It seems like the thirders seem to think that the question is not "What are the chances the coin came up heads?" but rather "What are the chances that I'm asking you this question under a scenario where the coin came up heads?" Because the question "What is the probability that the coin came up heads?" has nothing to do with you being woken up.
If the question is not "what are the chances that I'm asking you this question under a scenario where the coin came up heads" then what on earth is probability about??
It is asking a conditional probability: heads given you just woke up. Why should that be the same as the unconditional probability, which indeed would be 50%? Like, if I ask you "what are the chances that it is your birthday today" precisely when you are blowing out candles on a cake, the answer is no longer about 1 in 365.
@@landsgevaerAfter thinking on this more, my take on this problem is that the answer to the paradox is a reflection of how we _choose to define_ statistics, not a reflection of _the universe_. Statistics is a human invention to make predictions; you can make accurate predictions if you say this is conditional probability, you can make accurate predictions if you use a form of statistics that says the underlying probability is 2/3rds (think, Newtonian vs lagrangian mechanics). The paradox is a trap, because it asks "what are the odds of the coin flip?" but sleeping beauty's answer has no effect on the universe. If she said, "150%," or "hippopotamus," then who cares? So, maybe let's say that she's taking a test, and she only passes if she gets the answer right. What should she say to pass the test? Well, this is the same paradox, because how does the witch grade her answers? Let's presume the witch "just knows." How can you tell apart this paradox from one where the witch is actually wrong? We are in fact unable to construct a puzzle where the result of the question objectively matters. It's asking a nonsense question -- or at least, nonsense without providing a definition of statistics as a premise, and that definition could simply be anything. The correct answer *could* then be "hippopotamus." Moreover, by attaching a definition of statistics such that "the odds of a coin flip is always hippopotamus," the paradox is no longer interesting, or a paradox. If we want the paradox to apply to the real world, rather than a particular definition of statistics, we need an objective consequence. Something like, snow white can bet $5 on the outcome of the coin flip. In this scenario, snow white should bet it is tails. Neither interpretation of statistics disagrees with this. This version of the paradox is not interesting, and once again, not even a paradox. So in summary -- thank you for answering my question and providing a definition/interpretation of statistics that can be used to answer the puzzle with an answer of 1/2. I'm not a statistician so I didn't think of it in those terms, but thinking of it in those terms make sense. That's probably the "technically correct" answer. Some people probably find the "technically correct" answer illuminating (like stats students who have to learn "technically correct" language in order to properly converse with other statisticians), and I personally find the reduction of the paradox itself is also interesting, perhaps moreso.
@@MichaelFairhurst Thanks. There are (at least) two ways of thinking about what a probability means: frequentists look at it as a fraction, how often would it occur if the experiment were repeated; bayesians look at it as confidence, how certain am I that an outcome will occur. But leave aside that diversion. In either case, here it makes a difference from whose perspective you look at it: are you looking for the fraction of heads out of all the awakenings that sleeping beauty experiences, or out of all the times the coin was tossed by the witch (phrased in frequentist terms); or, how confident sleeping beauty can be that an awakening is heads, or how confident the witch can be that a toss is heads (in bayesian terms). They do not have to agree. In that sense I agree with you that there is no contradiction. Sleeping beauty sees heads 1/3 of the time she wakes up; the witch sees heads 1/2 of the time she tosses a coin. The question "what is THE probability of heads" is ill-defined. It is conditional on what 'universe' of events you calculate it relative to: all the tosses or all the awakenings. It depends on the measure you assign to events. And that we indeed "choose". Since you like physics here is an analogy: what is the probability of Schrödingers cat to be alive? From the researcher's perspective who opens the box it is 50%. From the cat's perspective who is in the box, it is 100%. :-) For the researcher, dead and alive have equal measure; for the cat, dead has measure zero because it will not be there to ever observe that. It can both reliably estimate that it is alive and be confident about that. (Disclaimer: I don't believe in heaven.)
If you program computers, such subtle distinctions of meaning can never be escaped. People often ask "why are programming languages not more intuitive and similar to our daily lives?" Well, this is why, because they cannot leave open room for interpretation.
Think more the Hardest way as how Graphic Interchange Format uses 256 colours Vs Think more the Hardest way as how it is not correct when an Anchor tag or tag in HTML is used to define tags in a webpage.
It’s this exactly. They need to precisely quantify what exactly needs be optimized. Speaking imprecisely, the argument about the problem seems to hinge around an implication that ascribing higher probability to the amnesiac run of the experiment when it’s true, so you’re right twice (or a million times) is worth more, or is in some sense “more correct,” than being right once in the non-amnesiac run and ascribing higher probability to that case. Weirdly this problem still gets a lot of traction along with things like the two envelopes problem (en.wikipedia.org/wiki/Two_envelopes_problem). It’s a problem of human intuition not exactly fitting with how one may formalize what a “problem” is.
@@trucid2 No it's not because there is no possible way of knowing the answer to that question, whereas this question there are only 2 known possible outcomes.
She's twice as likely to be woken up on the tails flip, but only because she is woken up twice on it. This doesn't change the fact that the coin has a 50/50 chance of landing H/T. The odds of her being awoken twice are equal to the odds of her being awoken once. Both 2/3 and 1/2 are correct, but to two different questions. Just because you record data twice on tails, doesn't mean the odds are higher it will land on tails - just that you're more likely to be recording data during tails flips. Can we all agree on that?
skillz -- "She's twice as likely to be woken up on the tails flip" That isn't quite true: the probability of 2 wakings on a tails flip is 1. A tails will always result in 2 wakings, likewise a head always results in 1 waking. As we know, half of all flips will be heads, but the real question isn't what are the odds of a heads, it's what proportion of wakings are caused by heads. She will wake once for heads and twice for tails. And since 2/3 of the wakings will be from tails and 1/3 from heads, she would be right to bet on tails. We could break it down For a tails - Monday's guess tails: right. Tuesday's guess tails: right. For a heads - Monday's guess is tails: incorrect. That's 2 right, 1 wrong for always guessing tails. For a tails, Monday guess heads: wrong; Tuesday guess heads: wrong. For heads, Monday guess heads: right. 1 right, 2 wrong when always guessing heads. In order for the odds to be 50/50 the two strategies would need to have the same rate of success. Recall that the question is not what are the odds of a heads, but what are the odds that upon waking that the toss was heads -- which on a functional level means what are the odds that you are being wakened following a heads?
skillzorz101 I am a Physicist and thus statistics is really my daily bread ;-) skillzorz101, You are exactly right in your interpretation. Both answers: 1/2 and 1/3 are correct, they're just answers to two completely different questions. The first question is about the odds of a coin flip. However the second question is about the probability of which data point are we looking at (from a single element set labeled "H" or from a multi element set labeled "T").
Where in the setup does she say that if Sleeping Beauty is going to be kept in bed till Tuesday, she's going to be asked to assess the probability of heads being the outcome of the coin toss on both Monday and Tuesday?
The question Sleeping Beauty is asked is "What do you think is the probability the coin came up heads?" Is the question ambiguous? I don't know, but I do know that some of the disagreement arises from people taking the question to be asking different things. Here are some possible rephrasings of the question that have different answers: Q: What is the probability that a flipped coin will come up heads? Q: If I flip a coin and ask one person to guess the outcome if it comes up heads and two people to guess the outcome if it comes up tails, which answer will result in more correct guesses? Q: Is Sleeping Beauty equally likely to be woken up (and asked the question) when the coin flip is heads as when it is tails? Q: If you run the experiment 100 times, and the coin lands heads 50 times and tails 50 times, how many of the 150 times Sleeping Beauty is asked the question will the coin have landed heads? Q: If you run the experiment 100 times, and the coin lands heads 50 times and tails 50 times, in how many of 100 trials will heads be the correct answer?
Yes, indeed. In fact, if she was allowed to make a fair bet (ie $1 stake, wins $2 if correct) every time she was woken up, she would clearly bet tails every time and no one would doubt her reasoning. To me, this indicates the real problem is with some ambiguity or equivocation in the phrasing of the question. I think I will post this in it own right too :-)
Think more the quickest intuitively way as R is for *Reception classes* again Vs Think more the quickest intuitively way as R is for *Regularity* again
Beaut is not as dumb as mathos and philosophers. She can tell from her leg stubble whether she's been asleep for 1 or 2 days. Drugging women for 'probability research', yeah, right. She's getting a lawyer.
The interesting thing is that even if she was able to tell whether it had been 2 days or not, her overall probablity of answering the question right does not change. If its Monday (2/3 of the questions) then she has a 50-50 shot of guessing the flip (so 2/3 * 1/2 = 1/3). If its Tuesday (1/3 of the questions) then she knows its tails (1/3 * 1 = 1/3). So her overall probability of getting it right is 2/3, exactly the same as if she just said tails every time.
The sleeping beauty problem is ambiguous because it does not say what sample space she is using. Probabilities are defined on a per sample space basis. The sample space of the coin toss is {H,T} and the sample space for the questions about the coin state is {MH,MT,UT} where H=heads, T=tails, M=Mondays and U=Tuesdays. The probability of heads for the first sample space is 1/2 and the probability of heads for the second sample space is 1/3, since they are both equiprobable sample spaces. To see equiprobability, just notice that out of every 1000 coin tosses about 500 will be heads, 500 will be tails, and about 1500 questions will be asked about 500 which will occur when it is Monday and heads, another 500 which will occur when it is Monday and tails, and the remaining 500 which will occur when it is Tuesday and tails. She should use the probability for the sample space she assumed and the problem doesn't tell what sample space that is. The problem is bad because it introduces two different sample spaces without clarifying which one is operative. For example, if the problem also stated that for betting purposes on repeated trials of the experiment she should bet as much money as possible then it would be clear that she should use the sample space for the questions about the coin state to get the probability. But if instead of that we added to the original problem that she give the probability for repeated tosses of the coin then money won or lost is irrelevant and she should use the sample space for the coin toss to get the probability. The sleeping beauty question is ambiguous because it is asking about belief in the frequency of the truth value of occurrences of the PROPOSITOIN "the coin landed heads" not the proposition that the coin's probability of landing heads is 1/2. That is, the question doesn't make clear if it is asking about the probability of the proposition being true during repeated coin tosses or if it is asking about the probability of the proposition being true during repeated questioning in many repetitions of the experiment. These are not the same thing because when the coin is tails she is questioned twice but when the coin is heads she is questioned only once. Now she knows the proposition is true one out of every three times she is asked and she is not going to mistake that for the fact that the coin comes up heads one out of every two times during coin tossing. So, for the proposition "the coin landed heads" the frequency of this proposition being true during repeated questioning in many repetitions of the experiment is different than the frequency of it being true during repeated coin tosses. If the coin toss actually came up heads then the proposition "the coin landed heads" is true but if the coin toss actually came up tails then the proposition "the coin landed heads" is false. How often the proposition is true or not depends on the circumstances. So adding two different prepositional phrases onto the original question highlights the ambiguity of that question: (case 1) What is your belief now for the proposition that "the coin landed heads" in the case of repeated questioning in repetitions of the experiment? (case 2) What is your belief now for the proposition that "the coin landed heads" in the case of repeated tosses of the coin? The conclusion: the sleeping beauty problem is ambiguous because case 1 and case 2 use different sample spaces and if one removes the phrase "questioning in repetitions of the experiment" from case 1 and removes the phrase "tosses of the coin" from case 2 then the ambiguity of the original question is exposed.
Louis Wilbur Spot on. It is depressing that so few people seem to understand this. The question is simply not well defined and people that give an answer (1/2 or 1/3) make additional implicit assumptions.
I wouldn't say it's ambiguous as it was presented in this video, because it was worded without reference to sample space and with reference to one variable only: the coin toss. The day of the week is not part of the question, so why would it ever be part of the answer?
@@nialltracey2599 I'm going to use your comment as an excuse to answer a 5 year old thread ;) I am really unsure why folks get so tossed about this, but I assume it's because of the dreadful way this problem is often presented. Louis Wilbur is obviously correct. A succinct way to explain it would be thus: a) When the coin is tossed, what are the chances it comes up heads? (50%) b) Of all the possible times Beauty is asked the question (3 in total: 1 in the heads scenario, 2 in the tails scenario), how many times will the correct answer be heads? (33,33%) Those two statements are both true AND NOT mutually exclusive.
Answer is obviously 1/3. Simple bayes. The sleeping beauty learned she doesn’t know which day of the week it is. Given the new information the answer is 1/3.
Think more the quickest intuitively way as Q is for *Quadratic equations* again Vs Think more the quickest intuitively way as Q is for *Quotient* again
I can see that a common response is, "Philosophers don't understand math, it's clearly 1/2." Sigh. Perhaps I should have clarified that *mathematicians* are divided on this puzzle, too, not just philosophers. For example, here are several mathematicians and physicists arguing the 1/3 position: 1. Nick Wedd is an International Math Olympiad winner, and a thirder: www.maproom.co.uk/sb.html#arg5 2. Jeffrey Rosenthal, winner of the "Nobel prize of statistics", is a thirder: probability.ca/jeff/ftpdir/beauty.ps 3. Another International Math Olympiad winner, Tanya Khovanova, is a thirder: blog.tanyakhovanova.com/2011/08/the-sleeping-beauty-problem/ 4. Physicist Sean Carroll is a thirder (but agrees it's controversial): www.preposterousuniverse.com/blog/2014/07/28/quantum-sleeping-beauty-and-the-multiverse/
Julia Galef Imagine you are the car in the Monty hall problem, and you have to guess which of the three doors you were placed behind... After the contestant makes their first guess and one of the doors is eliminated, you 'forget' which door he picked, all you know is which two doors are left and you are in one of them. Due to lack of information, you are facing a 50/50 choice. Meanwhile from the point of view of contestant the door he picked still has a 1/3 chance of being the right one, while the other remaining door now has a 2/3 chance. They are both right, depending on the frame of reference and what information you have available. As the sleeping beauty waking up, you would be facing a 1/3 chance of heads. As an observer, the chance of a fair coin flip is always 50/50. Basically the problem is poorly defined :P
Patternicity But the difference with the Monty Hall problem is that the circumstances are changed BEFORE the final choice is made, therefore altering the probability. With Beauty, everything happens AFTER the toss is made. She knows there will be different results depending on whether the coin falls heads or tails, but when she wakes, she has no knowledge of what the result was. Therfore, like the observer, the only reasonable answer she can give is 1/2.
Rob Smith She does have knowledge ABOUT the result, she knows that she is in one of three possible situations, 2/3 of which happen after the result of the coin flip is TAILS.
Patternicity By that logic, if the researcher gives her the amnesia drug for a thousand days, then the 50/50 coin flip is going to always end up tails. Which is flat out silly.
Thinking about this problem, I have a different perspective that clarifies the question for me: Flip a coin. If heads, take a picture of a dog. If tails, take two pictures of a cat. Question 1: what is the probability that the coin flipped heads? 1/2 Question 2: choose a picture at random. What is the probability it's a picture of a dog? 1/3
Choose a picture at random from what? Do you have 3 pictures? I'd say not. You have either 1 or 2, in any case you have only two options, either pick the 1 dog or pick 1 of the two cats.. So 1/2 is the prob
A closer analogy would be thus: The experimenter flips a fair coin, and takes a picture of a dog if it's heads or two pictures of a cat if it's tails. You, as a test subject, are about to be shown a picture. What is the probability that the coin toss that determined your picture was a heads?
Think more the wisest way as *Emphasis & Emotion* of *Communication Dynamics* Vs Think more the wisest way as *Who we are and what emphasis we put on our spoken thoughts, together from a critical part of our communication with others*
The confusion can be resolved as follows: It is a question of which timeline she is in, not which day. She was asked essentially what is the probability that you are in a timeline where heads came up. The probability of entering either timeline is 1/2. Once in a given timeline, the probability does not change. What changes is the number of times she will be asked the question. She will be asked the question more times in one timeline. That does not change the correct answer, it only changes how many times she will answer. She will say the correct answer more times if tails came up. But it is the same answer. If she gets a dollar each time she gets it right, she gets more money if tails came up. But the answer is the same. You can safely remove the Monday and Tuesday which changes nothing. If she were asked repeatedly on the same day with a dose of amnesia in between each time, it would boil down to: what are the odds that you will have been asked this question just once vs repeatedly? Answer: 1/2. If the question was: what are the odds that this is the only time you will have been asked this question, the answer would be 1/3. But that would not be analogous to the question in the premise.
" She will say the correct answer more times if tails came up. " So if you ask me what 2 + 2 is I'll say the correct answer more often if I say 4 rather than 5. We could talk about that in terms of "timelines" if we want but 4 is still the right answer.
@@sinistar99 that doesn’t really have anything to do with this problem, now does it? You may as well chalk this problem down to: if it’s tails and you guess tails, then you get a million points but if it’s heads and you guess heads then you only get one point. There are now two questions: 1) how do you play to maximise your points? 2) what’s the probability of getting heads?
Still 1 in 2 Ok. Let get to the point. I am the best at explaining things. It is easier to modify a problem to make you say arhhh i get it. First, let solve this problem. A god is bored and want to play a game. Out of 7 billion human in this world. He will choose only one human. This one chosen human will be awaken again and again 7 billion times. The rest of human will fall asleep only once. The god then telepatically explain the rule to all human on earth. And make all 7 billion people go to sleep alone in their personal space.. Now you woke up. What is the chance that you are this special human and this is just one of your 7 billion wake up circle? Is it 50% because the chance of falling in special 7 billion sleep is equally likely as the other normal sleep????!! ? I think now u can figure this sleeping beauty problem on your own. A few more point i want to add. Knowing more information does not always change probability. When confused, modify, exagerate the problem. It often help. Do not be misled. Be careful what has equal chance. What is not. All events do not need to always have equal chance. Such as wakeup event in this case.
Thank you finally someone with the correct answer for the correct reason. Just as I was about to go insane. Even if you get woken up a million times within the same timeline all of these times are still one entity. Once you enter the timeline all of the million days get activated at once like a button that lights up all of them in synchronicity. This doesn't change the probability which remains 50% even retrospectively. Seeing it like this I can't imagine how someone would count all waking events disconnected from each other.
The answer i believe is 1/2 whether you look at it from an outsider point of view or from beauty's perspective. Am going to change the experiment a bit just to make easier. Say that if the coin came up head, she would wake up on Sunday. Everything else holds the same. Thus, the question he asks her "what is the probability that the coin came up head? " is the same as "what is the probability that today is Sunday?". This change should not effect the probability at all. From beauty's perspective, she doesn't know whether she woke on Sunday from the Head scenario, Monday from tails scenario or Tuesday from tails scenario. What she knows is the probabilities of each one. She knows that since the coin is fair, there is a 50% chance that she woke up on Sunday. Also, that there is a 50% chance she woke up on Monday or Tuesday. Therefore, assuming probability of waking up on Monday is the same as waking up on Tuesday, then the probability she woke up on Monday is 25% and probability she woke up on Tuesday is 25٪ too. Thus, her answer is 50% either way. The mistake i think 1/3 advocaters is that they think the probability of her waking up on Sunday, monday and Tuesday is all the same.
The odds of it being Tuesday aren't the same as the odds of it being Monday. When she is asked, she can deduce that the probability of it being Monday with heads is twice as likely as the probability of it being either Monday with tails or Tuesday (with tails). So the correct answer is always 1/2. If you were asking her whether it was heads or tails, the answer would be tails 2/3rds of the time, but that's not the same as the question she is being asked.
The question is framed to make it misleading. The odds the toss came up heads is 1/2. The odds that this is your first waking are 2/3. The odds that this is Tuesday are 1/3.
I don't think this is about being misleading. I think this is about how one scenario of probability relates to the other. They've linked the results of a coin to a world that only has 1/3rd of a chance for you to be in when asked the question. So what happened to the probability of the coin? If we measure, it has objectively changed because of the new variables. If you take away the variables when answering it goes back to 1/2 but if that's really what they ask it would quite frankly be a pointless thing to demonstrate with a hypothetical scenario.
Wouldn't the odds of it being your first waking actually be 75% ? 50% = Heads > First Waking & 50% = Tails > 50/50 Chance of being the First Waking ~ 25% So 75% Chance of it being my First Waking
Actually this is a misleading question, in that the author is trying to illicit an answer to a different question than the one they asked. Specifically: 1. What is the probability that the coin was heads? Answer: 1/2 2. What is the probability that this is the second time you woke up since you last remember going to sleep? Answer: 2/3 Two completely different questions, but they are trying to illicit a response about the second and apply it to the first. "A" causes "B", therefore "B" causes "A" is a logical fallacy. A coin flip determined her sleeping, but her sleeping cannot influence the coin flip. No, probability is not influenced by measurement. It is not statistically correct to say that if I flip a fair coin 4 times and it comes up heads 3 times that therefore there is a 75% probability of heads. A finite sample size NEVER shows the actual probability of something occurring, because all it would take is one more data point to change the probability (which is ALWAYS the case). Probability is stated not as a finite dataset, but rather the limit x-->infinity of the probability function. If you flip a fair coin an infinite amount of times, the results would be 50/50, and anything less would not be a true representation of the dataset. The coin flip already occurred, and it has a 1/2 chance of either result. What day of the week is immaterial to the original probability of an event that has already occurred and could not have been influence based upon her perception. She knows nothing about the outcome of the coin flip. She only knows that she woke up, and that there is a 2/3 chance it is the first time since she last remembered going to sleep. None of that has any bearing on the probability of an outcome from the coin flip. To take it to an absurd level, say I flip a coin. If it was heads, 15 cats would be ran over and if tails 10 cats would be ran over. What is the probability that today is Tuesday? 1/4, because pancakes are purple when aliens wear tutus in the rain. Although I do agree, it is quite frankly a pointless thing to demonstrate with a hypothetical scenario.
The question is not "what is the probability of a coin being tossed to be heads" but "what is the probability the coin came up heads", which is different in the sense that she will be asked this question twice when it was tails but only once on heads. This obviously has to be factored in, otherwise you just disregard half of the scenario
Misleading is a stretch since you're presuming intent there. It is certainly ambiguous. Different readings may lead to different probability spaces (one that counts in units of "experiments" and one that counts "waking up events") which lead to the answers 1/2 or 1/3.
Has anyone noticed the way Julia subtly changes the question when she's explaining the 1/3 idea. She says something like "if the experiment is repeated many times and I find myself answering heads, I'll only be right only one third of the time" but the question wasn't "did it fall heads or tails", it was "what was the proability that it fell heads". If she answers 1/2, she'd be answering right 100% of the time (or 0% of the time if you think the answer's 1/3).
The question is “what is the probability that it DID fall heads”, given that this version of you is being asked the question. Her very existence in different states is related to the probabilities, so of course if you neglect that information the coin returns to being 50/50. If I throw a glass of water over you every time a coin is heads, and I throw it 1/100 times if the coin is tails, then if you saw me flip a fair hidden coin and then throw water at you, the chance of that FAIR coin having landed on heads is clearly higher, despite the coin itself having 50/50 odds. (You having water thrown on you here is analogous to some self-ignorant state of Alice being awoken, not knowing the day or what is about to happen to her. If I where not to throw water over you, then this is the equivalent of that version of Alice never existing, meaning you would know for a fact it was tails)
I think if we change the nature of the problem but keep the game principle the same we have an more intuitive answer: Imagine someone tosses a fair coin and asks you to determine the probability of whether it’s heads. They tell you have not tampered with it and will show you the actual result, and even toss the coin in front of you. You would naturally say 1/2 . They show you the result (heads or tails) and write it down as the last result on their log. [This is effectively what happens when sleeping beauty wakes up] Afterwards they show you the log for every flip you made in the past. Secretly they have removed half of the head results, but never the final result, So over time the spread shows 1/3 of heads now. You confront them and they admit it saying it all part of an experiment! [This is the same as being woken up twice for tails and once for heads] They toss the same coin and ask you to predict heads. As before they will show you the result and put that results as the last result in their log* . [You now know the rules of the game] You answer would remain as 1/2. The coin is a fair coin, you can see the toss and the result. If a betting person is betting on the log results with a bookmaker, then he/she would always bet on tails (as some of the historical heads are removed to keep the balance on 2/3 tails). Love to have other people's feedback. Ketan *Before the coin toss they will remove any excessive historical heads so that the split is always 1/3 heads 2/3 tails but only after they show you the last result. * PS even if the game is only played
Two different questions. What's the probability that a coin will come up heads? P = 1/2 What's the probability that Beauty will wake up on a heads up flip? P = 1/3
Infinity 1 Incorrect. Given that there is a coin toss, she has a 1/2 probability of being woken up on heads, and 1/2 on tails. But since there are two wake-up possibilities on tails, or heads, depending on which way round the story has it, there is a 1/4 chance of her having woken up on either day.
Holy Moly No, if it's heads she will only be interviewed on Monday. If it's tails she's interviewed on both Monday and Tuesday. There is only 3 possibilities she could be in: 1) It's Monday and it's heads. 2) It's Monday and it's tails. 3) It's Tuesday and it's tails.
Infinity 1, That was not the question though. The question was "What is the probability that the coin was heads?" You're answering the wrong thing. You just answered P(this waking event | H) but the question was P(H | this waking event) and these are not interchangeable. You're right that yours is 1/3, but the answer to the actual question asked (the second one, which is not your question) is 1/2, because by Bayes Theorem, it's P(H) * P(this waking event | H) / P(this waking event) which = (0.5 * 0.33(your portion)) / 0.33 = *0.5*
Not if she does so every night. But in such case she can simply count the movements. In the million iteration scenario, this would take quite a while and be error prone.
I think, Beauty knows there are three possible scenarios: * Heads, Monday - probability 1/2=.0.5 * Tails, Monday - probability 1/2*1/2=0.25 * Tails, Tuesday - probability 1/2*1/2=0.25 Why 1/2*1/2? Because the probability of a tails flip is 1/2, and the probability that she's woken on Monday is _also_ 1/2. This seems simple to me, so there's probably something wrong with it. What's wrong?
Max X _"Each situation will occur 5 times:"_ Yes, I see the reasoning. [Heads, Monday], [Tails, Monday] and [Tails, Tuesday] are all equally likely, assuming Beauty has no information other than the test procedure. But the question the researchers ask her isn't "What coin side/day combination you do think it is?", but "Did the coin flip heads or tails?". So the probabilities of what day it is are irrelevant. It comes down to [Heads] or [Tales], which is 50/50.
Id say you didnt take into consideration that she will wake up Monday AND Tuesday everytime its Tails. So like the Max buddy said, if you run the experiment youll notice how many times she wakes up in each situation
dude you are missing the major point here ... the coin is flipped once every week .. she , may answer it once per week if it was Heads , twice per week if it was Tails so if all her answer were Heads , if it was right , thats +1, if its wrong thats -2 think about it that way :)
Ibraheem Alshakly No, since the coin toss is fair, she is equally ,likely to wake up after a heads as after a tails. It doesn't matter that she will wake up twice if it is tails. That series of wake-ups is only as likely as the one wake-up after heads.
Think more the hardest way as how the errors in Execution error include unwanted output and stack overflow.. Vs Think more the hardest as how some common errors that we may find while compiling C++ program are discussed such as • main() • cout>>"Welcome"
Julia Galef ***** The philosophy subreddit is also discussing this, might be worth taking a look as it differs a bit :) www.reddit.com/r/philosophy/comments/3dw7ua/the_sleeping_beauty_problem/
Think more the Quickest Intuitively way as Ï(letter that comes after I in Khasi alphabet) is for *Ïing* which means 'House' Vs Think more the Quickest Intuitively way as Ñ(a letter that comes after N in Khasi alphabet) is for *Ñiew* which means 'count' again
Think more the wisest way as how *The most effective outcomes usually follow from conscious efforts to communicate effectively* Vs Think more the wisest way as how *Communication failures or misunderstandings are often very costly*
"There are three scenarios that I could be in that I would feel identical to each other" Not actually: the so called third state, Tuesday, is connected to the second state, Monday, and not with the flipping of a coin, that is, with a possible outcome of such flip. The only way in which you fit yourself in the Tuesday scenario is by first having reached the first Monday scenario, this one really connected to the coin flip and therefore probabilistic connected to it. The 'third option' is nuts.
Really enjoyed thinking about this. It’s annoying because I feel sure the thirders are wrong... although as you say they would think I am too! The probability of it being heads is 0.5; tails Monday is 0.25 and tails Tuesday 0.25. There is no new information in any of SB’s waking events. All making it a million days would do is reduce the probability of it being a given tails day to 0.5 / a million while heads probability stays at 0.5.
You may as well chalk this problem up to: if it’s tails and you guess correctly, you get a million points. Whereas, if it’s heads and you guess correctly then you only get one point. So there are quite obviously two questions now: 1) how do you play to maximise your score? 2) what’s the probability of the coin landing on heads. I hope this analogy helps you see that the question, as stated in the video, is asking us question 1 from my comment, whilst making us think it’s asking question 2. In reality, if the experimenters tracked all the results they will still find the probability of heads was always a half.
Think more the hardest way as how *The array is another kind of variable that is used extensively in C* Vs Think more the hardest way as how *There are several different ways to categorise arrays (e.g., integer arrays, character arrays, one dimensional arrays, multi-dimensional arrays)*
@@gdaaps Well, say that we made a bet where I was the sleeping beauty and you give me $1000 EVERY TIME I guess correctly, and I have to give you $1000 every time I'm wrong. I will always claim Tails and before going to sleep, there is 50% chance to lose $1000, and a 50% chance to win $2000, so still 1/2 probability of tails, but I get 2:1 odds in my favor! What if being correct every time gives me $1000 - and being wrong at least once gives YOU $1000? Then it's 1/2 probability again with normal 1:1 odds. If there's no bet - who cares? Who's counting and when? "What are the odds that it was tails" is a stupid question because the actual outcome is 1/2 in any which way, why would "asking twice when it's tails" make a difference? Even though you give two answers when it was tails, if there was no bet, the percentages you give out are meaningless, you might as well say 5/12 to be like in the middle or something.
The problem is that you're asking "What is the probability that the coin landed on heads?" and you're answering the question "What is the probability that I woke you up after the coin landing one heads?" You're conflating the two and they're similar enough that you can get away with it. The probability that the coin lands on heads is independent of how many times Sleeping Beauty wakes up, so why should that probability depend on how many states she can find herself in?
@@darbyl3872 That's not true. 0:56: "Ok Beauty, what do you think is the probability that the coin came up heads?" She is asked about the coin coming up heads, not about the probability of her being woken up after the coin came up heads. Again, the confusion comes from conflating two very similar concepts. You are asking about the probability of the coin coming up heads, but you're in actual fact trying to figure out the answer to the question: "What answer should i give to the question 'what face did the coin land on?' in order to have the best odds of being correct?" It's a completely different question.
@@darbyl3872 It doesn't matter who i am. It doesn't matter what state i am in. The probability that the coin landed on heads is still 1/2. You are asking one question and answering another.
No, you're ignoring both who is asked and when she is asked. Before the coin is tossed, and before she falls asleep, she must answer 1/2. After the coin has been tossed and after she is awakened, she knows an event may have have occurred *_other than the coin toss_* - she doesn't know if it's Monday and it was heads, it's Monday and it was tails, or it's Tuesday, it was tails, and her memory was erased. *_Now_* she must answer 1/3.
I think Cristi has this right. If the direct question Sleeping Beauty is asked is ‘What is the probability that the coin was heads?”, then the answer is always one half. However, if the questions is how many states might you be in at the time of the question, the answer is 3. But, there is a 50 percent chance it was heads and it is Monday, there is a 25 percent chance is was tails and it is Monday. And also 25 percent chance it was tails and it is Tuesday. It all depends on the exact wording of the question given to Sleeping Beauty.
@@darbyl3872 This is the same type of issues as asking: "If a coin landed on heads 9 times in a row, what are the odds it will land on heads again?" A lot of people will start calculating what the odds are of a coin landing on heads 10 times in a row, but the answer is still 1/2.
Think more the quickest intuitively way as Q is for *Qualified Dividends* again VS Think more the quickest intuitive way as Q is for *Quantitative Easing* again
In the video it is never stated that The Sleeping Beauty would be asked both on monday and on tuesday in case the coin comes up tails. Assuming the question is asking "given you just woke up in the puzzle scenario we're in, what is the chance that the coin flipped heads?" instead of just asking "what is the chance a fair coin flips heads?"(duh), then it would be important to know if she's asked twice or just on tuesday...
Anthropics reasoning is the most fascinating and frustrating thing. The implications are so large and the arguments are so persuasive yet you can’t help but feel like something is missing.
Yes, but the problem here is that this *isn't* anthropics, because we're dealing with known probabilities, and none of the outcomes involve the observe not existing. It strikes me that the whole 1/3 argument is an attempt to apply anthropics where it simply doesn't apply. It also makes anthropics seem counterintuitive by explaining it in terms that are counter to reality.
@@nialltracey2599 I disagree. Suppose the experiment was that after heads, sleeping beauty is woken up on Monday, but after tails she is never woken up. If the witch asks her, she can be absolutely sure that the coin "came up heads" (as phrased in the video), its probability is 100% then. Although she can still agree it is a fair coin with 50% probability if you toss it again (but her estimate of or confidence in "what happens next" is not the same as "what just came up"). That illustrates that the anthropic principle is at work. In the experiment of the video, it isn't as black and white, but still the probability is influenced by the fact that you are there to observe it. That is because to sleeping beauty it is a conditional probability.
Think more the Hardest way as how *Meaningfulness* relates to the overall meaning that a piece of writing has for the target audience.. Vs Think more the Hardest way as how Attainability relates to our setting a goal that could be achieved over a period of time, which again should also be measurable
she very subtly shifts the question, from "what is the probability the coin flipped heads" to "if you only guessed heads, what is the probability you would be correct" and this is where the divide arises.
Never mind I've worked it out. It's that if you measure the probability of her waking up in a world that is tails, tails is recorded twice and therefore is 2/3. If you measure the probability of flipping tails it is recorded once and therefore is 1/2
I agree. The coin flip is 1/2. Also probability theory is a hot mess. But for the sake of explanation in case any readers are curious: ---------- EXAMPLE: The coin is flipped. It landed heads. You wake up and are asked if it was has heads or tails. You say heads. Correct. The coin is flipped again. It lands tails. You wake up and are asked what it landed on. You say heads. Incorrect. You agree put back to sleep and made to forget your memory. The coin is not flipped. You wake up. You are asked about the coin. You say heads. Incorrect. SO Out of only TWO coin flips, you had to make THREE answers. Once the answer was heads, and twice it was tails. ---------- Exanation of the example: The 1/3rd idea comes from the fact that you are being asked about the coin multiple times. Like if the coin is tails, you will be waking up on both Monday and Tuesday. When you wake up on Monday, you would be right to answer 1/2. But then, unknown to you, when you wake up on Tuesday and are asked about the coin, the truth is that there is a 100% chance it was tails. (Because the coin was not flipped again). So, when you wake up, you have to ask yourself, "hmm.. was the coin even flipped last night? Or was it tails yesterday, and now I'm waking up for the second time (on Tuesday)?" So while I personally also adamantly say the probability of heads or tails is 1/2, other people say sleeping beauty gets asked about the coin when the answer is "tails" more often, so therefore the probability of tails is more likely.
@@fishamit Regardless of the outcome, each flip exists independently. It’s like choosing between three doors, where only one has a price and then changing doors to "increase" the odds of being right. In reality, the odds never change because each individual decision, each coin toss, is independent. I understand the origins of a lot of philosophical problems that are reasonable controversial, as well as basic statistical mathematics but this kind of argument really never resonated with me as the answer seems rather obvious.
Think more the hardest way of how I understandably keep in touch to you, not because of your beauty... Vs Think more the hardest way but of how you should understand what I'm actually fed-up for, and ended up sticking to you continuously
@Alfie Stoppani ahh thats the problem! I thought he would aks her at the end which would result in a probability of 1/2; I was confused what the debate was all about... but it seems that I have to take another approach. Thank you!
@@BenjusJamentus Still the coin has probability 1/2. Let's say instead of having a coin we had a 2/3 chance to be going on the path of the Monday and 1/3 for the Tuesday path. What is the probability now? The question is not what is the probability of her waking up and it having been heads or tails, but what the probability regardless of when walking up. They event is one, at the beginning, irrespectable of how many times the sleeping beauty wakes up. Other people have posted better answers, but wanted to add my 2 cents...
"- It just occurred to me... - Yes, Beauty? - It seems reasonable that it's Tuesday, the experiment's done, and tomorrow will be Wednesday... - Yes, naturally. - ...but there is a tiny, miniscule, yet finitely small chance that when I wake up tomorrow, it'll be Tuesday again and you'll ask me the same questions. - You mean that this day may happen twice? - Worse! It may very well be that that's all there ever is. I wake up every day, it is Tuesday, we go through our little routine, but I never remember any of the previous iterations, since everything reverts perfectly. This could go on indefinitely... - Nothing can go on indefinitely! Entropy would eventually put an end to that. - No! The very premise of the universe reverting assumes that entropy would roll back as well. - Alright, fine, so there's some pathetically small chance that you're stuck in an infinite loop... - I wouldn't call it small. The alternative is that I wake up once. The odds of that are... - 1 over infinity. - ...and the altenative is the rest of the domain. It is certain that I am trapped in a time loop! - Well, tomorrow you will know for sure that that doesn't happen to be the case, since it'll be Wednesday. - Possibly, but I won't know that I'm not trapped in an infinite loop of Wednesdays, now, will I? - ಠ_ಠ"
So it sounds like you would ask her the question every time she is woken up. (If you don't, then she should just answer 1/2.) If you do ask her the question every time she is woken up, then you have to look at her goal: - If her goal is to guess right most often on the final answer, then she should still stick with 1/2. - If her goal is to guess right most often, including when asked on the Tails Monday scenario, then should say 1/3.
The scenarios aren't all as likely to happen. Sleeping Beauty can't distinguish them from each other so from her perspective they could all be as likely, but in case the coin showed Tails, there is only a 50% chance of him asking her on Monday vs on Tuesday. They both have an accumulated probability of a 100% to happen if the coin shows Tails. The Heads scenario is 100% to happen if the coin shows Heads though, meaning it is twice as likey to happen as any single Tails scenario, since it doesn't have another condition to fulfill that would split its probability in two. The chances of the actual scenario happening stemming from the coin showing Heads is thus 1/2. There's a probability of 50% for it being Heads Monday, 25% for being Tails Monday and 25% for being Tails Tuesday.
I don't think you truly account for both Tails scenarios being guaranteed to happen when Tails is flipped because of the memory loss thing and them always waking her both times. If the coin is flipped to heads and tails once each, the question will have been asked 3 times. Heads twice is only 2 questions and if it lands tails twice, it will have been asked 4 times. This discrepancy of how often you're asked is what undoes your otherwise correct answer and is the reason it's far more likely you'll be answering after a tails flip. Imagine instead that they skip one of the tails days and picked one at random to ask you on and see how that applies to your scenario. That's where your solution would make perfect sense as far as I can tell. But not here.
Think more the Hardest way as how the structure described in the Poetics is considered to be the basis of modern screen play writing.. Vs Think more the Hardest way as how over the years, many filmmakers have tried to stretch the boundaries of what constitutes a 'movie'
plot twist: after disappearing for several weeks, sleeping beauty has been located & is recovering in a safe location, authorities are still searching for her abductor who posed as a scientist in order to roofie his victims.
You can't argue there are 3 cases in which the beauty can wake up - yes, she can wake up in 3 different situations but it's only 2 cases, as the 2 situations fall under the tails case, i.e. there are not three possible independent outcomes but only two. It's 1/2.
Agree, it doesn't matter if memento plays h/t and forget every single one of them the probability will always be 50 50. It doesn't matter if there is 3 outcomes, Javier Bardem can kill you, can decide in the last minute not to kill you, the police can stop him before he kills you, but the coin will be always have just 2 outcomes h/t
Think more the quickest intuitively way as J is for *Judgement summons* again VS Think more the quickest intuitively way as J is for *Jury Service* again
@@stevencraeynest7729 I'm not sure what Dana means actually. Certainly, at any given time, he has always been the clone (the one that survives), but that's not the same as him being the clone the _next_ time he does the trick.
@@luke-alex I meant that he is always the one that dies the next time he does it. Simply because it makes more sense to me that the original stays where it is, and a clone is created somewhere else, rather than the original being teleported and a clone being made on the original location
@@stevencraeynest7729 Don't worry, I understood you perfectly! My point was that I'm not sure what _Dana_ was trying to say. Note that, for the clone to be created (just before he falls through the trap-door into the tank), the information of him has to be transferred from there, to where the clone appears. You can conceptualise this transfer of information as teleportation. If the original person was instantly destroyed at the point of cloning, the whole process would be indistinguishable from teleportation. But of course, as you say, it's easier to not think about it like that.
This isn't a probability problem. It's an epistemological problem. And here's why. A probability calculation has two components. The first component is the actual ratio of total available outcomes to total possible outcomes: for example, a 1 out of 2 chance. That's the mathematical component. The second component is the way we define criteria for outcomes. The possible outcomes have to be materially differentiated from one another to be quantifiable, because if not, calculating a probability (component one) becomes meaningless. For example, what are the odds that Jason is shorter than Michael? Is it 50/50? You would think that it's 1 out or 2, like a coin flip, because there's only one out of two possible answers, yes or no. But just because there are two possible answers and only one can be chosen doesn't necessarily mean that the odds are 1 in 2. The problem is not in the math (component one). The problem is in not having enough information to say how Jason can or cannot be shorter than Michael. The problem, and thus the odds, are ill-defined. Indeed, to properly calculate the odds for the Jason/Michael example you would need two additional pieces of information: Michael's height and the average male height. If Michael is taller than the average height, then the possibility that Jason is shorter increases. If he is shorter than average, then the possibility decreases. The Sleeping Beauty problem suffers from the same lack of information. The probability component is sound. What are the odds that the coin flipped heads? 50/50. Because a coin toss is ALWAYS 50/50. Indeed, that's the very definition of a coin toss: a 1 out of 2 possibility. Where Numbers People appear to be flummoxed is in the second component, when the outcomes are defined. This isn't a probability problem (for as noted, a coin toss is always, by definition, 50/50). This is an epistemic problem. How do we know which outcome, heads or tails, we have arrived at? Well, by the way we have described the problem, it would seem we have merely muddied the description of each event. Indeed, if we strip away the extraneous putting back to sleep part, it should become clear that we have an ill-defined set of outcomes. In fact, we really have identical outcomes: heads it's Monday, tails it's Monday. This is because by making Sleeping Beauty decide that outcome (post-amnesia) we have thus made it not a probably problem but a problem of knowledge. Does Sleeping Beauty have the knowledge to figure out if she were anesthetized or not? The answer to that is, no, she does not have that knowledge. Therefore, in Sleeping Beauty's mind it's anyone's guess if the coin landed heads or tails. But here's the thing, she has an equal chance of being right with either guess, since a coin toss is always, by definition, 50/50, that means that Sleeping Beauty can take a guess as to whether the coin was heads or tails, and she still has a 1 in 2 chance of being correct, because that outcome is clearly defined.
Why is it that modern philosophy has a tendency to hijack one's intuitions to create a problem, when the issue itself stems from poor definitions and subjective interpretation? It seems like most of these infamous problems aren't actually problems at all, like I believe you've demonstrated.
Tal Moore Conditional probability though. It's not "what is the P(H)?" It's "what is P(H|Woken up)?" The Monty Hall problem also takes advantage of this concept.
Ryan Brady But that's not what the question asks. The question asks what is P(H)? What is P(H|Woken up) could only apply if being woken up is a knowable piece of information, and it's not. That's the whole point. You've taken a knowable bit of information--the odds of a coin flip--and tacked on an unknowable bit of information--odds of being woken up on Monday or Tuesday.
If the question is Which is the probability the coin landed on heads? the answer is always 50%, no matter if you ask this million times, as increasing the amount of questions on the other side doesn't increase the chance of the coin falling onto that side. The amount of wrong answers if the question is changed _Which side you believe the coin fell on?_ increases by adding more questions on another side if you keep answering Heads, but that is different statistical question entirely. The whole statistical analysis proposed in 1/3 answer is a red herring used to confuse you the half way forgetting what the original question was.
I'm not sure if this is the dumbest thing ever but I just feel like: the chances of the coin landing on heads is 1/2, but the chances of you waking up on a heads landing is 1/3
Sleeping Beauty does have new information, namely that she just woke up. So the question is "what is the probability that the coin came up heads given that you just woke up?" and that's 1/3.
I think the debate is based around different interpretations of the question. There are two questions to ask here, and the answers are different: 1. What is the probability that the coin came up heads? 2. What is the probability that this instance of you waking is happening in a timeline where the coin came up heads? I hope everyone agrees that the answer to question #1 is 1/2. The coin is stated to be fair, so the probability of either outcome is 1/2. (Actually, it's very slightly less since 'on edge' is a possibility but a negligible one.) The answer to question 2 feels to me like it requires more thought, but hopefully everyone eventually agrees that the answer is 1/3 -- there are 3 occasions on which she gets woken up, and only one of them happens in the 'coin = heads' timeline.
Putting Beauty to sleep, then waking her up, wiping out those memories then waking her up again is the low-tech logical equivalent of uploading her mind to a computer and then running two instances of her. Combining that with the intuition pump Julia mentioned, the scenario becomes this: The scientist gives the uploaded Beauty strong evidence that he's going to run one instance of her on a computer in New York if the coin comes up heads, or a million instances of her on a computer in Tokyo if it comes up tails. Then he stops Beauty's current instance, flips the coin, and does exactly that. When Beauty becomes aware again, what should she (rationally) believe is the probability that she's in Tokyo?
Think more the quickest intuitively way as R is for *Relative reference* again Vs Think more the quickest intuitively way as R is for *Random Access Memory* again
The experiment rules clearly imply that the whole ordeal ends on Tuesday the latest (so no 3rd or millionth waking up). And the coin is only tossed once, on Monday. I'm confused.
Every time she awakens, she is in one of four possible outcomes: 1) Heads Monday- being asked to guess the coin flip. 2) Heads Tuesday- in the comfort of her own bed at home--the experiment was over yesterday. 3) Tails Monday- being asked to guess the coin flip. 4) Tails Tuesday- being asked to guess the coin flip. Looking at it this way, Sleeping Beauty has, in fact, been given an additional piece of information upon waking. She has a 1/4 chance that she is in the Monday of a Heads Universe, and a 2/4 chance that she is in a Monday OR Tuesday of a Tails Universe. The problem is only confusing because we neglect the Tuesday of the Heads universe, which has a 1/4 chance that has been eliminated by the NEW INFORMATION Sleeping Beauty got upon waking...namely, that she is still IN the experiment. Using this new info, she is right to say that the odds the coin landed heads is 1/3, even though in the past, before she had this info, before the coin was flipped, the odds WERE 50/50.
If she is awoken an infinite number of times, then that is a different problem. Her guess in that impossible scenario should be that she is almost certainly in a tails universe, and may God have mercy on her soul.
RUclips has a glitch where OPs have only four visible lines, oftentimes the viewer is not given the "read more" option, so I am reposting my OP solution here; Every time she awakens, she is in one of four possible outcomes: 1) Heads Monday- being asked to guess the coin flip. 2) Heads Tuesday- in the comfort of her own bed at home--the experiment was over yesterday. 3) Tails Monday- being asked to guess the coin flip. 4) Tails Tuesday- being asked to guess the coin flip. Looking at it this way, Sleeping Beauty has, in fact, been given an additional piece of information upon waking. She has a 1/4 chance that she is in the Monday of a Heads Universe, and a 2/4 chance that she is in a Monday OR Tuesday of a Tails Universe. The problem is only confusing because we neglect the Tuesday of the Heads universe, which has a 1/4 chance that has been eliminated by the NEW INFORMATION Sleeping Beauty got upon waking...namely, that she is still IN the experiment. Using this new info, she is right to say that the odds the coin landed heads is 1/3, even though in the past, before she had this info, before the coin was flipped, the odds WERE 50/50.
I agree. If the question is the probability of heads then it should be 1/2 because that is based on the result of the coin toss. If the question is what is the probability that she wakes up on Monday and the coin toss was heads then it would be 1/3.
I am confused why nearly nobody (With the exception of Louis Wilbur here in the comments) does get the only real and correct answer: The problem is ill defined and does not have a correct answer. Anybody who claims it is either 1/2 or 1/3 makes additional implicit assumptions about the experiment/problem. In the original formulation the sample space is ill defined and relies on the outcome of the coin toss in question. This cannot be modeled using probability theory and it also does not have a "real physical model". The Sleeping Beauty Problem cannot be turned into a real experiment. While there might be more complete answers than the answer given above, the answer clearly is neither 1/2 nor 1/3 since classical probabilities are not and cannot defined in the context of the Sleeping Beauty Problem. It is easy to turn the implicit assumptions (people make) into explicit rules and turn the problem into a well defined one. Then the answer is typically 1/2 or 1/3 depending on the exact assumptions/rules. Mathematics (and logic) are full of such subtleties, where suddenly additional assumptions from metaphysics/philosophy might enter the picture and confuse people. But mathematicians have become extremely good to to find those and make them explicit. Just look up "Skolem's paradox", Banach-Tarski paradox or reverse mathematics to get an idea how rigorous and formal logical thinking has to be to stay consistent in some cases. Jeffrey Rosenthal: "Despite these caveats, I hope and believe that there is merit in providing a simple, short, direct argument that 1/3 is the correct answer, using solid mathematical foundations with few assumptions and little philosophical ambiguity." Jeffrey seems to think, for whatever reason, that the question has an answer outside of mathematics (which I do not believe), but he is clearly aware that there is no purely mathematical solution.I disagree. Probability is a (purely) mathematical concept. Either there is a purely mathematical solution and a proof or there is no answer to the question.
Consider this: P(M) =Monday = 2/3; P(H) = Heads = 1/2. We can write P(H) = P(H|M)P(M) / P(M|H) = P(H|M) * 2/3 / 1. Now things get tricky. If we accept that P(H) is clearly 1/2, then P(H|M) = 3/4. But then, if we are restricting our probability space to only Monday, there are two possibilities, one of which has a Heads, so 3/4 feels just a wrong result. Hence P(H|M) = 1/2 and P(H) = 1/3.
Hey guys, if you want to discuss this video (and others) more broadly, or if you just want a place to discuss rationality in general, check out our subreddit at www.reddit.com/r/rationallyspeaking Hope to see you there!
Think more the hardest way as I is for Illustration, Illusion, Investment, Intuitive, Inheritance, Income, Intern, Interrupt, Interest, Insecure etc Vs Think more the harder way as 10 words that starts with I
I'm a 1/2-er. It seems to me that the 1/3 answer is predicated on a specious desire to get the answer right as many times as possible. Since SB has more chances to get it right by saying "tails", she is better of from that point of view picking tails. So the double tails awakening is amplifying the apparent value of the tails answer. If you gave me evens on heads but 2:1 on on tails, I'm going to pick tails every time, thank you very much! But that's not because tails is more likely, just that you have increased the payout for tails. That's exactly what SB is being offered. I think the logical flaw is similar to the "we are almost certainly living in a simulation" argument. Asking a question multiple times does not change the probability of an answer being correct.
Phil Hibbs I agree 100%. If Beauty is offered $100 for every time she guesses correctly whether it's heads or tails, she'll get double the money for the correct tail answer (assuming they ask her question every day). But as you say, that doesn't alter the probability of the coin falling heads in the first place.
@@robsmith2047 But that's not what the question is. The question is not "What is the probability of heads?" but "What is the probability I'm in a situation where heads came up?" In the first case you don't need anthropic reasoning, in the second you do.
Surely a premise to this scenario is that we're using a 50/50 coin, right? So we already know that SB's got 1/2 chance of being on one probabilistic arm, and 1/2 chance on the other. But to work out which stage she might be at requires the attribution of probabilistic weights to each stage. It's fallacious to automatically attribute equal weight to 'Heads', 'Tails, woken once', 'Tails, woken twice', 'Tails woken thrice', etc. If we know the scenario uses a 50/50 coin then branch A demands 1/2 probability; and branch B 1/2 probability. That remaining 1/2 is split between the remaining stages of being awoken and amnesiatised. It would be possible for SB to work out the chances she'd been subject to a Head or Tail, if she knew the number of times she'd been sent back to sleep and amnesiatised. But if she hadn't been told that number, or if it hadn't been decided beforehand, she couldn't really know. The experimenter could know the probabilities however (not that they'd need them), but she couldn't. E.g. if she knew she'd be amnesiatised three times in total, then that means when she wakes up she knows that there's a 50% chance of Heads, 12.5% Tails first time, 12.5% Tails second time (amnesiatised once), 12.5% Tails third time (amnesiatised twice), and 12.5% Tails fourth time (amnesiatised thrice). Which means she has a 1 in 8 chance of being the subject of each Tails toss. That's a 5/8 chance of it being Monday, and a 1/8 chance each of it being Tuesday, Wednesday or Thursday. Which means 'winning the money' means she should definitely go with 'Monday'. I imagine that if mathematicians disagree on the 'answer' to this scenario, then it is because they disagree on its definition - not because it doesn't have an answer, in any sufficiently well-defined form.
notsyort It seems to be that it is philosophers, not mathematicians, who are arguing about this. At least those mentioned in the Wikipedia article on the subject are philosophers, as far as I can tell.
notsyort " If we know the scenario uses a 50/50 coin then branch A demands 1/2 probability" Not if you're sampling more from one branch than the other. The process isn't just the flip, it's the flip plus the obviously biased sampling from the flips.
buybuydandavis It's not possible, insofar as Julia's defined it, for either branch to be 'sampled' less than every time. Are you saying Sleeping Beauty dies sometimes? Or that the experimenter ignores flips sometimes, and flips again? And what difference would that make, anyway? We can't get around the 50/50 thing, unless we introduce extra elements into the scenario. Attempting to breach the known 50/50 limitation sounds very much more like the kind of faux-Philosophy that Michael thinks this scenario belongs to, than genuine Mathematics. Summing 'awakenesses', for example, is simply meaningless.
notsyort I thought the point about having SB guess what the outcome was, brought home the point about multiple 'samples' on one branch quite well. Given 2000 runs of the experiment, SB would be right around 1000 times if she guessed heads when she woke up, but she would wake up around 3000 times.
Think more the Hardest way as *• Break the Ice:* which implies to break down social formality and stiffness.. Vs Think more the Hardest way as *•By hook or by crook:* which means by whatever means necessary - be they fair or foul.
The question is "What do you think is the probability that the coin came up heads?" The act of polling the result (asking sleeping beauty about the probability) doesn't affect the initial event. Meaning that regardless of how many times she is put to sleep, woken, and asked the coin was still only flipped once, and therefore there are still only two outcomes -> Heads or Tails. Even by her own perception, in this scenario, there are only two final outcomes: She wakes up and its heads or she wakes up and its tails. It's the same reason why regardless of how many times you get heads in a row, the probability of another heads is still 50%. The probability is tied to the event, not the perception of the event. It seems like the halve-nots are misinterpreting the question and falling into the gamblers fallacy. A question that would produce the result 1/3 would have to be phrased, "What do you think is the probability that you have woken up after the coin came up heads?" This changes the event being polled from the coin coming up heads, with a 1/2 probability, to the event of waking up after heads, which has a 1/3 probability.
@@wassupusa I understand the Monty Hall problem. Even if you are looking at it as the three possible outcomes of {Mo,H}, {Mo,T},{Tu,T}; The case of {Tu, T} only exists on Tuesday, and SB knows she is more likely to be woken on a Monday. This updates her three outcomes to {Mo, H, 1/2}, {Mo, T, 1/2}, and {Tu, T, 1/3}. There is only one case where the correct answer is 1/3, where there are two cases that have the correct answer of 1/2. So the answer will always result in 1/2 as it will always be the answer that is most likely to be correct.
Think more the quickest Intuitively way as P is for *Programming Language* again Vs Think more the quickest Intuitively way as P is for *Pythagoras* again
The two answers differ in perspective, whether they admit it or not. The 50% perspective is from the coin's perspective. The 33% perspective is from Sleeping Beauty's perspective.
The 50% perspective isn't the coin's perspective, it's the world perspective. I would call it an objective truth The 33% perspective is subjective because it's based on Sleeping Beauty's feeling sort of. The fact that she could feel the same waking up in 3 different scenarios dosen't alter the fact that in the world the coin still has a 50% chance to have ended up on one side or the other.
@@botcontador3286 The monty hall problem is unambigous. In this problem, the contradicting facts that the coin toss is fair AND Tuesday has to be tails are making it so that there is no correct answer.
@@busTedOaS The fact that the experiment is implied to terminate on Tuesday. This implies that the coin flip is fixed to fall on tails on Tuesday. This alters the probability of the coin flip being heads to a third. However this is contradictory since the coin flip is already stated to be fair. You can't have it both ways.
@@olavbakke2889 What's the problem it being Head+Tuesday, are you saying that's an impossible situation? My understand is you just wouldn't wake the subject.
i think it would depend on when the question is asked, this seems to have not been stated. if the question is asked only on heads and the last time she would wake up the it would be 1/2. but if the question is asked every time she is woken up, then it is 1/wake-ups. in either case the answer is 1/times-question-asked, which seems right as that would follow P(A)=A/(Sample-Size).
The question could be reformulated to : "can you guess if it was Heads or tails?" In wich case you will win 66% of the time if you ALLWAYS choose Tails.
Only if each individual waking event ie Monday heads, Tuesday heads and Tuesday tails are equally likely ie 1/3. And they aren’t. Because of the amnesiac, Monday tails and Tuesday tails both have probability 0.25 while heads has 0.5.
@@nandc2009 good point, but then again it comes down to what the question is exactly. what you're suggesting is correct if the question is "how likely is it to get Tails\Heads" and not "wich one was is in THIS particular flip".
@@hwiz8282 I think the question is 'How likely was heads in this particular flip - ie what is the credence/probability that the coin landed heads'? I think tails comes up 50% of the time, and 50% of the times she wakes up when it has been tails is on Monday and another 50% on Tuesday, so both those waking events are 0.25 probability.
I think the real problem is in the definition of the problem; particularly, what is the precise question that SB has to answer. If the question is equivalent to asking what the probability of a fair coin coming up heads was, then the answer has to be 0.5; there is no way that the number of subsequent awakenings can affect that. The thirders, though, seem to be answering a different question, but I'm not exactly sure what it is. It appears to somehow involve the number of awakenings, and the assumption that the probability of this one being any particular awakening is equally likely and independent. But that assumption is not correct. There will be a second awakening only half the time, and it will only happen after there has already been a first awakening. Consider how a computer simulation of this problem could be done. For each trial, the computer would use a pseudo-random-number generator and store the results, heads or tails. Then it could put itself to sleep and wake itself up according to the rules of the problem. At the end of each trial, it would return the results of the PRNG. Finally, after all the trials are turn, it would report appropriate statistics. Obviously, assuming no bugs in the program or operating system or hardware, nothing that happens after the PRNG is run can change its result, regardless of whether the program sleeps once, twice, or a million times.
Think more the hardest way as why we are acquainted with various types of oral Communications right from our childhood, like talking with friends, quarrel over pitty issues, discuss current affairs, narrates stories to children where we are involved in a process known as Oral communication and for such Communication that we do not need any form of training... Vs Think more the hardest way as why Oral communications are in the form of staff meetings, presentations, debates, speeches, discussion, telephonic chats, classroom lectures or simple informal conversations
I hated probability in school....I'm sure this won't aggravate me though...I'm older and more patient now. EDIT: *Fuck*...I really wanted to get sleep tonight. sigh.
My initial reaction is that both choices are right given the right framing of the question and that this isn't a paradox. When asked what is the probable outcome of a fair coin toss, we all agree the outcome is 1/2. That's pure probability. But Sleeping Beauty is waking up with *evidence.* There's more information here. Waking up makes it not the same puzzle. All that matters is that the question is different. "What was the likelihood this fair coin tossed at some point was heads?" is still 1/2. "What is the likelyhood you were woken up by a heads coin toss?" is a question including more evidence. It's just a different scenario. Both answers are right given the framing/context, fundamentally because they are different questions.
Not really. The answer is in the question. This is not a problem, it's a riddle and who doesn't love riddles? If the question is; "what is the probability that the coin came up heads. " Well single toss, 2 outcomes, 1/2 is the correct answer. If instead the question is; "what is the possibility that sleeping beauty woke up to the coin came up heads;" then the answer is 1/3 as she woke up twice to tails and once to heads. The only problem is the ambiguity of the question. Is the header part of the question or not? Also note; that it is ok for riddles to have multiple answers.
I think everyone can agree that they should divide the two camps into the “halves and the halve nots”.
I can't sleep on that pun! 🤣
I'd like a swig of that potion that makes me forget things, please...
Haha... you go to the HEAD of the class.
Hmmm... Now you have me wondering what being in the halve not camp enTAILS...
Stop the madness!!!
It seems like the thirders seem to think that the question is not "What are the chances the coin came up heads?" but rather "What are the chances that I'm asking you this question under a scenario where the coin came up heads?" Because the question "What is the probability that the coin came up heads?" has nothing to do with you being woken up.
If the question is not "what are the chances that I'm asking you this question under a scenario where the coin came up heads" then what on earth is probability about??
It is asking a conditional probability: heads given you just woke up. Why should that be the same as the unconditional probability, which indeed would be 50%?
Like, if I ask you "what are the chances that it is your birthday today" precisely when you are blowing out candles on a cake, the answer is no longer about 1 in 365.
@@landsgevaerAfter thinking on this more, my take on this problem is that the answer to the paradox is a reflection of how we _choose to define_ statistics, not a reflection of _the universe_. Statistics is a human invention to make predictions; you can make accurate predictions if you say this is conditional probability, you can make accurate predictions if you use a form of statistics that says the underlying probability is 2/3rds (think, Newtonian vs lagrangian mechanics).
The paradox is a trap, because it asks "what are the odds of the coin flip?" but sleeping beauty's answer has no effect on the universe. If she said, "150%," or "hippopotamus," then who cares? So, maybe let's say that she's taking a test, and she only passes if she gets the answer right. What should she say to pass the test? Well, this is the same paradox, because how does the witch grade her answers? Let's presume the witch "just knows." How can you tell apart this paradox from one where the witch is actually wrong? We are in fact unable to construct a puzzle where the result of the question objectively matters. It's asking a nonsense question -- or at least, nonsense without providing a definition of statistics as a premise, and that definition could simply be anything. The correct answer *could* then be "hippopotamus." Moreover, by attaching a definition of statistics such that "the odds of a coin flip is always hippopotamus," the paradox is no longer interesting, or a paradox.
If we want the paradox to apply to the real world, rather than a particular definition of statistics, we need an objective consequence. Something like, snow white can bet $5 on the outcome of the coin flip. In this scenario, snow white should bet it is tails. Neither interpretation of statistics disagrees with this. This version of the paradox is not interesting, and once again, not even a paradox.
So in summary -- thank you for answering my question and providing a definition/interpretation of statistics that can be used to answer the puzzle with an answer of 1/2. I'm not a statistician so I didn't think of it in those terms, but thinking of it in those terms make sense. That's probably the "technically correct" answer. Some people probably find the "technically correct" answer illuminating (like stats students who have to learn "technically correct" language in order to properly converse with other statisticians), and I personally find the reduction of the paradox itself is also interesting, perhaps moreso.
@@MichaelFairhurst Thanks. There are (at least) two ways of thinking about what a probability means: frequentists look at it as a fraction, how often would it occur if the experiment were repeated; bayesians look at it as confidence, how certain am I that an outcome will occur.
But leave aside that diversion. In either case, here it makes a difference from whose perspective you look at it: are you looking for the fraction of heads out of all the awakenings that sleeping beauty experiences, or out of all the times the coin was tossed by the witch (phrased in frequentist terms); or, how confident sleeping beauty can be that an awakening is heads, or how confident the witch can be that a toss is heads (in bayesian terms). They do not have to agree. In that sense I agree with you that there is no contradiction.
Sleeping beauty sees heads 1/3 of the time she wakes up; the witch sees heads 1/2 of the time she tosses a coin. The question "what is THE probability of heads" is ill-defined. It is conditional on what 'universe' of events you calculate it relative to: all the tosses or all the awakenings. It depends on the measure you assign to events. And that we indeed "choose".
Since you like physics here is an analogy: what is the probability of Schrödingers cat to be alive? From the researcher's perspective who opens the box it is 50%. From the cat's perspective who is in the box, it is 100%. :-) For the researcher, dead and alive have equal measure; for the cat, dead has measure zero because it will not be there to ever observe that. It can both reliably estimate that it is alive and be confident about that. (Disclaimer: I don't believe in heaven.)
If you program computers, such subtle distinctions of meaning can never be escaped. People often ask "why are programming languages not more intuitive and similar to our daily lives?" Well, this is why, because they cannot leave open room for interpretation.
Think more the Hardest way as how Graphic Interchange Format uses 256 colours
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Think more the Hardest way as how it is not correct when an Anchor tag or tag in HTML is used to define tags in a webpage.
Seems to me that the debate is more about how to understand the question rather than the likelyhood of the cointoss itself:)
It's one of those "If a tree falls in the forest and there is no one to hear it, does it make a sound?" type of questions.
Always has been
That's true of a lot of thought experiments. For example "unstoppable force vs immovable object."
It’s this exactly. They need to precisely quantify what exactly needs be optimized. Speaking imprecisely, the argument about the problem seems to hinge around an implication that ascribing higher probability to the amnesiac run of the experiment when it’s true, so you’re right twice (or a million times) is worth more, or is in some sense “more correct,” than being right once in the non-amnesiac run and ascribing higher probability to that case.
Weirdly this problem still gets a lot of traction along with things like the two envelopes problem (en.wikipedia.org/wiki/Two_envelopes_problem). It’s a problem of human intuition not exactly fitting with how one may formalize what a “problem” is.
@@trucid2 No it's not because there is no possible way of knowing the answer to that question, whereas this question there are only 2 known possible outcomes.
Think more when I learnt to satisfy with my own worth....
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Think more when I learnt to be happy with my own value...
She's twice as likely to be woken up on the tails flip, but only because she is woken up twice on it. This doesn't change the fact that the coin has a 50/50 chance of landing H/T. The odds of her being awoken twice are equal to the odds of her being awoken once. Both 2/3 and 1/2 are correct, but to two different questions. Just because you record data twice on tails, doesn't mean the odds are higher it will land on tails - just that you're more likely to be recording data during tails flips. Can we all agree on that?
skillzorz101 Yeah i agree, was just about to comment "Why not both?"
skillz -- "She's twice as likely to be woken up on the tails flip"
That isn't quite true: the probability of 2 wakings on a tails flip is 1. A tails will always result in 2 wakings, likewise a head always results in 1 waking. As we know, half of all flips will be heads, but the real question isn't what are the odds of a heads, it's what proportion of wakings are caused by heads. She will wake once for heads and twice for tails. And since 2/3 of the wakings will be from tails and 1/3 from heads, she would be right to bet on tails.
We could break it down
For a tails - Monday's guess tails: right. Tuesday's guess tails: right.
For a heads - Monday's guess is tails: incorrect.
That's 2 right, 1 wrong for always guessing tails.
For a tails, Monday guess heads: wrong; Tuesday guess heads: wrong.
For heads, Monday guess heads: right.
1 right, 2 wrong when always guessing heads.
In order for the odds to be 50/50 the two strategies would need to have the same rate of success. Recall that the question is not what are the odds of a heads, but what are the odds that upon waking that the toss was heads -- which on a functional level means what are the odds that you are being wakened following a heads?
skillzorz101 I am a Physicist and thus statistics is really my daily bread ;-)
skillzorz101, You are exactly right in your interpretation. Both answers: 1/2 and 1/3 are correct, they're just answers to two completely different questions. The first question is about the odds of a coin flip. However the second question is about the probability of which data point are we looking at (from a single element set labeled "H" or from a multi element set labeled "T").
Where in the setup does she say that if Sleeping Beauty is going to be kept in bed till Tuesday, she's going to be asked to assess the probability of heads being the outcome of the coin toss on both Monday and Tuesday?
I was also thinking that. If she is asked twice, then this is simply an alternative wording of the Monty Hall Problem.
The question Sleeping Beauty is asked is "What do you think is the probability the coin came up heads?" Is the question ambiguous? I don't know, but I do know that some of the disagreement arises from people taking the question to be asking different things. Here are some possible rephrasings of the question that have different answers:
Q: What is the probability that a flipped coin will come up heads?
Q: If I flip a coin and ask one person to guess the outcome if it comes up heads and two people to guess the outcome if it comes up tails, which answer will result in more correct guesses?
Q: Is Sleeping Beauty equally likely to be woken up (and asked the question) when the coin flip is heads as when it is tails?
Q: If you run the experiment 100 times, and the coin lands heads 50 times and tails 50 times, how many of the 150 times Sleeping Beauty is asked the question will the coin have landed heads?
Q: If you run the experiment 100 times, and the coin lands heads 50 times and tails 50 times, in how many of 100 trials will heads be the correct answer?
I am not to read your novel in the comments section.
@@ranevc Then jog off.
100% agree and this goes for so many of these misleading, so called "philosophical" questions.
@@ranevc If you were afraid you'd be wasting time, I assure you, time was wasted because of you replying.
Yes, indeed.
In fact, if she was allowed to make a fair bet (ie $1 stake, wins $2 if correct) every time she was woken up, she would clearly bet tails every time and no one would doubt her reasoning. To me, this indicates the real problem is with some ambiguity or equivocation in the phrasing of the question. I think I will post this in it own right too :-)
Think more the quickest intuitively way as R is for *Reception classes* again
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Think more the quickest intuitively way as R is for *Regularity* again
Beaut is not as dumb as mathos and philosophers. She can tell from her leg stubble whether she's been asleep for 1 or 2 days. Drugging women for 'probability research', yeah, right. She's getting a lawyer.
Nope, they shaved her in her sleep, anticipating this problem.
@@brianarbenz7206 oh, you cunning bastard!!!
Sounds like the case for calling a lawyer is getting stronger.
The interesting thing is that even if she was able to tell whether it had been 2 days or not, her overall probablity of answering the question right does not change. If its Monday (2/3 of the questions) then she has a 50-50 shot of guessing the flip (so 2/3 * 1/2 = 1/3). If its Tuesday (1/3 of the questions) then she knows its tails (1/3 * 1 = 1/3). So her overall probability of getting it right is 2/3, exactly the same as if she just said tails every time.
She consented to the experiment
The sleeping beauty problem is ambiguous because it does not say what sample space she is using. Probabilities are defined on a per sample space basis. The sample space of the coin toss is {H,T} and the sample space for the questions about the coin state is {MH,MT,UT} where H=heads, T=tails, M=Mondays and U=Tuesdays. The probability of heads for the first sample space is 1/2 and the probability of heads for the second sample space is 1/3, since they are both equiprobable sample spaces. To see equiprobability, just notice that out of every 1000 coin tosses about 500 will be heads, 500 will be tails, and about 1500 questions will be asked about 500 which will occur when it is Monday and heads, another 500 which will occur when it is Monday and tails, and the remaining 500 which will occur when it is Tuesday and tails.
She should use the probability for the sample space she assumed and the problem doesn't tell what sample space that is. The problem is bad because it introduces two different sample spaces without clarifying which one is operative. For example, if the problem also stated that for betting purposes on repeated trials of the experiment she should bet as much money as possible then it would be clear that she should use the sample space for the questions about the coin state to get the probability. But if instead of that we added to the original problem that she give the probability for repeated tosses of the coin then money won or lost is irrelevant and she should use the sample space for the coin toss to get the probability. The sleeping beauty question is ambiguous because it is asking about belief in the frequency of the truth value of occurrences of the PROPOSITOIN "the coin landed heads" not the proposition that the coin's probability of landing heads is 1/2. That is, the question doesn't make clear if it is asking about the probability of the proposition being true during repeated coin tosses or if it is asking about the probability of the proposition being true during repeated questioning in many repetitions of the experiment. These are not the same thing because when the coin is tails she is questioned twice but when the coin is heads she is questioned only once.
Now she knows the proposition is true one out of every three times she is asked and she is not going to mistake that for the fact that the coin comes up heads one out of every two times during coin tossing. So, for the proposition "the coin landed heads" the frequency of this proposition being true during repeated questioning in many repetitions of the experiment is different than the frequency of it being true during repeated coin tosses. If the coin toss actually came up heads then the proposition "the coin landed heads" is true but if the coin toss actually came up tails then the proposition "the coin landed heads" is false. How often the proposition is true or not depends on the circumstances. So adding two different prepositional phrases onto the original question highlights the ambiguity of that question:
(case 1)
What is your belief now for the proposition that "the coin landed heads" in the case of repeated questioning in repetitions of the experiment?
(case 2)
What is your belief now for the proposition that "the coin landed heads" in the case of repeated tosses of the coin?
The conclusion: the sleeping beauty problem is ambiguous because case 1 and case 2 use different sample spaces and if one removes the phrase "questioning in repetitions of the experiment" from case 1 and removes the phrase "tosses of the coin" from case 2 then the ambiguity of the original question is exposed.
Louis Wilbur Spot on. It is depressing that so few people seem to understand this. The question is simply not well defined and people that give an answer (1/2 or 1/3) make additional implicit assumptions.
This is a fantastic explanation.
I wouldn't say it's ambiguous as it was presented in this video, because it was worded without reference to sample space and with reference to one variable only: the coin toss.
The day of the week is not part of the question, so why would it ever be part of the answer?
@@nialltracey2599 I'm going to use your comment as an excuse to answer a 5 year old thread ;)
I am really unsure why folks get so tossed about this, but I assume it's because of the dreadful way this problem is often presented. Louis Wilbur is obviously correct. A succinct way to explain it would be thus:
a) When the coin is tossed, what are the chances it comes up heads? (50%)
b) Of all the possible times Beauty is asked the question (3 in total: 1 in the heads scenario, 2 in the tails scenario), how many times will the correct answer be heads? (33,33%)
Those two statements are both true AND NOT mutually exclusive.
Answer is obviously 1/3. Simple bayes. The sleeping beauty learned she doesn’t know which day of the week it is. Given the new information the answer is 1/3.
Think more the quickest intuitively way as Q is for *Quadratic equations* again
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Think more the quickest intuitively way as Q is for *Quotient* again
I can see that a common response is, "Philosophers don't understand math, it's clearly 1/2."
Sigh. Perhaps I should have clarified that *mathematicians* are divided on this puzzle, too, not just philosophers. For example, here are several mathematicians and physicists arguing the 1/3 position:
1. Nick Wedd is an International Math Olympiad winner, and a thirder:
www.maproom.co.uk/sb.html#arg5
2. Jeffrey Rosenthal, winner of the "Nobel prize of statistics", is a thirder:
probability.ca/jeff/ftpdir/beauty.ps
3. Another International Math Olympiad winner, Tanya Khovanova, is a thirder:
blog.tanyakhovanova.com/2011/08/the-sleeping-beauty-problem/
4. Physicist Sean Carroll is a thirder (but agrees it's controversial):
www.preposterousuniverse.com/blog/2014/07/28/quantum-sleeping-beauty-and-the-multiverse/
Julia Galef I think anyone familiar with the concept of conditional probability should come to the 1/3 conclusion.
Julia Galef Imagine you are the car in the Monty hall problem, and you have to guess which of the three doors you were placed behind... After the contestant makes their first guess and one of the doors is eliminated, you 'forget' which door he picked, all you know is which two doors are left and you are in one of them. Due to lack of information, you are facing a 50/50 choice. Meanwhile from the point of view of contestant the door he picked still has a 1/3 chance of being the right one, while the other remaining door now has a 2/3 chance. They are both right, depending on the frame of reference and what information you have available. As the sleeping beauty waking up, you would be facing a 1/3 chance of heads. As an observer, the chance of a fair coin flip is always 50/50. Basically the problem is poorly defined :P
Patternicity But the difference with the Monty Hall problem is that the circumstances are changed BEFORE the final choice is made, therefore altering the probability. With Beauty, everything happens AFTER the toss is made. She knows there will be different results depending on whether the coin falls heads or tails, but when she wakes, she has no knowledge of what the result was. Therfore, like the observer, the only reasonable answer she can give is 1/2.
Rob Smith She does have knowledge ABOUT the result, she knows that she is in one of three possible situations, 2/3 of which happen after the result of the coin flip is TAILS.
Patternicity By that logic, if the researcher gives her the amnesia drug for a thousand days, then the 50/50 coin flip is going to always end up tails. Which is flat out silly.
Thinking about this problem, I have a different perspective that clarifies the question for me:
Flip a coin. If heads, take a picture of a dog. If tails, take two pictures of a cat.
Question 1: what is the probability that the coin flipped heads? 1/2
Question 2: choose a picture at random. What is the probability it's a picture of a dog? 1/3
Choose a picture at random from what? Do you have 3 pictures?
I'd say not. You have either 1 or 2, in any case you have only two options, either pick the 1 dog or pick 1 of the two cats.. So 1/2 is the prob
@@p4xx07 Sorry, maybe I wasn't clear. You have three pictures - one is of a dog, and two are of a cat.
A closer analogy would be thus:
The experimenter flips a fair coin, and takes a picture of a dog if it's heads or two pictures of a cat if it's tails. You, as a test subject, are about to be shown a picture. What is the probability that the coin toss that determined your picture was a heads?
@@danwylie-sears1134 That seems to be Question 1 from my original comment
@@DarthCalculus You never have 3 pictures. If coin was heads, you have 1 picture. If coin was tails, you have 2 pictures.
Think more the wisest way as *Emphasis & Emotion* of *Communication Dynamics*
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Think more the wisest way as *Who we are and what emphasis we put on our spoken thoughts, together from a critical part of our communication with others*
The confusion can be resolved as follows: It is a question of which timeline she is in, not which day. She was asked essentially what is the probability that you are in a timeline where heads came up. The probability of entering either timeline is 1/2. Once in a given timeline, the probability does not change. What changes is the number of times she will be asked the question. She will be asked the question more times in one timeline. That does not change the correct answer, it only changes how many times she will answer. She will say the correct answer more times if tails came up. But it is the same answer. If she gets a dollar each time she gets it right, she gets more money if tails came up. But the answer is the same. You can safely remove the Monday and Tuesday which changes nothing. If she were asked repeatedly on the same day with a dose of amnesia in between each time, it would boil down to: what are the odds that you will have been asked this question just once vs repeatedly? Answer: 1/2. If the question was: what are the odds that this is the only time you will have been asked this question, the answer would be 1/3. But that would not be analogous to the question in the premise.
" She will say the correct answer more times if tails came up. "
So if you ask me what 2 + 2 is I'll say the correct answer more often if I say 4 rather than 5. We could talk about that in terms of "timelines" if we want but 4 is still the right answer.
Very well stated. This cleared things up for me nicely.
@@sinistar99 that doesn’t really have anything to do with this problem, now does it?
You may as well chalk this problem down to: if it’s tails and you guess tails, then you get a million points but if it’s heads and you guess heads then you only get one point. There are now two questions: 1) how do you play to maximise your points? 2) what’s the probability of getting heads?
Still 1 in 2
Ok. Let get to the point. I am the best at explaining things.
It is easier to modify a problem to make you say arhhh i get it.
First, let solve this problem.
A god is bored and want to play a game. Out of 7 billion human in this world. He will choose only one human. This one chosen human will be awaken again and again 7 billion times. The rest of human will fall asleep only once. The god then telepatically explain the rule to all human on earth. And make all 7 billion people go to sleep alone in their personal space..
Now you woke up. What is the chance that you are this special human and this is just one of your 7 billion wake up circle? Is it 50% because the chance of falling in special 7 billion sleep is equally likely as the other normal sleep????!! ? I think now u can figure this sleeping beauty problem on your own.
A few more point i want to add. Knowing more information does not always change probability.
When confused, modify, exagerate the problem. It often help. Do not be misled. Be careful what has equal chance. What is not. All events do not need to always have equal chance. Such as wakeup event in this case.
Thank you finally someone with the correct answer for the correct reason. Just as I was about to go insane. Even if you get woken up a million times within the same timeline all of these times are still one entity. Once you enter the timeline all of the million days get activated at once like a button that lights up all of them in synchronicity. This doesn't change the probability which remains 50% even retrospectively. Seeing it like this I can't imagine how someone would count all waking events disconnected from each other.
The answer i believe is 1/2 whether you look at it from an outsider point of view or from beauty's perspective.
Am going to change the experiment a bit just to make easier. Say that if the coin came up head, she would wake up on Sunday. Everything else holds the same. Thus, the question he asks her "what is the probability that the coin came up head? " is the same as "what is the probability that today is Sunday?". This change should not effect the probability at all.
From beauty's perspective, she doesn't know whether she woke on Sunday from the Head scenario, Monday from tails scenario or Tuesday from tails scenario. What she knows is the probabilities of each one. She knows that since the coin is fair, there is a 50% chance that she woke up on Sunday. Also, that there is a 50% chance she woke up on Monday or Tuesday. Therefore, assuming probability of waking up on Monday is the same as waking up on Tuesday, then the probability she woke up on Monday is 25% and probability she woke up on Tuesday is 25٪ too.
Thus, her answer is 50% either way. The mistake i think 1/3 advocaters is that they think the probability of her waking up on Sunday, monday and Tuesday is all the same.
The odds of it being Tuesday aren't the same as the odds of it being Monday. When she is asked, she can deduce that the probability of it being Monday with heads is twice as likely as the probability of it being either Monday with tails or Tuesday (with tails). So the correct answer is always 1/2. If you were asking her whether it was heads or tails, the answer would be tails 2/3rds of the time, but that's not the same as the question she is being asked.
The question is framed to make it misleading. The odds the toss came up heads is 1/2. The odds that this is your first waking are 2/3. The odds that this is Tuesday are 1/3.
I don't think this is about being misleading. I think this is about how one scenario of probability relates to the other. They've linked the results of a coin to a world that only has 1/3rd of a chance for you to be in when asked the question. So what happened to the probability of the coin? If we measure, it has objectively changed because of the new variables. If you take away the variables when answering it goes back to 1/2 but if that's really what they ask it would quite frankly be a pointless thing to demonstrate with a hypothetical scenario.
Wouldn't the odds of it being your first waking actually be 75% ?
50% = Heads > First Waking
&
50% = Tails > 50/50 Chance of being the First Waking ~ 25%
So 75% Chance of it being my First Waking
Actually this is a misleading question, in that the author is trying to illicit an answer to a different question than the one they asked. Specifically:
1. What is the probability that the coin was heads? Answer: 1/2
2. What is the probability that this is the second time you woke up since you last remember going to sleep? Answer: 2/3
Two completely different questions, but they are trying to illicit a response about the second and apply it to the first.
"A" causes "B", therefore "B" causes "A" is a logical fallacy. A coin flip determined her sleeping, but her sleeping cannot influence the coin flip.
No, probability is not influenced by measurement. It is not statistically correct to say that if I flip a fair coin 4 times and it comes up heads 3 times that therefore there is a 75% probability of heads. A finite sample size NEVER shows the actual probability of something occurring, because all it would take is one more data point to change the probability (which is ALWAYS the case).
Probability is stated not as a finite dataset, but rather the limit x-->infinity of the probability function. If you flip a fair coin an infinite amount of times, the results would be 50/50, and anything less would not be a true representation of the dataset.
The coin flip already occurred, and it has a 1/2 chance of either result. What day of the week is immaterial to the original probability of an event that has already occurred and could not have been influence based upon her perception. She knows nothing about the outcome of the coin flip. She only knows that she woke up, and that there is a 2/3 chance it is the first time since she last remembered going to sleep. None of that has any bearing on the probability of an outcome from the coin flip.
To take it to an absurd level, say I flip a coin. If it was heads, 15 cats would be ran over and if tails 10 cats would be ran over. What is the probability that today is Tuesday? 1/4, because pancakes are purple when aliens wear tutus in the rain.
Although I do agree, it is quite frankly a pointless thing to demonstrate with a hypothetical scenario.
The question is not "what is the probability of a coin being tossed to be heads" but "what is the probability the coin came up heads", which is different in the sense that she will be asked this question twice when it was tails but only once on heads. This obviously has to be factored in, otherwise you just disregard half of the scenario
Misleading is a stretch since you're presuming intent there. It is certainly ambiguous. Different readings may lead to different probability spaces (one that counts in units of "experiments" and one that counts "waking up events") which lead to the answers 1/2 or 1/3.
Has anyone noticed the way Julia subtly changes the question when she's explaining the 1/3 idea. She says something like "if the experiment is repeated many times and I find myself answering heads, I'll only be right only one third of the time" but the question wasn't "did it fall heads or tails", it was "what was the proability that it fell heads". If she answers 1/2, she'd be answering right 100% of the time (or 0% of the time if you think the answer's 1/3).
The question is “what is the probability that it DID fall heads”, given that this version of you is being asked the question. Her very existence in different states is related to the probabilities, so of course if you neglect that information the coin returns to being 50/50.
If I throw a glass of water over you every time a coin is heads, and I throw it 1/100 times if the coin is tails, then if you saw me flip a fair hidden coin and then throw water at you, the chance of that FAIR coin having landed on heads is clearly higher, despite the coin itself having 50/50 odds.
(You having water thrown on you here is analogous to some self-ignorant state of Alice being awoken, not knowing the day or what is about to happen to her. If I where not to throw water over you, then this is the equivalent of that version of Alice never existing, meaning you would know for a fact it was tails)
I think Beauty's answer would be, in the case of tails, "Coin? What coin? What are you talking about? Who are you?"
I think if we change the nature of the problem but keep the game principle the same we have an more intuitive answer:
Imagine someone tosses a fair coin and asks you to determine the probability of whether it’s heads. They tell you have not tampered with it and will show you the actual result, and even toss the coin in front of you.
You would naturally say 1/2 . They show you the result (heads or tails) and write it down as the last result on their log. [This is effectively what happens when sleeping beauty wakes up]
Afterwards they show you the log for every flip you made in the past. Secretly they have removed half of the head results, but never the final result, So over time the spread shows 1/3 of heads now. You confront them and they admit it saying it all part of an experiment! [This is the same as being woken up twice for tails and once for heads]
They toss the same coin and ask you to predict heads. As before they will show you the result and put that results as the last result in their log* .
[You now know the rules of the game]
You answer would remain as 1/2. The coin is a fair coin, you can see the toss and the result.
If a betting person is betting on the log results with a bookmaker, then he/she would always bet on tails (as some of the historical heads are removed to keep the balance on 2/3 tails).
Love to have other people's feedback.
Ketan
*Before the coin toss they will remove any excessive historical heads so that the split is always 1/3 heads 2/3 tails but only after they show you the last result.
* PS even if the game is only played
Two different questions.
What's the probability that a coin will come up heads? P = 1/2
What's the probability that Beauty will wake up on a heads up flip?
P = 1/3
Infinity 1 Exactly.
Well said! I completely agree.
Infinity 1 Incorrect. Given that there is a coin toss, she has a 1/2 probability of being woken up on heads, and 1/2 on tails. But since there are two wake-up possibilities on tails, or heads, depending on which way round the story has it, there is a 1/4 chance of her having woken up on either day.
Holy Moly No, if it's heads she will only be interviewed on Monday. If it's tails she's interviewed on both Monday and Tuesday. There is only 3 possibilities she could be in:
1) It's Monday and it's heads.
2) It's Monday and it's tails.
3) It's Tuesday and it's tails.
Infinity 1, That was not the question though. The question was "What is the probability that the coin was heads?" You're answering the wrong thing.
You just answered P(this waking event | H) but the question was P(H | this waking event) and these are not interchangeable.
You're right that yours is 1/3, but the answer to the actual question asked (the second one, which is not your question) is 1/2, because by Bayes Theorem, it's P(H) * P(this waking event | H) / P(this waking event) which = (0.5 * 0.33(your portion)) / 0.33 = *0.5*
If she has evacuated her bowels or peed herself, then it's probably Tuesday
Great alternative stradegy
Biology just solved a probability problem. Brilliant!
@@uptown3636 biology is a probability factory, as you well know.
Not if she does so every night. But in such case she can simply count the movements. In the million iteration scenario, this would take quite a while and be error prone.
Or wakes up with a voracious appetite and B.O.
Think more the hardest way as "Chota" which means small in Hindi
Vs
Think more the hardest way as "Dekho" which means look in Hindi
I think, Beauty knows there are three possible scenarios:
* Heads, Monday - probability 1/2=.0.5
* Tails, Monday - probability 1/2*1/2=0.25
* Tails, Tuesday - probability 1/2*1/2=0.25
Why 1/2*1/2? Because the probability of a tails flip is 1/2, and the probability that she's woken on Monday is _also_ 1/2.
This seems simple to me, so there's probably something wrong with it. What's wrong?
Max X
_"Each situation will occur 5 times:"_
Yes, I see the reasoning. [Heads, Monday], [Tails, Monday] and [Tails, Tuesday] are all equally likely, assuming Beauty has no information other than the test procedure.
But the question the researchers ask her isn't "What coin side/day combination you do think it is?", but "Did the coin flip heads or tails?".
So the probabilities of what day it is are irrelevant. It comes down to [Heads] or [Tales], which is 50/50.
Id say you didnt take into consideration that she will wake up Monday AND Tuesday everytime its Tails. So like the Max buddy said, if you run the experiment youll notice how many times she wakes up in each situation
dude you are missing the major point here ...
the coin is flipped once every week .. she , may answer it once per week if it was Heads , twice per week if it was Tails
so if all her answer were Heads , if it was right , thats +1, if its wrong thats -2
think about it that way :)
Ibraheem Alshakly No, since the coin toss is fair, she is equally ,likely to wake up after a heads as after a tails. It doesn't matter that she will wake up twice if it is tails. That series of wake-ups is only as likely as the one wake-up after heads.
Ha ha, "...Rest assured...". That's my favorite part.
Think more the hardest way as how the errors in Execution error include unwanted output and stack overflow..
Vs
Think more the hardest as how some common errors that we may find while compiling C++ program are discussed such as
• main()
• cout>>"Welcome"
FYI, there is a large discussion thread about this video on Reddit: www.reddit.com/r/math/comments/3915xh/the_sleeping_beauty_problem/
***** Ah, I wasn't aware -- thanks for letting me know!
Julia Galef ***** The philosophy subreddit is also discussing this, might be worth taking a look as it differs a bit :)
www.reddit.com/r/philosophy/comments/3dw7ua/the_sleeping_beauty_problem/
I have never before felt this blue balled by an informational video.
+++
Explain?
Explanation : this lady is very sexually attractive, especially compared to other informational video presenters.
No I actually meant that the information provided was just enough to get interested in knowing more.
Sure you did. Choice of analogy was a coincidence... ;-)
Think more the Quickest Intuitively way as Ï(letter that comes after I in Khasi alphabet) is for *Ïing* which means 'House'
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Think more the Quickest Intuitively way as Ñ(a letter that comes after N in Khasi alphabet) is for *Ñiew* which means 'count' again
Julia, you are the beauty that makes me donot want to sleep , just to hear you and see you talking is the gift of heaven....
1/2 of the time, she will be awoken after a head is flipped.
The other 2/3 of the time she will be awoken after tails is flipped.
I like it :)
This should be the final response. Let's stop now...
1/2 + 2/3 = 7/3 = 116.6%... something doesn't add up
@@Dragon30ficationXD I think you mean 7/6. And it would round up to 116.7%
@@sk8rdman yeah my bad
Think more the wisest way as how *The most effective outcomes usually follow from conscious efforts to communicate effectively*
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Think more the wisest way as how *Communication failures or misunderstandings are often very costly*
This is the most intelligent and rigorously thoughtful set of responses I have ever see posted on RUclips.
Grammar!
@@ranevc ‘set of responses’ is singular
@@earlystrings1 True, but the word should be seen not see.
I love this channel.
Are we conflating the odds its Monday or Tuesday with the odds of the coin toss?
Think more the quickest intuitively way as B is for *Band Theory* again
Vs
Think more the quickest intuitively way as B is for *Boiling point* again
"There are three scenarios that I could be in that I would feel identical to each other" Not actually: the so called third state, Tuesday, is connected to the second state, Monday, and not with the flipping of a coin, that is, with a possible outcome of such flip. The only way in which you fit yourself in the Tuesday scenario is by first having reached the first Monday scenario, this one really connected to the coin flip and therefore probabilistic connected to it.
The 'third option' is nuts.
Really enjoyed thinking about this. It’s annoying because I feel sure the thirders are wrong... although as you say they would think I am too! The probability of it being heads is 0.5; tails Monday is 0.25 and tails Tuesday 0.25. There is no new information in any of SB’s waking events. All making it a million days would do is reduce the probability of it being a given tails day to 0.5 / a million while heads probability stays at 0.5.
You may as well chalk this problem up to: if it’s tails and you guess correctly, you get a million points. Whereas, if it’s heads and you guess correctly then you only get one point. So there are quite obviously two questions now: 1) how do you play to maximise your score? 2) what’s the probability of the coin landing on heads.
I hope this analogy helps you see that the question, as stated in the video, is asking us question 1 from my comment, whilst making us think it’s asking question 2. In reality, if the experimenters tracked all the results they will still find the probability of heads was always a half.
Think more the hardest way as how *The array is another kind of variable that is used extensively in C*
Vs
Think more the hardest way as how *There are several different ways to categorise arrays (e.g., integer arrays, character arrays, one dimensional arrays, multi-dimensional arrays)*
My brain is so firmly in the 1/2 camp that it is prepared to fight the one-thirders to the death in hand to hand combat. It’s actually frightening me.
clearly wrong... just make the experiment... the pure thinking is for idiots; real men make experiments.
@@gdaaps Well, say that we made a bet where I was the sleeping beauty and you give me $1000 EVERY TIME I guess correctly, and I have to give you $1000 every time I'm wrong.
I will always claim Tails and before going to sleep, there is 50% chance to lose $1000, and a 50% chance to win $2000, so still 1/2 probability of tails, but I get 2:1 odds in my favor!
What if being correct every time gives me $1000 - and being wrong at least once gives YOU $1000? Then it's 1/2 probability again with normal 1:1 odds.
If there's no bet - who cares? Who's counting and when?
"What are the odds that it was tails" is a stupid question because the actual outcome is 1/2 in any which way, why would "asking twice when it's tails" make a difference?
Even though you give two answers when it was tails, if there was no bet, the percentages you give out are meaningless, you might as well say 5/12 to be like in the middle or something.
Me too my friend. This lady should give up philosophy and get a real job.
The problem is that you're asking "What is the probability that the coin landed on heads?" and you're answering the question "What is the probability that I woke you up after the coin landing one heads?" You're conflating the two and they're similar enough that you can get away with it. The probability that the coin lands on heads is independent of how many times Sleeping Beauty wakes up, so why should that probability depend on how many states she can find herself in?
@@darbyl3872 That's not true.
0:56: "Ok Beauty, what do you think is the probability that the coin came up heads?"
She is asked about the coin coming up heads, not about the probability of her being woken up after the coin came up heads.
Again, the confusion comes from conflating two very similar concepts. You are asking about the probability of the coin coming up heads, but you're in actual fact trying to figure out the answer to the question: "What answer should i give to the question 'what face did the coin land on?' in order to have the best odds of being correct?" It's a completely different question.
@@darbyl3872 It doesn't matter who i am. It doesn't matter what state i am in. The probability that the coin landed on heads is still 1/2. You are asking one question and answering another.
No, you're ignoring both who is asked and when she is asked. Before the coin is tossed, and before she falls asleep, she must answer 1/2. After the coin has been tossed and after she is awakened, she knows an event may have have occurred *_other than the coin toss_* - she doesn't know if it's Monday and it was heads, it's Monday and it was tails, or it's Tuesday, it was tails, and her memory was erased. *_Now_* she must answer 1/3.
I think Cristi has this right. If the direct question Sleeping Beauty is asked is ‘What is the probability that the coin was heads?”, then the answer is always one half. However, if the questions is how many states might you be in at the time of the question, the answer is 3. But, there is a 50 percent chance it was heads and it is Monday, there is a 25 percent chance is was tails and it is Monday. And also 25 percent chance it was tails and it is Tuesday. It all depends on the exact wording of the question given to Sleeping Beauty.
@@darbyl3872 This is the same type of issues as asking: "If a coin landed on heads 9 times in a row, what are the odds it will land on heads again?" A lot of people will start calculating what the odds are of a coin landing on heads 10 times in a row, but the answer is still 1/2.
Think more the quickest intuitively way as Q is for *Qualified Dividends* again
VS
Think more the quickest intuitive way as Q is for *Quantitative Easing* again
In the video it is never stated that The Sleeping Beauty would be asked both on monday and on tuesday in case the coin comes up tails. Assuming the question is asking "given you just woke up in the puzzle scenario we're in, what is the chance that the coin flipped heads?" instead of just asking "what is the chance a fair coin flips heads?"(duh), then it would be important to know if she's asked twice or just on tuesday...
That's not relevant.
@@zyaicob it is
Anthropics reasoning is the most fascinating and frustrating thing. The implications are so large and the arguments are so persuasive yet you can’t help but feel like something is missing.
Yes, but the problem here is that this *isn't* anthropics, because we're dealing with known probabilities, and none of the outcomes involve the observe not existing.
It strikes me that the whole 1/3 argument is an attempt to apply anthropics where it simply doesn't apply.
It also makes anthropics seem counterintuitive by explaining it in terms that are counter to reality.
@@nialltracey2599 I disagree. Suppose the experiment was that after heads, sleeping beauty is woken up on Monday, but after tails she is never woken up.
If the witch asks her, she can be absolutely sure that the coin "came up heads" (as phrased in the video), its probability is 100% then. Although she can still agree it is a fair coin with 50% probability if you toss it again (but her estimate of or confidence in "what happens next" is not the same as "what just came up").
That illustrates that the anthropic principle is at work. In the experiment of the video, it isn't as black and white, but still the probability is influenced by the fact that you are there to observe it. That is because to sleeping beauty it is a conditional probability.
Think more the Hardest way as how *Meaningfulness* relates to the overall meaning that a piece of writing has for the target audience..
Vs
Think more the Hardest way as how Attainability relates to our setting a goal that could be achieved over a period of time, which again should also be measurable
she very subtly shifts the question, from "what is the probability the coin flipped heads" to "if you only guessed heads, what is the probability you would be correct" and this is where the divide arises.
Agreed, the ambiguity of the question is the puzzle, not the reasoning.
Can you explain the difference to me?
Never mind I've worked it out. It's that if you measure the probability of her waking up in a world that is tails, tails is recorded twice and therefore is 2/3. If you measure the probability of flipping tails it is recorded once and therefore is 1/2
Well put. Thank you!
Regardless of how many times she is woken up. Could be a million, doesn't matter.
The coin is only flipped once. Ever. 1/2
Exactly - I don't get how this is controversial. If there is one coin toss, it's 1/2, regardless of how many times she is woken up
I agree. The coin flip is 1/2. Also probability theory is a hot mess.
But for the sake of explanation in case any readers are curious:
----------
EXAMPLE:
The coin is flipped.
It landed heads.
You wake up and are asked if it was has heads or tails.
You say heads. Correct.
The coin is flipped again.
It lands tails.
You wake up and are asked what it landed on.
You say heads. Incorrect.
You agree put back to sleep and made to forget your memory.
The coin is not flipped.
You wake up. You are asked about the coin.
You say heads. Incorrect.
SO
Out of only TWO coin flips, you had to make THREE answers. Once the answer was heads, and twice it was tails.
----------
Exanation of the example:
The 1/3rd idea comes from the fact that you are being asked about the coin multiple times.
Like if the coin is tails, you will be waking up on both Monday and Tuesday.
When you wake up on Monday, you would be right to answer 1/2.
But then, unknown to you, when you wake up on Tuesday and are asked about the coin, the truth is that there is a 100% chance it was tails. (Because the coin was not flipped again).
So, when you wake up, you have to ask yourself, "hmm.. was the coin even flipped last night? Or was it tails yesterday, and now I'm waking up for the second time (on Tuesday)?"
So while I personally also adamantly say the probability of heads or tails is 1/2, other people say sleeping beauty gets asked about the coin when the answer is "tails" more often, so therefore the probability of tails is more likely.
@@fishamit Regardless of the outcome, each flip exists independently. It’s like choosing between three doors, where only one has a price and then changing doors to "increase" the odds of being right. In reality, the odds never change because each individual decision, each coin toss, is independent.
I understand the origins of a lot of philosophical problems that are reasonable controversial, as well as basic statistical mathematics but this kind of argument really never resonated with me as the answer seems rather obvious.
Think more the hardest way of how I understandably keep in touch to you, not because of your beauty...
Vs
Think more the hardest way but of how you should understand what I'm actually fed-up for, and ended up sticking to you continuously
I didn't realise the examiner would ask her the question if it was tails on a Monday. That needed to be made clear.
Depends, will she be told beforehand that she will be asked that question afterwards? If she isn't told, then nothing about her situation changes.
@Alfie Stoppani ahh thats the problem! I thought he would aks her at the end which would result in a probability of 1/2; I was confused what the debate was all about... but it seems that I have to take another approach. Thank you!
@@mike140298 no, but she knows that she is asked the question twice when it was tails
@@BenjusJamentus Still the coin has probability 1/2. Let's say instead of having a coin we had a 2/3 chance to be going on the path of the Monday and 1/3 for the Tuesday path. What is the probability now?
The question is not what is the probability of her waking up and it having been heads or tails, but what the probability regardless of when walking up. They event is one, at the beginning, irrespectable of how many times the sleeping beauty wakes up. Other people have posted better answers, but wanted to add my 2 cents...
"- It just occurred to me...
- Yes, Beauty?
- It seems reasonable that it's Tuesday, the experiment's done, and tomorrow will be Wednesday...
- Yes, naturally.
- ...but there is a tiny, miniscule, yet finitely small chance that when I wake up tomorrow, it'll be Tuesday again and you'll ask me the same questions.
- You mean that this day may happen twice?
- Worse! It may very well be that that's all there ever is. I wake up every day, it is Tuesday, we go through our little routine, but I never remember any of the previous iterations, since everything reverts perfectly. This could go on indefinitely...
- Nothing can go on indefinitely! Entropy would eventually put an end to that.
- No! The very premise of the universe reverting assumes that entropy would roll back as well.
- Alright, fine, so there's some pathetically small chance that you're stuck in an infinite loop...
- I wouldn't call it small. The alternative is that I wake up once. The odds of that are...
- 1 over infinity.
- ...and the altenative is the rest of the domain. It is certain that I am trapped in a time loop!
- Well, tomorrow you will know for sure that that doesn't happen to be the case, since it'll be Wednesday.
- Possibly, but I won't know that I'm not trapped in an infinite loop of Wednesdays, now, will I?
- ಠ_ಠ"
if only i had a " sleeping beauty time loop" like that during my exams lol
So it sounds like you would ask her the question every time she is woken up. (If you don't, then she should just answer 1/2.)
If you do ask her the question every time she is woken up, then you have to look at her goal:
- If her goal is to guess right most often on the final answer, then she should still stick with 1/2.
- If her goal is to guess right most often, including when asked on the Tails Monday scenario, then should say 1/3.
The coin turns up tails but on the Monday Beauty withdraws her consent to participate in the experiment and goes home.
The scenarios aren't all as likely to happen. Sleeping Beauty can't distinguish them from each other so from her perspective they could all be as likely, but in case the coin showed Tails, there is only a 50% chance of him asking her on Monday vs on Tuesday. They both have an accumulated probability of a 100% to happen if the coin shows Tails. The Heads scenario is 100% to happen if the coin shows Heads though, meaning it is twice as likey to happen as any single Tails scenario, since it doesn't have another condition to fulfill that would split its probability in two. The chances of the actual scenario happening stemming from the coin showing Heads is thus 1/2. There's a probability of 50% for it being Heads Monday, 25% for being Tails Monday and 25% for being Tails Tuesday.
I don't think you truly account for both Tails scenarios being guaranteed to happen when Tails is flipped because of the memory loss thing and them always waking her both times. If the coin is flipped to heads and tails once each, the question will have been asked 3 times. Heads twice is only 2 questions and if it lands tails twice, it will have been asked 4 times. This discrepancy of how often you're asked is what undoes your otherwise correct answer and is the reason it's far more likely you'll be answering after a tails flip. Imagine instead that they skip one of the tails days and picked one at random to ask you on and see how that applies to your scenario. That's where your solution would make perfect sense as far as I can tell. But not here.
Think more the Hardest way as how the structure described in the Poetics is considered to be the basis of modern screen play writing..
Vs
Think more the Hardest way as how over the years, many filmmakers have tried to stretch the boundaries of what constitutes a 'movie'
plot twist: after disappearing for several weeks, sleeping beauty has been located & is recovering in a safe location, authorities are still searching for her abductor who posed as a scientist in order to roofie his victims.
You can't argue there are 3 cases in which the beauty can wake up - yes, she can wake up in 3 different situations but it's only 2 cases, as the 2 situations fall under the tails case, i.e. there are not three possible independent outcomes but only two. It's 1/2.
Agree, it doesn't matter if memento plays h/t and forget every single one of them the probability will always be 50 50. It doesn't matter if there is 3 outcomes, Javier Bardem can kill you, can decide in the last minute not to kill you, the police can stop him before he kills you, but the coin will be always have just 2 outcomes h/t
Think more the quickest intuitively way as J is for *Judgement summons* again
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Think more the quickest intuitively way as J is for *Jury Service* again
Brings to mind for me the movie the Prestige, where Hugh Jackson's character doesn't know which one he'll be when he does his trick.
What? He kills himself every time. He's always the clone
@@Lucas-fo8ci you probably mean he's always the one that dies
@@stevencraeynest7729 I'm not sure what Dana means actually. Certainly, at any given time, he has always been the clone (the one that survives), but that's not the same as him being the clone the _next_ time he does the trick.
@@luke-alex I meant that he is always the one that dies the next time he does it.
Simply because it makes more sense to me that the original stays where it is, and a clone is created somewhere else, rather than the original being teleported and a clone being made on the original location
@@stevencraeynest7729 Don't worry, I understood you perfectly! My point was that I'm not sure what _Dana_ was trying to say.
Note that, for the clone to be created (just before he falls through the trap-door into the tank), the information of him has to be transferred from there, to where the clone appears. You can conceptualise this transfer of information as teleportation. If the original person was instantly destroyed at the point of cloning, the whole process would be indistinguishable from teleportation. But of course, as you say, it's easier to not think about it like that.
This isn't a probability problem. It's an epistemological problem. And here's why.
A probability calculation has two components. The first component is the actual ratio of total available outcomes to total possible outcomes: for example, a 1 out of 2 chance. That's the mathematical component.
The second component is the way we define criteria for outcomes. The possible outcomes have to be materially differentiated from one another to be quantifiable, because if not, calculating a probability (component one) becomes meaningless. For example, what are the odds that Jason is shorter than Michael? Is it 50/50? You would think that it's 1 out or 2, like a coin flip, because there's only one out of two possible answers, yes or no. But just because there are two possible answers and only one can be chosen doesn't necessarily mean that the odds are 1 in 2.
The problem is not in the math (component one). The problem is in not having enough information to say how Jason can or cannot be shorter than Michael. The problem, and thus the odds, are ill-defined. Indeed, to properly calculate the odds for the Jason/Michael example you would need two additional pieces of information: Michael's height and the average male height. If Michael is taller than the average height, then the possibility that Jason is shorter increases. If he is shorter than average, then the possibility decreases.
The Sleeping Beauty problem suffers from the same lack of information. The probability component is sound. What are the odds that the coin flipped heads? 50/50. Because a coin toss is ALWAYS 50/50. Indeed, that's the very definition of a coin toss: a 1 out of 2 possibility. Where Numbers People appear to be flummoxed is in the second component, when the outcomes are defined.
This isn't a probability problem (for as noted, a coin toss is always, by definition, 50/50). This is an epistemic problem. How do we know which outcome, heads or tails, we have arrived at? Well, by the way we have described the problem, it would seem we have merely muddied the description of each event. Indeed, if we strip away the extraneous putting back to sleep part, it should become clear that we have an ill-defined set of outcomes. In fact, we really have identical outcomes: heads it's Monday, tails it's Monday.
This is because by making Sleeping Beauty decide that outcome (post-amnesia) we have thus made it not a probably problem but a problem of knowledge. Does Sleeping Beauty have the knowledge to figure out if she were anesthetized or not? The answer to that is, no, she does not have that knowledge. Therefore, in Sleeping Beauty's mind it's anyone's guess if the coin landed heads or tails.
But here's the thing, she has an equal chance of being right with either guess, since a coin toss is always, by definition, 50/50, that means that Sleeping Beauty can take a guess as to whether the coin was heads or tails, and she still has a 1 in 2 chance of being correct, because that outcome is clearly defined.
Why is it that modern philosophy has a tendency to hijack one's intuitions to create a problem, when the issue itself stems from poor definitions and subjective interpretation? It seems like most of these infamous problems aren't actually problems at all, like I believe you've demonstrated.
MyOnlyFarph Philosophers like to work at the frontiers of our reasoning. That's where all the puzzles, paradoxes and mysteries lie.
Tal Moore Conditional probability though. It's not "what is the P(H)?" It's "what is P(H|Woken up)?" The Monty Hall problem also takes advantage of this concept.
MyOnlyFarph Perfect response.
Ryan Brady But that's not what the question asks. The question asks what is P(H)? What is P(H|Woken up) could only apply if being woken up is a knowable piece of information, and it's not. That's the whole point. You've taken a knowable bit of information--the odds of a coin flip--and tacked on an unknowable bit of information--odds of being woken up on Monday or Tuesday.
If the question is Which is the probability the coin landed on heads? the answer is always 50%, no matter if you ask this million times, as increasing the amount of questions on the other side doesn't increase the chance of the coin falling onto that side. The amount of wrong answers if the question is changed _Which side you believe the coin fell on?_ increases by adding more questions on another side if you keep answering Heads, but that is different statistical question entirely. The whole statistical analysis proposed in 1/3 answer is a red herring used to confuse you the half way forgetting what the original question was.
I'm not sure if this is the dumbest thing ever but I just feel like: the chances of the coin landing on heads is 1/2, but the chances of you waking up on a heads landing is 1/3
No you're 100% correct, this is the dumbest thing ever. Both of your statements pretty much explain the entire controversy.
Sleeping Beauty does have new information, namely that she just woke up. So the question is "what is the probability that the coin came up heads given that you just woke up?" and that's 1/3.
To me this seems obvious. I don't understand the 1/2 answer. Makes no sense at all.
@@astromastro6026 If you flip a coin, but dont look at it. What is the probability that it flipped heads?
@@DeusExAstra 1/2. What difference does it make if I looked at it or not?
I think the debate is based around different interpretations of the question. There are two questions to ask here, and the answers are different:
1. What is the probability that the coin came up heads?
2. What is the probability that this instance of you waking is happening in a timeline where the coin came up heads?
I hope everyone agrees that the answer to question #1 is 1/2. The coin is stated to be fair, so the probability of either outcome is 1/2. (Actually, it's very slightly less since 'on edge' is a possibility but a negligible one.)
The answer to question 2 feels to me like it requires more thought, but hopefully everyone eventually agrees that the answer is 1/3 -- there are 3 occasions on which she gets woken up, and only one of them happens in the 'coin = heads' timeline.
Putting Beauty to sleep, then waking her up, wiping out those memories then waking her up again is the low-tech logical equivalent of uploading her mind to a computer and then running two instances of her. Combining that with the intuition pump Julia mentioned, the scenario becomes this:
The scientist gives the uploaded Beauty strong evidence that he's going to run one instance of her on a computer in New York if the coin comes up heads, or a million instances of her on a computer in Tokyo if it comes up tails. Then he stops Beauty's current instance, flips the coin, and does exactly that. When Beauty becomes aware again, what should she (rationally) believe is the probability that she's in Tokyo?
I just like looking at her. Half of the time I have no idea whats going on.
Think more the quickest intuitively way as R is for *Relative reference* again
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Think more the quickest intuitively way as R is for *Random Access Memory* again
The experiment rules clearly imply that the whole ordeal ends on Tuesday the latest (so no 3rd or millionth waking up). And the coin is only tossed once, on Monday. I'm confused.
That doesn't mean the lab rat knows that.
It's the frequentist interpretation. You imagine repeating the whole thing many times. Each experiment ends on Tuesday and next week there's another.
Every time she awakens, she is in one of four possible outcomes:
1) Heads Monday- being asked to guess the coin flip.
2) Heads Tuesday- in the comfort of her own bed at home--the experiment was over yesterday.
3) Tails Monday- being asked to guess the coin flip.
4) Tails Tuesday- being asked to guess the coin flip.
Looking at it this way, Sleeping Beauty has, in fact, been given an additional piece of information upon waking. She has a 1/4 chance that she is in the Monday of a Heads Universe, and a 2/4 chance that she is in a Monday OR Tuesday of a Tails Universe. The problem is only confusing because we neglect the Tuesday of the Heads universe, which has a 1/4 chance that has been eliminated by the NEW INFORMATION Sleeping Beauty got upon waking...namely, that she is still IN the experiment. Using this new info, she is right to say that the odds the coin landed heads is 1/3, even though in the past, before she had this info, before the coin was flipped, the odds WERE 50/50.
If she is awoken an infinite number of times, then that is a different problem. Her guess in that impossible scenario should be that she is almost certainly in a tails universe, and may God have mercy on her soul.
RUclips has a glitch where OPs have only four visible lines, oftentimes the viewer is not given the "read more" option, so I am reposting my OP solution here;
Every time she awakens, she is in one of four possible outcomes:
1) Heads Monday- being asked to guess the coin flip.
2) Heads Tuesday- in the comfort of her own bed at home--the experiment was over yesterday.
3) Tails Monday- being asked to guess the coin flip.
4) Tails Tuesday- being asked to guess the coin flip.
Looking at it this way, Sleeping Beauty has, in fact, been given an additional piece of information upon waking. She has a 1/4 chance that she is in the Monday of a Heads Universe, and a 2/4 chance that she is in a Monday OR Tuesday of a Tails Universe. The problem is only confusing because we neglect the Tuesday of the Heads universe, which has a 1/4 chance that has been eliminated by the NEW INFORMATION Sleeping Beauty got upon waking...namely, that she is still IN the experiment. Using this new info, she is right to say that the odds the coin landed heads is 1/3, even though in the past, before she had this info, before the coin was flipped, the odds WERE 50/50.
Think more when it's impossible for me to share you with others...
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Think more when it's impossible for you to share me with others...
Phew! I thought this was going to be about the morality of kissing an unconscious woman with no consent. 😂
The original version of the fairytale was way worse. I don't even know if the comment filters allow me to mention it at all.
In the original, she was more than kissed. She gave birth and was woken by the nursing babies. (twins)
this is exactly where my head was at.
"I want a third pill!" Zizek is thirder confirmed.
Zizek is a walking third pill, a force of nature 😂😂
😂
Think more being corporate means being calm and quiet...
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Think more being justified means follow the clock's timing
This girl is a beauty.
isn't it irrelevant how many times she has woken up? in either case, there was only 1 flip so 1/2 always?
I agree. If the question is the probability of heads then it should be 1/2 because that is based on the result of the coin toss. If the question is what is the probability that she wakes up on Monday and the coin toss was heads then it would be 1/3.
Think more the hardest way as A is for Questions
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Think more the hardest way as R is for Reference
The girl in the pic: I'm not sure, Dr. Cosby. I only remember something like this.
Was a Jello Pudding pop not a coin.
I am confused why nearly nobody (With the exception of Louis Wilbur here in the comments) does get the only real and correct answer: The problem is ill defined and does not have a correct answer. Anybody who claims it is either 1/2 or 1/3 makes additional implicit assumptions about the experiment/problem.
In the original formulation the sample space is ill defined and relies on the outcome of the coin toss in question. This cannot be modeled using probability theory and it also does not have a "real physical model". The Sleeping Beauty Problem cannot be turned into a real experiment.
While there might be more complete answers than the answer given above, the answer clearly is neither 1/2 nor 1/3 since classical probabilities are not and cannot defined in the context of the Sleeping Beauty Problem.
It is easy to turn the implicit assumptions (people make) into explicit rules and turn the problem into a well defined one. Then the answer is typically 1/2 or 1/3 depending on the exact assumptions/rules.
Mathematics (and logic) are full of such subtleties, where suddenly additional assumptions from metaphysics/philosophy might enter the picture and confuse people. But mathematicians have become extremely good to to find those and make them explicit. Just look up "Skolem's paradox", Banach-Tarski paradox or reverse mathematics to get an idea how rigorous and formal logical thinking has to be to stay consistent in some cases.
Jeffrey Rosenthal:
"Despite these caveats, I hope and believe that there is merit in providing a simple, short, direct argument that 1/3 is the correct answer, using solid mathematical foundations with few assumptions and little philosophical ambiguity."
Jeffrey seems to think, for whatever reason, that the question has an answer outside of mathematics (which I do not believe), but he is clearly aware that there is no purely mathematical solution.I disagree. Probability is a (purely) mathematical concept. Either there is a purely mathematical solution and a proof or there is no answer to the question.
Consider this: P(M) =Monday = 2/3; P(H) = Heads = 1/2. We can write P(H) = P(H|M)P(M) / P(M|H) = P(H|M) * 2/3 / 1. Now things get tricky. If we accept that P(H) is clearly 1/2, then P(H|M) = 3/4. But then, if we are restricting our probability space to only Monday, there are two possibilities, one of which has a Heads, so 3/4 feels just a wrong result. Hence P(H|M) = 1/2 and P(H) = 1/3.
Hey guys, if you want to discuss this video (and others) more broadly, or if you just want a place to discuss rationality in general, check out our subreddit at www.reddit.com/r/rationallyspeaking
Hope to see you there!
If I flip a coin on Sunday what difference does it make if I ask you on Monday or Tuesday "was it heads or tails?"
Think more the hardest way as I is for Illustration, Illusion, Investment, Intuitive, Inheritance, Income, Intern, Interrupt, Interest, Insecure etc
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Think more the harder way as 10 words that starts with I
I'm a 1/2-er. It seems to me that the 1/3 answer is predicated on a specious desire to get the answer right as many times as possible. Since SB has more chances to get it right by saying "tails", she is better of from that point of view picking tails. So the double tails awakening is amplifying the apparent value of the tails answer.
If you gave me evens on heads but 2:1 on on tails, I'm going to pick tails every time, thank you very much! But that's not because tails is more likely, just that you have increased the payout for tails. That's exactly what SB is being offered.
I think the logical flaw is similar to the "we are almost certainly living in a simulation" argument. Asking a question multiple times does not change the probability of an answer being correct.
Phil Hibbs I agree 100%. If Beauty is offered $100 for every time she guesses correctly whether it's heads or tails, she'll get double the money for the correct tail answer (assuming they ask her question every day). But as you say, that doesn't alter the probability of the coin falling heads in the first place.
@@robsmith2047 But that's not what the question is. The question is not "What is the probability of heads?" but "What is the probability I'm in a situation where heads came up?" In the first case you don't need anthropic reasoning, in the second you do.
Ha! You were wrong. We are in a simulation. Asking a question multiple times does change the probability of an answer being correct. ;D
Surely a premise to this scenario is that we're using a 50/50 coin, right? So we already know that SB's got 1/2 chance of being on one probabilistic arm, and 1/2 chance on the other. But to work out which stage she might be at requires the attribution of probabilistic weights to each stage.
It's fallacious to automatically attribute equal weight to 'Heads', 'Tails, woken once', 'Tails, woken twice', 'Tails woken thrice', etc. If we know the scenario uses a 50/50 coin then branch A demands 1/2 probability; and branch B 1/2 probability. That remaining 1/2 is split between the remaining stages of being awoken and amnesiatised. It would be possible for SB to work out the chances she'd been subject to a Head or Tail, if she knew the number of times she'd been sent back to sleep and amnesiatised. But if she hadn't been told that number, or if it hadn't been decided beforehand, she couldn't really know. The experimenter could know the probabilities however (not that they'd need them), but she couldn't.
E.g. if she knew she'd be amnesiatised three times in total, then that means when she wakes up she knows that there's a 50% chance of Heads, 12.5% Tails first time, 12.5% Tails second time (amnesiatised once), 12.5% Tails third time (amnesiatised twice), and 12.5% Tails fourth time (amnesiatised thrice). Which means she has a 1 in 8 chance of being the subject of each Tails toss. That's a 5/8 chance of it being Monday, and a 1/8 chance each of it being Tuesday, Wednesday or Thursday. Which means 'winning the money' means she should definitely go with 'Monday'.
I imagine that if mathematicians disagree on the 'answer' to this scenario, then it is because they disagree on its definition - not because it doesn't have an answer, in any sufficiently well-defined form.
notsyort It seems to be that it is philosophers, not mathematicians, who are arguing about this. At least those mentioned in the Wikipedia article on the subject are philosophers, as far as I can tell.
notsyort " If we know the scenario uses a 50/50 coin then branch A demands 1/2 probability"
Not if you're sampling more from one branch than the other. The process isn't just the flip, it's the flip plus the obviously biased sampling from the flips.
buybuydandavis
It's not possible, insofar as Julia's defined it, for either branch to be 'sampled' less than every time. Are you saying Sleeping Beauty dies sometimes? Or that the experimenter ignores flips sometimes, and flips again? And what difference would that make, anyway? We can't get around the 50/50 thing, unless we introduce extra elements into the scenario.
Attempting to breach the known 50/50 limitation sounds very much more like the kind of faux-Philosophy that Michael thinks this scenario belongs to, than genuine Mathematics. Summing 'awakenesses', for example, is simply meaningless.
notsyort I thought the point about having SB guess what the outcome was, brought home the point about multiple 'samples' on one branch quite well. Given 2000 runs of the experiment, SB would be right around 1000 times if she guessed heads when she woke up, but she would wake up around 3000 times.
embargokong But SB is not asked to guess the outcome. She is asked what the probability of heads was. Different questions.
Think more the Hardest way as *• Break the Ice:* which implies to break down social formality and stiffness..
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Think more the Hardest way as *•By hook or by crook:* which means by whatever means necessary - be they fair or foul.
The question is "What do you think is the probability that the coin came up heads?" The act of polling the result (asking sleeping beauty about the probability) doesn't affect the initial event. Meaning that regardless of how many times she is put to sleep, woken, and asked the coin was still only flipped once, and therefore there are still only two outcomes -> Heads or Tails. Even by her own perception, in this scenario, there are only two final outcomes: She wakes up and its heads or she wakes up and its tails. It's the same reason why regardless of how many times you get heads in a row, the probability of another heads is still 50%. The probability is tied to the event, not the perception of the event. It seems like the halve-nots are misinterpreting the question and falling into the gamblers fallacy. A question that would produce the result 1/3 would have to be phrased, "What do you think is the probability that you have woken up after the coin came up heads?" This changes the event being polled from the coin coming up heads, with a 1/2 probability, to the event of waking up after heads, which has a 1/3 probability.
I feel like a thorough understanding of the Monty Hall problem would help you a lot
@@wassupusa I understand the Monty Hall problem. Even if you are looking at it as the three possible outcomes of {Mo,H}, {Mo,T},{Tu,T}; The case of {Tu, T} only exists on Tuesday, and SB knows she is more likely to be woken on a Monday. This updates her three outcomes to {Mo, H, 1/2}, {Mo, T, 1/2}, and {Tu, T, 1/3}. There is only one case where the correct answer is 1/3, where there are two cases that have the correct answer of 1/2. So the answer will always result in 1/2 as it will always be the answer that is most likely to be correct.
you're the waking beauty thats for sure
SIMP
Think more the quickest Intuitively way as P is for *Programming Language* again
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Think more the quickest Intuitively way as P is for *Pythagoras* again
The two answers differ in perspective, whether they admit it or not.
The 50% perspective is from the coin's perspective.
The 33% perspective is from Sleeping Beauty's perspective.
The 50% perspective isn't the coin's perspective, it's the world perspective. I would call it an objective truth The 33% perspective is subjective because it's based on Sleeping Beauty's feeling sort of. The fact that she could feel the same waking up in 3 different scenarios dosen't alter the fact that in the world the coin still has a 50% chance to have ended up on one side or the other.
@@ozymandias2726 you used more words to repeat what I said... good job.
@@ChristnThms I was more specific than you were.
@@ozymandias2726 no. You used more words. Good job.
@@ChristnThms You do you and I do me and we're all going to be fine. Granted I don't fear using words.
Looks like the monty hall problem when looking for the first time, the amnesia part make it more of a phylosophical problem.
I am 5 years in the future and, while watching it, also thought of monty hall problem.
@@botcontador3286 The monty hall problem is unambigous. In this problem, the contradicting facts that the coin toss is fair AND Tuesday has to be tails are making it so that there is no correct answer.
@@olavbakke2889 Interesting! What's the contradiction? What part of the setup would be impossible to carry out?
@@busTedOaS The fact that the experiment is implied to terminate on Tuesday. This implies that the coin flip is fixed to fall on tails on Tuesday. This alters the probability of the coin flip being heads to a third.
However this is contradictory since the coin flip is already stated to be fair. You can't have it both ways.
@@olavbakke2889 What's the problem it being Head+Tuesday, are you saying that's an impossible situation? My understand is you just wouldn't wake the subject.
i think it would depend on when the question is asked, this seems to have not been stated. if the question is asked only on heads and the last time she would wake up the it would be 1/2. but if the question is asked every time she is woken up, then it is 1/wake-ups. in either case the answer is 1/times-question-asked, which seems right as that would follow P(A)=A/(Sample-Size).
The question could be reformulated to : "can you guess if it was Heads or tails?"
In wich case you will win 66% of the time if you ALLWAYS choose Tails.
Only if each individual waking event ie Monday heads, Tuesday heads and Tuesday tails are equally likely ie 1/3. And they aren’t. Because of the amnesiac, Monday tails and Tuesday tails both have probability 0.25 while heads has 0.5.
Sorry I meant Monday heads, Monday tails and Tuesday tails
@@nandc2009 good point, but then again it comes down to what the question is exactly. what you're suggesting is correct if the question is "how likely is it to get Tails\Heads" and not "wich one was is in THIS particular flip".
@@hwiz8282 I think the question is 'How likely was heads in this particular flip - ie what is the credence/probability that the coin landed heads'? I think tails comes up 50% of the time, and 50% of the times she wakes up when it has been tails is on Monday and another 50% on Tuesday, so both those waking events are 0.25 probability.
I think the real problem is in the definition of the problem; particularly, what is the precise question that SB has to answer. If the question is equivalent to asking what the probability of a fair coin coming up heads was, then the answer has to be 0.5; there is no way that the number of subsequent awakenings can affect that.
The thirders, though, seem to be answering a different question, but I'm not exactly sure what it is. It appears to somehow involve the number of awakenings, and the assumption that the probability of this one being any particular awakening is equally likely and independent. But that assumption is not correct. There will be a second awakening only half the time, and it will only happen after there has already been a first awakening.
Consider how a computer simulation of this problem could be done. For each trial, the computer would use a pseudo-random-number generator and store the results, heads or tails. Then it could put itself to sleep and wake itself up according to the rules of the problem. At the end of each trial, it would return the results of the PRNG. Finally, after all the trials are turn, it would report appropriate statistics. Obviously, assuming no bugs in the program or operating system or hardware, nothing that happens after the PRNG is run can change its result, regardless of whether the program sleeps once, twice, or a million times.
Think more the hardest way as why we are acquainted with various types of oral Communications right from our childhood, like talking with friends, quarrel over pitty issues, discuss current affairs, narrates stories to children where we are involved in a process known as Oral communication and for such Communication that we do not need any form of training...
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Think more the hardest way as why Oral communications are in the form of staff meetings, presentations, debates, speeches, discussion, telephonic chats, classroom lectures or simple informal conversations
I hated probability in school....I'm sure this won't aggravate me though...I'm older and more patient now. EDIT: *Fuck*...I really wanted to get sleep tonight. sigh.
My initial reaction is that both choices are right given the right framing of the question and that this isn't a paradox. When asked what is the probable outcome of a fair coin toss, we all agree the outcome is 1/2. That's pure probability. But Sleeping Beauty is waking up with *evidence.* There's more information here. Waking up makes it not the same puzzle. All that matters is that the question is different. "What was the likelihood this fair coin tossed at some point was heads?" is still 1/2. "What is the likelyhood you were woken up by a heads coin toss?" is a question including more evidence. It's just a different scenario. Both answers are right given the framing/context, fundamentally because they are different questions.
How is there any more evidence or additional information? She already knows she will wake up before the coin is flipped
Think more the Hardest way as E is for Ethical
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Think more the Hardest way as F is for Fundamental
All I see is an awaken beauty!
anyone here cuz of Zero Time Dilemma? xD
yup, just got here xd
holy shit ... this shit is deep
Think more the quickest intuitively way as G is for *Gideon* again
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Think more the quickest intuitively way as G is for *Genesis* again
Silly people: "My answer is right!" "No, mine is!"
Smart people: "My interpretation of the question is right!" "No, mine is!"
🤨
Not really. The answer is in the question. This is not a problem, it's a riddle and who doesn't love riddles? If the question is; "what is the probability that the coin came up heads. " Well single toss, 2 outcomes, 1/2 is the correct answer. If instead the question is; "what is the possibility that sleeping beauty woke up to the coin came up heads;" then the answer is 1/3 as she woke up twice to tails and once to heads.
The only problem is the ambiguity of the question. Is the header part of the question or not? Also note; that it is ok for riddles to have multiple answers.