she wakes up is not related to the original coin flip and will not change the probability.
50/50 seems obvious to me. Which makes me think I'm not understanding something. But your description fits. The question now is... why are people picking 1/3rd?
Thank you! Totally agree and you saved me some time typing this exact message
That's because they ARE two different problems. I don't think he meant that they were the same. If he did, he is wrong because MH has an actual, non-ambiguous answer whereas this [currently] does not.
Edit: you're also misunderstanding the question. He's not recording each tails flip twice more than he should; he's doing that because if it IS tails she will wake up twice but will have no way of knowing which day it is, or that she has woken up before or will wake up again (to put that another way: she doesn't know if it was heads or tails). They are recording two tally marks because she woke up twice: monday AND tuesday. So because she has no memory of the first time (if it is tuesday) and has no way of knowing there will be a second time (if it is monday), and 2/3 scenarios are tails scenarios, 1/3 is a totally valid answer. She doesn't know if it's Monday or Tuesday, but she _can_ know that if it WAS tails, she will wake up twice while heads she will wake up once. That very well may affect the answer.. we just don't know at this juncture in time.
"what day is today?" is actually not an interesting question at all, and you can find the answer quite easily: given that she is being asked what day it is (and she is never asked on tuesday if it was heads), then it's 2/3 likely to be Monday, and 1/3 likely to be Tuesday. Idk what you're missing, but if you're analogizing this to the Monty Hall problem then that might be doing more harm than good!
yeah, I agreed with this etirely, if heads you pour 1 liter in a tub, and tails you pour two in another tub, the second tub will be twce as full about,, but not twice as likely to be chosen, it's still fifty fifty. I thought out of all people, veritasum would be a halfer, and I'm really perplexed that it''s this big of an imbalance between the agreeers and the disagreeers.
The dilemma becomes much easier if you realize that "what are the odds that something WILL happen" and "what are the odds that something DID happen" are two completely different concepts. One is prediction, one is deduction.
And really, "the odds that something happened in the past" isn't all that meaningful. It already happened, or not. There is no probability associated with it.
The question to Sleeping Beauty could be phrased as odds, if you can relate it to the future: "Given all that you know right now, what are the odds that you will be correct if you conclude that the coin landed on heads?" This is a much easier question to digest, the answer is based 100% upon deduction, and zero percent upon the toss of a coin.
There is quite an attractive quantum rabbit-hole here, but it is Super Bowl Sunday, and I have no time! :-)
Great comment
I think it depends on what exactly is at stake. In the Brazil example, if you have to bet $1 on who the winner is when you wake up, you take Canada because there's a chance you'll get woken up 30x in a row and lose a dollar each time. On the other hand, if your life was at stake for getting the answer wrong, you take Brazil.
@@billybob-uz6wz that only works because you can only lose your life once. If instead it was shortened (or prolonged) by a year each time you should still answer Canada.
@@AbsoluteHuman Indeed. More broadly it depends on the consequence of being right/wrong and if it compounds. In your scenario if Brazil loses and you pick Canada, you lose 1 year of your life and that's the worst case scenario for picking Canada--but 20% of the time you gain 30 years. If you pick Brazil the best you can get is 1 more year, but you lose 30 years 20% of the time. Basically, without knowing the parameters and end goal, there's no real solution here.
This! This is the answer. By casting the question in terms of "probability" the questioner is trying to confuse the discussion. There are no probabilities about the past. Asking someone something they don't know is similarly wrong-headed. They can use deduction to maximize the payout on a bet they make with you over the question, but that's not knowing AND not probability.
There's no relevance to additional instances because the multiple experiences exists within a 50% probability across the possibility of heads or either tails. You're being given irrelevant information, then taking that information and treating it as a factor in how a coin has been flipped. For example, on the tails side, you would be woken up ten thousand times and you're currently the seven hundredth time to answer the question, your response will always be that the coin was only ever flipped once.
The part where you said that if you choose 1/2 you are more often correct about the outcome of the coin tosses, but if you choose 1/3 you answer the question correctly more often made a lot of sense at first, but then I realized it's not the same question at all. The football/soccer question is asking which outcome actually happened, but the Sleeping Beauty question is asking what the likelihood is of a specific outcome happening. Sleeping Beauty may be woken up twice when the coin landed tails, but both times it was still a result of the same initial coin toss, which had a 1/2 probability of landing on either side. If she had been asked whether or not it did land on heads, then the football/soccer analogy would apply.
The thing is that in the soccer question even if Canada won and it's Tuesday you still don't remember previously being awakened nor do you know what day it is. So you should (and probably would) still guess Brazil won.
Exactly, the question didn't ask what the probability of the outcome was for that particular day, it just asked what the probability was if it landing on heads or tails.
Exactly! That bothered me during the whole video. Why would she ever be right answering 1/3, even if the coin happened to show tails?
It's a matter of perspective:
The odds of her flipping heads is 1/2.. but the odds of her waking up on Monday is 1/3. But the result of waking up on Monday can only happen if she flips heads. So you must assume the probability of the consequences rather than just the result of the coin toss. Therefor it is 1/3 and there is no debate about it; because she is waking up more often than the coin is tossed, so that changes the likelihood of the standard 50/50 chance. The fact that waking her up is a qualifier, completely changes the standard parameters of a 50/50 coin toss.
It's easier to think of it as Monday is heads/ Monday is tails and Tuesday is NO TOSS. So every time she is woken up, she has to think, is it a heads or tails Monday, or is it no toss Tuesday. So she has 3 results to consider. The answer is 1/3, because there are 3 possibilities.
But what is very interesting: From the perspective of the "coin toss man"-- he gets to go directly from toss to toss, so even though tails has multiple consequences, it does not effect his odds one way or another. So from the perspective of the coin toss man, it is 1/2 chance. But that is not the question. The question is from HER perspective of having to wake up and answer, when she has to wake up more times than the coin is tossed; so from her perspective, it's 1/3 chance, because of the "waking up" prerequisite.
@@Sirithil The comparison with the soccer game was extremely misleading in my opinion, because it was a completely different question. The initial question was about the _probability,_ but for the soccer match he asked about the _outcome._
I feel like no matter how many times they wake you up, it still was 50/50 whether you're gonna sleep one day or million
Correct.
It is a 50/50 chance on sleeping 1 day or a million.
However, each time she wakes up its a 1/1000000 chance of being tails
@@joshpollnitz1618 Each time she wakes up it's 50/50 chance she either exists in the heads sequence or the tails sequence.
The sequence is one complete package. It doesn't matter how long the sequence is, it's still one package. The reason being that the conditional statement is the fair coin which only has two outcomes at 50% chance each.
You are confusing the probability of the coin coming up heads _before_ it was flipped with the probability that it _did_ comes up heads, after it was flipped and the observer has some amount of information pertaining to what the outcome was. These are two very different things. The answer to the first is always 50%, but the answer to the second is not necessarily 50%, depending on what information the observer has observed. This is the fundamental principle behind Bayesian probability.
@@therainman7777 Sigh.
Leftism is a religion upon which the idea that an individual's perceptions can dictate reality is the central object of divine worship, thus the solipsistic imposition of the self's mind onto the world makes the Leftist a god in his own eyes.
Pay attention to me. There is no secondary set of probabilities. There exists no such thing as metaphysical probability. That's a perceptual lie that the illogical parts of the human brain concoct in order to alleviate the pain and stress caused by the sheer gravity of reality pressing on you from all sides forcing you to accept that you cannot escape objective truth.
There is no such thing as "probability after the fact" nor is there any such thing as a "choose your own adventure" kind of probability. Probability is determined by material scientific forces that can be measured and predicted to a certain level of accuracy. In this case though we have from the outset a theoretical "perfect" object in the form of the fair coin. We know from the get go what the probabilities are because those probabilities are deterministic and they are determined by a physical object whose attributes are already established -- the coin is an ideal one.
According to the parameters set by the use of such an ideal instrument we are bound to make the conclusion IMPOSED by the parameters set by the ideal instrument. You cannot escape the physical imposition of physical laws even when those laws are purely theoretical because those laws still follow the rules of logic. The coin is set to have a 50/50 chance to land either heads or tails therefore those are THE ONLY outcomes you can measure in the experiment. Nothing else besides those two outcomes exist.
Period. End of story. --
Having notifs on made me saw 0 agrees 0 disagrees and I thought this was gonna be a video about how you can never have everyone agree on anything XD
I know, I was confused too
Same
6:10 - relates to my argument why the Universe exists at all: the probability of there being something is infinitely higher than there being nothing because there's infinite variations of somethingness as opposed to only one possible state of nothigness. 😁
That's a tricky one, really twisted my brain at first, but after a while I came to conclusion there is no paradox :) The core of the problem is that we compare two completely different values - on one hand we have a toin coss probability which is we know to be 50% for both heads and tails, but the second measurement is basically "how many times the princess was right" and it actually has nothing to do with coin toss probability, it only depends on how many times she was asked... So if we ask her (aka wake her up) 1 time for heads and 1 time for tails, it would be 50% guess for right answer, and same holds for any amount of questions as long as its the same for heads and tails. But in that ptoblem we ask her 2 times more questions for tails than for heads, so she can safely say "its tails" every time and she will be right 66% of times and it has nothing to do with toin coss chance which still remains at 50%.
Now if we take that example form one of the articles of asking her a million times for tails - everything will still hold up, the heads/tails is 50% as always but she can always say "tails" and be right 99.999% chance because of the amount of questions she gets when its tails.
So, yeah, don't thank me thouthands of matematicians all over the world, just remember this post when next Noble prize in math is being issued!
Thank you!
thanks for the simpler explanation, I didn't get it by watching the video the first time
No. That's not the question she is asked. They ask her what she believes the probability is. They do not ask her which side the coin landed on. Her answer is "1/2" or "1/3", not "heads" or "tails".
The video says this exact same thing at 8:02
What makes you think this comment, let alone "solving" this math problem, will win you a Nobel?
The difficult thing with this problem is defining the question you want to answer.
If Beaty has to make a guess for heads or tails when she awakes and she always guesses heads, what is the chance she is right?
Is it about how many times she answers right(1/3 of the time) or about how many weeks she is right(1/2 of the weeks)?
I think you've found the cause of the controversy spot-on.
The question is spelled out in the video.
"What do you believe is the probability that the coin came up heads?"
Yes once you realize this the video becomes boring
Will she answer right 1/3 of the time though? Not when it turned out to be heads! That's the problem. It is N=1. In that case the so called 2/3 of potential actually didn't turn out to be 2/3
@@jogadorjnc That's not spelling it out, that's intentionally obfuscating it. Probability of correctly guessing heads is _never_ less than 1/2; only the probability of getting _more right answers_ by guessing heads declines.
Until _that_ question is what's actually asked, this is just another cheap semantic deception.
What would I say? I would say “the question ambiguous, please clarify”
No the question is clear, after being woken up what are the odds that the coin flip landed on heads/tails. The answer's 50/50.
@@a-walpatches6460 The answer becomes 1/3 assuming you know that you get woken up twice a week on tails, and only once a week on heads. I think that's where the lack of clarity comes into the mix. If you know that on heads you only get woken up on mondays, that means that if you guessed heads 3 times in a row you would likely be wrong 2 out of 3 times, because you get woken up twice as often on tails.
Or no, I think you are right after all. I think my maths is right, but it answers a different question from the one being asked. "The probability that the coin came up heads" is simply 1 in 2, and will always be. The other question might be phrased as "What do you think was the result of the coin toss", or "Based on your limitd information, do you find it more likely that heads or tails was rolled?"
@@holysecret2 Same answer for those questions, 50/50. It doesn't matter how many times you get woken up, each time you're woken up is the same from your perspective, 50/50.
@@a-walpatches6460 Nope 50/50 only makes sense if you're asked what is the probability the coin *will* be heads or tails, but that is the wrong question since the event already happened, you now live in the timeline created by the outcome and you're more likely to be awake in that timeline. So she's really answering what are the chances it *was* heads or tails and information about that outcome has leaked via the fact she's more likely to be in the second timeline by virtue of being awake.
@@neodonkey Nope, the odds of being in either timeline remain 50%. This isn't like the Monty Hall problem.