To try everything Brilliant has to offer-free-for a full 30 days, visit brilliant.org/AnotherRoof You’ll also get 20% off an annual premium subscription. ⬣ *CORRECTIONS, CLARIFICATIONS, AND COMMON COMMENTS* ⬣ SURVEY forms.gle/EtfqTEP4dxJhZiT19 1. 18:45 a remark about the Bayesian argument used in [1]. This methodology aims to not only prove that the probability of the coin toss is greater than 0.5, but provide evidence for the fact that the probability is 0.508 as predicted by Diaconis et al. in [4]. Explaining this in detail is beyond the scope of this video, where I merely wanted to present convincing reasons why the coin tosses collected likely had p > 0.5. 2. It is likely that the odd-numbered house fact only applies in the UK or in countries with similar numbering conventions. I should have stated this in the video! James Grime made a video about it back in the day: ruclips.net/video/wydlZ9lcEiQ/видео.html 3. At 12:12 I say "520 out of 1000" but the caption says "out of 100". Sorry! 4. There is a typo at 17:20. It's 350,757 flips, not 350,737. I checked those numbers *so many times* so can't believe I didn't spot that >_
The 'fair coin flip' method I was taught is to have both participants simultaneously flip a coin, with each player choosing matches/mismatches instead of heads/tails. Even if both players are cheating and using one sided coins or whatever, unless they have knowledge of how the other will cheat, it still ends up being 50:50 in practice.
That is a failure of measurement! There will always be bias! Question is... How many decimals does it take. Why don't they make a perfect flipping machine in a vacuum to do the flipping. Do two sets. One starting heads. One starting tails. A million flips total or a piece. Coin used are changed to brand new ones every 1,000 flips. The flipping machine has to mimic a human thumb, and arm. Good luck making a perfect repeatable machine to do that. Seriously no one has made a flipping machine? Surely a gambler has to try to get some advantage.
@dianapennepacker6854 if you broaden to talking about random number generation in general, Alpha Phoenix made a muon-powered random number generator. That's about as non-biased as is quantumly possible
A previous Ignobel winner, Bert Tolkamp, works in the same building that I do. His team showed that cows that are lying down are more likely to stand up the longer they've been lying, but you can't predict when a cow that is standing up will lie down based on how long it's been standing already. (Statistically, this means that the wait time before lying down is exponentially distributed, but the wait time before standing up is not.)
@@Noname-67 it means cows have memory, in contrast to radioactive atoms, where if you have a U238 atom from the 1st pop-III (iirc) stars made 12 Billion years ago, and I have one made in a reactor this morning..."which one will decay 1st?" is truly a coin-toss.
Fun video! Regarding your request to viewers to toss 10 coins: there's a risk that people will not report boring results, such as 5-5. But one out of 512 people trying this will get 10 heads or 10 tails, and they're very likely to report it. In short, there's a strong risk of reporting bias.
With a US quarter, I just did 24 flips, I started on heads for half and tails for the other half, all my flips have that satisfying ring to them. I started on heads where 11 of the 12 flips landed back on heads. Then tails, 9 of the 12 stayed on tails. Maybe I am closer to a robot with my style or that was really weird.
Actually, it's debatable as to whether comments help videos in the algorithm; the key factors are click-through rate and watch time. The rough consensus is that comments don't matter and I agree. Anyway, I asked viewers to fill in a survey -- links are in the description and in the pinned comment. If viewers want to comment their results, that's their choice.
In high school, there was a period where people were gambling in various ways with quarters. When it came to coin flips, a few of us knew how to legitimately (not using the magic trick) cleanly flip them rather consistently, so we knew what we'd get based on which side was facing up. The funniest thing is that when our math class had everyone perform 100 coin flips for a probability lesson, two of us tried to find ways to get more accurate "random" results. (We didn't want to stand out with results that were "too perfect", but didn't know how "off" would look best.)
I’m a bad roller AND I used to guess at night which of 2 keys went in my door. They looked identical at first and it turns out I’d guess wrong 96% of the time out of 600 tries over a 3 year period. The neighbors often hear me quietly cussing trying to get in my house.
@@bryanbryan2968 This wasn't guessing. It was maintaining a consistency in how you flipped the coin, to the point that like a machine flipper, the result of the flip was determined by which side of the coin was face up before the flip. We didn't have the 100% precision of a machine flipper, but we were maybe at around 75%?
@@BainesMkII Oh yeah. Definitely fixing the results which, for what we hope, will be random results, but are not. ‘Ideally’, you should get about 50/50 as mentioned in the last part of the video, as I recall. It could even be subconscious, as in my case, and presumably more conscious in yours. Interesting test showing how things are not random, though we were always taught a coin flip seems like it is.
I remember doing this in middle school (86-87). The trick was to pull your flip and let it land on the ground. Convince those who are gambling that letting it land on the ground avoids any magic or slight of hand. Basically rapidly lifting your hand up but the coin doesn't flip above the hand. The result was it landed on the starting face nearly every flip.
"This was originally just a joke but then I got bored" is the source of almost all of my favorite video essays wildly overanalyzing things like video games and whatever. Don't you dare ever suppress that prediction!
@12:30 It's worth noting that "i.e., a 0.1 chance that the coin is in fact fair" is a very common but _completely incorrect_ interpretation of p-values. We have concluded that P(520 of 1000 successes | fair coin) = 0.1 (i.e., "probability of 520 of 1000 successes under the assumption that the coin is fair is 0.1"). This is very distinct from the statement that P(fair coin | 520 of 1000 successes) = 0.1 (i.e., "probability of the coin being fair given the observation of 520 of 1000 successes is 0.1"), which is the "i.e." statement at the bottom. In general, P(A | B) and P(B | A) are two arbitrarily different quantities (related by something called Bayes' Theorem, which in general says P(A | B) = P(B | A) * P(A) / P(B)). Imagine a circumstance where we have 10000 coins. One is magically rigged to always land on heads, and the rest are fair, and they're otherwise indistinguishable and mixed uniformly together. You pick a coin and flip it 10 times and it lands on all heads. The probability that the coin you picked is rigged is just ~0.1, not ~0.999 (which is 1 minus the "p-value" here for rejecting the null-hypothesis that "the coin you picked is fair") (I am rounding 1024 to 1000 and also rounding 1/9999 to 1/10000 and I think maybe doing some other rounding). The struck-through part below is incorrect (kept for context of replies), it's actually just impossible to interpret p-values in a mathematically useful way and p-values are stupid. See mjeffery's reply below. -A result with a given p-value is -_-some updated to your belief about the likelihood.-_- The p-value tells you how much to update it by (so we got from 1 in 10000 odds of the null-hypothesis being false to 1 in 10 odds, a 1000x improvement from our p-value of 0.001), but if the thing being tested is astronomically unlikely to begin with, this hardly lets you be confident.- -Of course, in this case, we know exactly the prior probability of the null-hypothesis, which isn't usually true, making interpreting p-values in the real world much harder. But in general, when you see a study with a p-value of 0.05, the correct interpretation is not "this is true with probability 95%", it's -*-"this is 20 times more likely to be true than I thought it was."-*- If the study is "some random food cures cancer"... well, there's a whole lot of foods and most of them probably don't cure cancer.- Footnote: the easy way to do the math here is in terms of odds. We started with 1:9999 odds of rigged:fair. The likelihood of 1/1024 tells us that we have an odds-ratio of 1024. This tells us that we have 1024:9999 posterior odds of a rigged coin. (This is exactly equivalent to Bayes theorem, where you would have P(rigged | 10 heads) = P(10 heads | rigged) * P(rigged) / P(10 heads) = P(10 heads | rigged) * P(rigged) / (P(10 heads | rigged) * P(rigged) + P(10 heads | fair) * P(fair)) = 1 * 0.0001 / (1 * 0.0001 + 1/1024 * 0.9999) = 1024/(9999 + 1024)
Useful comment, but I was initially confused what statement you were responding to, since you didn't quote it in your reply. (Maybe timestamps can be a bit finnicky.)
This is mistaking p-values with Bayes Factors - a p-value is always relative to some base (null) hypothesis, and is defined as the likelihood that the observed results are obtained if the null hypothesis is true. You’re correct about the odds calculations, and that a common and incorrect interpretation of a .05 p-value is as a likelihood of untruth. The idea that you can use the p-value to weight your belief in a particular model also assumes that the space of possible models is both finite and known, which is an ongoing question in statistics. Source: I’m a predominately Bayesian biostatistician who taught mixed research methods and statistics.
@@ForsakenDAemon trying to get my head around all of this - Mr Roof has mistakenly said that we can use the measured result to calculate the likelihood that the null hypothesis is true/false. Yaksher is unknowningly describing a Bayesian approach which says that p-values are used to update our beliefs. So both are mistaken. What is the correct interpretation?
@@parkerstroh6586 The first paragraph from Forsaken's comment tells you how to interpret a p-value. I'm not following exactly what people are responding to otherwise. (Seems I missed the Bayesian part of the original comment as I was focused on figuring out the original mistake.) Spitballing, it seems people have added in extra context to try to explain a concept, but that context isn't exactly right. (Which I've noticed happens with statistics decently often since a lot of terms are very particular in how they work.)
If I'm not mistaken, the slightly greater probability of an odd numbered house only holds in places where houses are numbered sequentially starting on one side of the street, that is where houses 1 and 2 are adjacent. In places like Canada and the US, odd numbered houses are on one side of the street, and even numbered houses are on the opposite side, so we would except the probabilities to be equal for a sufficiently large number of houses.
And in Australia (or, Queensland at least; not sure about other states), houses are numbered by their theoretical standard allotment facing, so if a block is wider than that the road numbers will skip (this is to future-proof road numbers from potential subdivisions), and if your block is _narrower_ you get A, B, C, D, etc (because the future-proofing doesn't always work). If you have multiple residences within the same building on a single allotment, then it's 'Unit 1', 'Unit 2', etc.
So you can have a road where the numbers go 1, 2, 3, 4, 6, 7, 8A, 8B, 9 (Units 1-20), 10, 12, 14, 15. Odds and evens are on opposite sides of the road here, so in this example you've got a very dense road frontage on the even side, and a sparse development on the odd side that has a large block of units built on one of the properties. It's not at all uncommon to see this; it can make finding people's houses a pain if they've told you the street number and haven't given any landmarks to find the place by.
When I flip a coin, I can make it land 'starting side up' about 80% of the time. For years I just though I had some weird talent which allowed me to win any coin toss.... but no... science. Thanks science (grumble...)
Well TBF if you're both calling and flipping then the other person just deserved to lose. The sneaky way to do it is tell them to call it as you're showing the side facing up. A lot of people simply pick that side then you flip it onto the back of your other hand. If they call the bottom, it's on you to decide to just go for it or if they'll complain about not flipping it over after the catch.
14:14 Here's my guess as a statistical practitioner. When I have to do this kind of "power analysis" I usually just program a simulation and try different values until I find the sample size needed to reject the null at some given effect size.
great video, great explanation, great everything! i expected you to have millions of subs from the quality of this one, here’s to getting you one step closer 🙏
As a data scientist, my insight into why they chose that number of flips is that nobody does power analyses before studies like they're supposed to do.
@@SoftBreadSoft No it's not, it's just a hot-job-field-degree. Actual academic disciplines are math, computer science, physics, statistics. Data Science is just "here's the bare minimum of t-tests you need to know so you can import sklearn and plug things into it wildly".
@@alessandrorossi1294 It is. There are researchers producing theories and models, and professors teaching, i.e. it is an academic field. like the other fellow said it does depend on the college whether they focus on the theory/modeling side of things or the applied engineering side of things, which yeah those curriculums could/should be a specialization of computer/software engineering.
You need a mechanical flipper to test this. Build it with a randomised starting energy and only count flips which had a minimum number of flips before landing.
i've done 10 flips with each of the six types of australian coin, including the dodecagon that is the 50 cent coin! this is the real research that needed to be done
14:07 Besides deciding the p value you want to consider a result significant, you also have to decide on the power of the test. The power is the ability of the test to find a difference using your data, if in fact one exists. The closer your two values are together, the more samples you need to be able to find that difference. Statistics programs can give you want sample size you need to have a particular power. In this case, if your significance level is at 3 sigma (~p of 0.001), and you want about a 99% power to find a difference between 0.5 and 0.51, you need about 250,000 coin flips
There is still always the chance a coin flip ends up on it's edge. I witnessed it once on a hard floor. The coin landed and bounced onto it's side and slowly rolled into a wall where it stopped and stayed on edge. The odds are extremely low, but I can say they aren't zero.
The most funny thing about stochastics is, that it will never will tell you anything about individuals, other than how atypical they are. This includes individuals rolling dice or flipping coins. So technically the distribution of how well specific individuals fit the expectation in their results is in itself a bell curve. Which means that people exist that can't roll dice for shit. Their whole life. It's not particularly LIKELY, but that doesn't mean they don't.
Before watching this I’m going to guess that the actual physical construction of the coin will matter quite a bit. A US Quarter may be weighted slightly to one side as compared to a Pound Sterling or Penny Edit: not even ten minutes in and I was wrong lol
Problem: The researchers all operated by flipping and CATCHING the coins, ending the rotation immediately and confirming that the coin spends more time on its starting side than the other side. However, coin flips are not always caught - they are often dropped, or allowed to land naturally, where they bounce and further randomize.
Before watching the video, the issue I have with studying actual coin flips is that they technically aren't random at all. If we were capable of understanding every single thing about the coin flip, from the force applied to the coin the angle of the flip, distance from the ground, any interference in the room etc. then we can calculate where the coin would land 100% of the time. It's a bit like how a roulette wheel isn't technically random. For all intents and purposes it is since people can't predict where it will land, but there absolutely is a way to determine it, it's just not something we're capable of doing at the moment. So the issue with having several hundred people flip a bunch of coins is that all these minute factors that impact a coin flip will change very slightly from person to person. They each aren't in control of the flips enough to dictate exactly where it will land, but the flip itself is still impacted by these things. This is why I don't like the term "fair coin flip", because it's not really fair, it's just that we're incapable of accurately measuring all the things that impact the flip from the outset.
@@RAGEAlanBun Without reading your whole comment, the problem with pointing out that coin flips aren't random is that this is addressed in the video and the study is very much about what happens when people flip coins.
Take-away: calculating the probability of heads/tails requires defining the path from initial orientation through rotational vectors, and accounting for resistance or modifiers like temperature, pressure, air currents, etc.
@ that is true, but consider the perception of the coin - it balances out and you can see how “fair” is the spin by the radius of the circle it makes. Then the chance is down to when it loses the energy and drops on the table.
Spinned coins should then be biased to weight I’d assume. Probably the heavier face of the coin would end up pulled down by gravity more times that the other?
technically speaking, even a flip isnt a perfect 50/50 without human intervention. It is incredibly rare outside of the contexts where the outcome is orchestrated, but landing on the edge is always a remote possibility. It comes from the nature of being a 3d object: 3 dimensions means a minimum of 3 "faces", attributed directly to the x, y, and z coordinates. cylinders might just be the only 3d shape with exactly 3 faces simply because the rounded shape is a bit difficult to properly define, so i am just simplifying it into a single face. By that same logic, a coin already balanced on its edge to produce a spin is even more likely to remain on that edge depending on the energy added into the mechanical system of spinning the coin. Just like flipping a coin, it is human error that forces the coin to pick a side. The difference is that it seems much easier to rig a spin in your own favor compared to a coin flip. Even acknowledged in this study is the idea that isolating variables for a coin flip to properly test is practically impossible, simply by nature of the comparisons to "perfect flipping" machines that can orchestrate the desired outcome with 100% accuracy. But spinning a coin? Comparatively, the action requires a lot more attention, all while being much easier to hide an intentional tilt in the coin in the initial action to start the spin. Add in the direction of spin and rigging the action seems much more plausible than the mechanisms out of a coin flipper's control like involuntary micro spasms or the force of the initial liftoff. Just look at the overemphasized example of a wobbly coin flip, it starts to look an awful lot like the last few moments of spinning a coin on a table, where the outcome is already determined it is just a matter of waiting for it to slow enough to identify it.
Randomness often can come from outside. An electronic dice doesn't need to be internally random. It can simply be fast so that a human can't reliably stop it on any particular number.
The fun thing is that statistically, a coin has three possible results. Because it can also land on the rim. Now the chance to land on that side is extremely small, but not zero. But fun to see that coin flips are just like toast falling off the table. It's all a result of time, distance, and speed.
Worth mentioning that coins are not symmetrical. The embossing on the heads is not the same pattern, nor does it have the same amount or weight of embossed material as the tails side. So why would you expect the results to be 50/50:in the first place?
I once read a paper which concluded that with American coins, heads is slightly more probably than tails. They concluded that the results are due to the different style of markings that the heads side of the coin has from the tails side of the coin.
For 13:20: Sq.rt.(250,000)/2 = 250 per stdev Fair coin = 125,000 125,000 + 250 = 127,500 125,250 / 250,000 = 0.501 So at n = 250,000, you can claim precision to +-0.001 with 68% confidence. It's kind of arbitrary, but I think it makes sense to suggest 250,000 flips to get to 0.001 precision with ~68% confidence.
12:30 The last line is wrong: a 10% chance that the null hypothesis leads to some result does not mean that there's a 10% chance that the null hypothesis is true given this result. This is a simple case of conditional probabilities. Let A be the event that we get a result this significant, and let B be the event that the null hypothesis is true. What we have worked out is that P(A|B) = 0.1 (the probability of this result given the null hypothesis is true is 0.1), but you're saying that this means that P(B|A) = 0.1 (the probability of the null hypothesis being true given this result is 0.1) as well. This is not the case. Bayes' theorem states that P(B|A) = P(A|B) P(B)/P(A). So the actual value also depends on the unconditional probabilities of the null hypothesis being true, and of getting a result of this significance (specifically, the ratio between these). Of course, we don't really know these values, so that's why we don't usually claim anything about the chance of the null hypothesis being true given this result (we basically just don't know with this information). This sort of mistake with switching round conditional probabilities can lead to real problems. For example, it's the difference between the probability that some medical test comes back positive given that you have a disease (which we certainly want to be very close to a 100% chance), and the probability that you have this disease given that the test comes back positive (which is what you actually want to know, but could be very small if the unconditional probability of having the disease is tiny, and there is any chance of a false positive).
I'm not sure exactly how the authors calculated a sample size of 250,000. One thing you need to take into account is also the power of the test (i.e., the probability of rejecting the null given that the alternative of .5027 is correct). A common value for power is .80. Using power = .80, alpha = .05 and alternative proportion of .5027, the number I obtain is 212,019. See the R code below: library(pwr) power
Interesting. What is the power of test measuring in terms of the probability that the null hypothesis would be rejected in favor of the alternative? (Statistics isn’t my strong suit, have never heard of this before)
@@photoniccannon2117, good question. Power is a probability that you will reject the null hypothesis (e.g., the coin is fair) given that the alternative hypothesis (e.g., the probability of heads is .55). Power depends on three quantities: the assumed effect size (e.g., how different is the assumed probability of heads from fair?), the number of samples, and the statistical significance level (typically set to .05, which is a 5% chance of rejecting the null hypothesis given that it is true). The basic relationships are intuitive. An effect that is large (e.g., 80% heads) is easier to identify than one that is small (e.g., 51% heads), the more samples you collect, the easier it is to find a difference, and increasing alpha makes it easier to identify an effect (but also find a false effect). RUclips sometimes removes comments with links. So I recommend seaching "rpsychologist statistical power" to find a visual illustration using a continuous variable as opposed to a discrete count. If you select "solve for Power", you can change the significance level, sample size, and effect size to see how it changes power .
I'm not a rocket surgeon, so take this with a corn of pepper, but, isn't the reason it doesn't matter what side it started on have a lot to do with the fact that at the very start the first side is already "half flipped"... ie- a flipped coin (usually) starts parallel to the ground. So, it only takes a quarter-rotation for it not to be on top any more, and then the second side gets a full half-rotation before giving up the top. From then on they trade full half-rotations on top. So, in your chalk diagram at 0:50 the very first pink bar at the top left should be half the height of all the other bars that follow (pink and blue). Which begs the question... if you start the coin standing straight up and down so the first side gets a full-half rotation before not being on top, THEN does the first side have a slightly higher probability? It must, right?
Methinks the protocol may entail a possibly-cumulative but easily-removed bias, namely, start half the tosses on heads. I used a 1922 Peace dollar weighing 26.63 grams which is easy for me to handle and, being silver, provides a pleasant "ching" upon launch and some Doppler effect from the spinning flight, with a muffled thud when caught. 10 tosses starting heads -- 2 heads and 8 tails, surprising 10 tosses starting tails -- 6 heads and 4 tails Bonus: 100 tosses starting 50 heads and 50 tails, results exactly 50/50. Keeping track of the individual trials might be interesting as a follow-on. Good vid, quirky edutainment at its best. P.S. I should've read the description first about the survey. I'll go do the survey but leaving my comment anyway.
I have a biased for opposite of start when catching, mostly due to I trained some time ago to do that. I am more 50/50 when landing on the ground or table. For the number I got out of 10 landing in hand: it was 9 opposite 1 starting.
Congrats on the Ig-Nobel :-) I have to also let you know that I live on a strange road (I cannot use the word 'odd') where all eight house numbers are even!
I did 10 flips with my right hand using a US penny from 2022 with a shield on the tails side. I started with heads and maintained the same face from previous flips. Started with H, then THTTHHHHHH. 3 tails, 7 heads 6 same-sides results out of 10
The Sides each Have very small different mass due to different images. The one with the larger mass will theoretically. End up on the down side and the smaller mass on top, which is side that wins.
Biostats instructor here... I am just itching to see the heads and tails tallies... Gonna plug those shits into a Chi-square test so quick... Edit( 18:16 ): X-squared = 83.104, df = 1, p-value < 2.2e-16, reject the null of 50/50 chance.
I did the survey on an Australian 20 cent and got HTHTHTHTHT starting on heads, so that’s only 1 of ten flips that ended on the same side as they started. Interesting. This clearly means that either there is actually a less likely chance that you flip a coin and it lands on the same side and all the maths is wrong, or that my Australian money is skewed. Clearly with such a high sample size of ten one of these is the case and not simply luck. Seriously though cool video!
Got to explain the angle, if it is as I think, then if force coin flip to hit way more closer to the edge would make way less biased. And yes I might later but not now look at it myself. And what did the paper say about this? This should be a huge point in the paper. As well as tilt hand rotated from from thumb as well.
Its because of flippers. Wr have a tendancy to try and do things the same way. The energy put in dictates rotation. I can consistently flip a coin to either heads or tails after just a few flips to calibrate.
I can say it's harder to get good at not dropping the coin than it is encouraging the outcome. I can convincingly flip a quarter 10 times and be pretty confident in it landing heads nearly every time. I'm much less confident that it won't land awkwardly then roll over or simply bounce off. For whatever reason, a lot of people calling the side get fooled into picking the one they can see and then lose when the person flips it onto their other hand. If anyone volunteers to flip for something, the safer bet is to just tell them to let it fall to the floor.
Did they consider the average force applied to flip the coin and the height at which the coin drops relative to the height of the hand that flipped it?
Is it 50-50 whether a street ends in an odd or even house? Under some assumptions is this the same thing as how a random number is more likely to start in a 1 than a 9?
Two possibilities: 1. The flipper getting one result and inadvertently putting the wrong answer down. I know I would be a bad candidate for the coin test for that reason. 2. The roughness of one side vs a smother side may affect the aerodynamics of the coin.
Doing my part for science: Used a 2 Belgian euro coin. Started with heads side up, keeping the result for the next flip as discussed in the video. Sequence: HHHHTHHHTH Same side landings: 6, different side landings: 4 Self-report on flipping technique: very poor. Dropped coin multiple times, on one occasion (flip 3 or 4), I believe the coin didn't actually flip at all. EDIT: I forgot there was a form, and have now filled that in as well with these results.
@Kounomura that's true. Landing on its edge would also make it not 50/50 as there are now 3 states: head, tails and edge. So the paradigm 50/50 is actually invalid. Good point.
Well the more coins you flip, the less likely it becomes for the results to be exactly 50/50 (only considering even numbers of flips because it’s impossible for odd numbers). But as it gets less likely for it to be exactly 50/50, it becomes more likely for it to be arbitrarily close to 50/50 as a proportion of the total number of coins thrown. Not close to 50/50 in terms of by how many flips off it is. I’m pretty sure. I might be wrong.
And the distribution isn’t 50/50 because a coin landing on its edge is a possibility for most coin designs… A rounded edge can bias the results by shifting the edge possibly to either heads or tails…
I noticed this as a child. Heads had a slight bias and people start with heads. I just never knew why. I was told it was a random distribution. This is one of the world's greatest lies.
The more I see about coin flipping the more I get convinced that there is a significant chance that the outcome of a coin flip is not random. For every coin flip there are at least four possible outcomes, it land heads up, it land tails up, it lands on its side or it does not land at all. The fifth possible outcome we can ignore, that the coin had not flipped at all. I predict that even if a totally impartial coin flipping machine would be made, that the outcome can be manipulated. The cumber of flips can be regulated by the controlling the force, the distance is also of influence to the number of flips. True randomness will only appear if and when it is done manually and if the starting position is unknown and the force of the flip is unknown and also the distance between the flipper and the catching hand is unknown. I must be done by two individuals both with their eyes closed. If one coin flipping machine would be made, the outcome could be ordered, every coin would be after one flip at exactly the opposite side up as the original position was. The number of flips can be controlled and therefore it is not random. Football referees that flip a coin at the start of the game are just clever bastards who are in favour of one of the two teams. So clever that they van hide their preferences.
8:40 Are they actually independent though? If someone picks up a coin to flip it again, depending on the way they do that they may favour keeping it same side up (or flipping it over) before the next flip, which would mean that for any one person flipping a coin multiple times in sequence you can't assume the flips are independent right? Since the previous result affects which side starts up on the next flip. Am I missing something?
As long as there is no heads-tails bias (and we have extremely strong evidence to support this) then starting from the same side as the last flip doesn't influence the outcome.
Flip the coin with your non dominant hand, while standing. The coin must reach a height of 2 meters, and fall to the ground. The side up is then recorded. A rotating laser will determine if the coin breached the 2 meter mark if needed. The ground will be a standard Astro turf as used in most football stadiums. This way, the experiment can be conducted with on going results for decades to come.
it just depends on all the angles of forces working on the coin and the way you are holding it. So you could say it is undefined by just randomly flipping it without precision
If you look carefully. Most of the flip video the main side of the coin is tossed and when catched. The person uses the same side and flip again. So due to the fact the main side is over 50%. And when they flip second time they used the catched side as main side. It reinforces this bias. And repeat this 27 times. Most likely the start side is the same and hence easier to be over 50%. They should control the starting side with equal heads and tails for this study.
You don't have to look carefully -- this was the protocol as explained in the video. Since the results show 50/50 heads vs tails we can conclude there isn't a bias there.
interesting note, the paper says Canadian quarters are biased towards tails. 52% or something like that. I just flipped one 10 times from tails and got heads 8 times. Statistics are fun. Changed sides to the side it's biased against.
Is there a number of times you can flip a coin to be 99.999% random, knowing the starting position, if you only count the last flip? For example, if you need to make a random decision, you flip a coin 10 times in a row, with the starting position for each flip the same as how the previous flip landed, then you count the 10th flip as the one result .
I'm a magician. I can flip a coin on the side I want it almost all the time and have it look perfectly legit. I do f up sometimes, cos of being a meatbag.
"Every coin flip is deterministic." Mostly true, but there are instances where a true random result is possible due to quantum effects, like if the coin bounced perfectly balanced off of a corner of its edge, but these circumstances would be rare.
It seems to me that the only way to get procession is to have an object with the center of mass which is NOT at its physical center. Thus, a coin should follow procession. Also, IF the coin were biasly flipped, and IF that coin were perfectly balanced, center of mass is the same point as the physical center, then the coin should flip concentrically.
Would spinning it on its side be more random? Since it is not starting on a particular side. Y’all know the flick spinning of quarters game where everyone at the table tries to keep it spinning by flicking it while it spins.
The quantum test would be to have 10 people in a room and all just focusing on a particular side of the coin. Flip the coin 100 times where everyone is thinking heads. Then another 100 flips where everyone thinks tails. Maybe what the people are thinking about changes the chance of landing on a certain side.
the 5% significance level is a holdover from when these calculations had to be done by hand, which lead to just selecting a few values to use depending of the sensibility of the data. For instance you want to be more sure about medical data than general population data, so you may use 1% when dealing with that. With the ease of calculation of the probability of having the experimental results or more extreme under null hypothesis (we call that the p-value), these fixed levels are not really needed anymore, unless you happen to get a borderline p-value like 0.035 or something, when you would need to think if 3.5% chance of error is acceptable based on the kind of data you're working with.
Bro says his probability skills are subpar and immediately after effortlessly writes out a probability equation that 99.99% of humans couldn’t possibly create
There is a way of determining the outcome of a coin toss which magicians use. After the toss that the magician does they run their fingers very quickly on the underside of the tossed coin and turn it over and call it out correctly. The secret? Use a coin with a rougher tails side (which is usually the case). If it feels rough on the finger scrape then it is tails otherwise heads. When I do it with a UK £2 coin I can call it correctly > 90% of the time.
@@dexterdixon2000 No, but it lessens it! Assume the same-side bias is 51% and say we start on heads. What's the probability of getting heads at the end of both flips? There are two options for getting heads. Either we get TH or HH. To get TH, that means we got a different side landing (0.49), then we turn it upside down onto heads, then a same-side landing (0.51), so P(TH)=0.49x0.51. To get HH we get a same side landing (0.51), then turn it upside down onto tails, followed by a different side landing (0.49), so P(HH)=0.51x0.49. All in all, 0.49x0.51 + 0.51+0.49 = 0.4998. this is less biased than 0.51 but still biased. Interesting idea though, got me thinking!
Genuinely, a guy called Robert Matthews investigated this in 1996 and showed that butter-side down is more likely. And yes, he won the Ig Nobel prize 😆
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⬣ *CORRECTIONS, CLARIFICATIONS, AND COMMON COMMENTS* ⬣
SURVEY
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1. 18:45 a remark about the Bayesian argument used in [1]. This methodology aims to not only prove that the probability of the coin toss is greater than 0.5, but provide evidence for the fact that the probability is 0.508 as predicted by Diaconis et al. in [4]. Explaining this in detail is beyond the scope of this video, where I merely wanted to present convincing reasons why the coin tosses collected likely had p > 0.5.
2. It is likely that the odd-numbered house fact only applies in the UK or in countries with similar numbering conventions. I should have stated this in the video! James Grime made a video about it back in the day:
ruclips.net/video/wydlZ9lcEiQ/видео.html
3. At 12:12 I say "520 out of 1000" but the caption says "out of 100". Sorry!
4. There is a typo at 17:20. It's 350,757 flips, not 350,737. I checked those numbers *so many times* so can't believe I didn't spot that >_
Another typo at 12:12 unless you have some sort of hydra coin :D
@@warvinn Well spotted! I'll add that now.
is it becoming obvious one have throw the dime twice - after the first throw the partipicant switch side
In case anyone reads these before going to the form, the form has reached the max submissions and won't tell you until you submit.
@@richardhosler7301 Thanks for flagging! I'll fix momentarily. Who knew MS Forms was so stingy with response numbers??
The 'fair coin flip' method I was taught is to have both participants simultaneously flip a coin, with each player choosing matches/mismatches instead of heads/tails. Even if both players are cheating and using one sided coins or whatever, unless they have knowledge of how the other will cheat, it still ends up being 50:50 in practice.
That is a failure of measurement! There will always be bias! Question is... How many decimals does it take.
Why don't they make a perfect flipping machine in a vacuum to do the flipping.
Do two sets. One starting heads. One starting tails. A million flips total or a piece. Coin used are changed to brand new ones every 1,000 flips.
The flipping machine has to mimic a human thumb, and arm. Good luck making a perfect repeatable machine to do that.
Seriously no one has made a flipping machine? Surely a gambler has to try to get some advantage.
@dianapennepacker6854 if you broaden to talking about random number generation in general, Alpha Phoenix made a muon-powered random number generator. That's about as non-biased as is quantumly possible
With minimal practice, you can spin the coin when you throw it, and it will not flip in the air. You control the outcome.
Give it a few tries.
@@dianapennepacker6854what?
flipping machine isn't necessary, you dummy!
@@delwoodbarker Sure, but you don't control your opponents outcome, and unless you control both, you can't bias the result.
A previous Ignobel winner, Bert Tolkamp, works in the same building that I do. His team showed that cows that are lying down are more likely to stand up the longer they've been lying, but you can't predict when a cow that is standing up will lie down based on how long it's been standing already.
(Statistically, this means that the wait time before lying down is exponentially distributed, but the wait time before standing up is not.)
Incredible!
@@AnotherRoof uhh, I think you meant, "Holy Cow!"?
So I suppose that it means for a cow, lying down for a long amount of time is uncomfortable, but standing up is not.
@@Noname-67 it means cows have memory, in contrast to radioactive atoms, where if you have a U238 atom from the 1st pop-III (iirc) stars made 12 Billion years ago, and I have one made in a reactor this morning..."which one will decay 1st?" is truly a coin-toss.
@@DrDeuteronhonest question: how do we know whether it truly is random or there is some way to predict it and we just don't know it yet?
"this year's Ig Nobel prize awarded to this flipping paper"
2:23
Are you flipping kidding me?
Should have heard what they said about the research into human mating habits.
Ignoble Shirley?
Great video! Thanks for covering our paper and reaching out with the questions!
Thanks for helping out towards such vital research! Out of curiosity, were you a stable tosser or a wobbly tosser in the end?
@@Imperial_Squid a bit wobbly, 50.5% with 10,148/20,100 same sides
Thanks so much for conducting this experiment and helping me!
"Wobbly Tosser" sounds like the most British insult I've ever heard.
Calling someone a wobbly tosser is one of the best ways to say your British without saying you're British!
Just by spending a few years in England, that felt weirdly personal..
@@dannydetonatorAre you implying you're a wobbly tosser?
All these flipping participants are wobbly tossers!
Fun video! Regarding your request to viewers to toss 10 coins: there's a risk that people will not report boring results, such as 5-5. But one out of 512 people trying this will get 10 heads or 10 tails, and they're very likely to report it. In short, there's a strong risk of reporting bias.
Completely agree -- the survey is more a matter of curiosity and not a publishable piece of research!
With a US quarter, I just did 24 flips, I started on heads for half and tails for the other half, all my flips have that satisfying ring to them. I started on heads where 11 of the 12 flips landed back on heads. Then tails, 9 of the 12 stayed on tails. Maybe I am closer to a robot with my style or that was really weird.
one time in a math class we were supposed to flip coins n look at the results, but i was too lazy so i just reported that i got heads each time lmao
He doesn't care what people comment, it's called a "call to action" that content creators use cynically to drive up comments and please the algo.
Actually, it's debatable as to whether comments help videos in the algorithm; the key factors are click-through rate and watch time. The rough consensus is that comments don't matter and I agree. Anyway, I asked viewers to fill in a survey -- links are in the description and in the pinned comment. If viewers want to comment their results, that's their choice.
18:08 this "started as a joke, got bored, actually did it" speaks to me on a whole new level 😂
In high school, there was a period where people were gambling in various ways with quarters. When it came to coin flips, a few of us knew how to legitimately (not using the magic trick) cleanly flip them rather consistently, so we knew what we'd get based on which side was facing up. The funniest thing is that when our math class had everyone perform 100 coin flips for a probability lesson, two of us tried to find ways to get more accurate "random" results. (We didn't want to stand out with results that were "too perfect", but didn't know how "off" would look best.)
I did this as well. I won 65 percent of the time
I’m a bad roller AND I used to guess at night which of 2 keys went in my door. They looked identical at first and it turns out I’d guess wrong 96% of the time out of 600 tries over a 3 year period. The neighbors often hear me quietly cussing trying to get in my house.
@@bryanbryan2968 This wasn't guessing. It was maintaining a consistency in how you flipped the coin, to the point that like a machine flipper, the result of the flip was determined by which side of the coin was face up before the flip.
We didn't have the 100% precision of a machine flipper, but we were maybe at around 75%?
@@BainesMkII Oh yeah. Definitely fixing the results which, for what we hope, will be random results, but are not. ‘Ideally’, you should get about 50/50 as mentioned in the last part of the video, as I recall. It could even be subconscious, as in my case, and presumably more conscious in yours. Interesting test showing how things are not random, though we were always taught a coin flip seems like it is.
I remember doing this in middle school (86-87). The trick was to pull your flip and let it land on the ground. Convince those who are gambling that letting it land on the ground avoids any magic or slight of hand. Basically rapidly lifting your hand up but the coin doesn't flip above the hand. The result was it landed on the starting face nearly every flip.
"This was originally just a joke but then I got bored" is the source of almost all of my favorite video essays wildly overanalyzing things like video games and whatever. Don't you dare ever suppress that prediction!
@12:30 It's worth noting that "i.e., a 0.1 chance that the coin is in fact fair" is a very common but _completely incorrect_ interpretation of p-values. We have concluded that P(520 of 1000 successes | fair coin) = 0.1 (i.e., "probability of 520 of 1000 successes under the assumption that the coin is fair is 0.1"). This is very distinct from the statement that P(fair coin | 520 of 1000 successes) = 0.1 (i.e., "probability of the coin being fair given the observation of 520 of 1000 successes is 0.1"), which is the "i.e." statement at the bottom. In general, P(A | B) and P(B | A) are two arbitrarily different quantities (related by something called Bayes' Theorem, which in general says P(A | B) = P(B | A) * P(A) / P(B)).
Imagine a circumstance where we have 10000 coins. One is magically rigged to always land on heads, and the rest are fair, and they're otherwise indistinguishable and mixed uniformly together. You pick a coin and flip it 10 times and it lands on all heads.
The probability that the coin you picked is rigged is just ~0.1, not ~0.999 (which is 1 minus the "p-value" here for rejecting the null-hypothesis that "the coin you picked is fair") (I am rounding 1024 to 1000 and also rounding 1/9999 to 1/10000 and I think maybe doing some other rounding).
The struck-through part below is incorrect (kept for context of replies), it's actually just impossible to interpret p-values in a mathematically useful way and p-values are stupid. See mjeffery's reply below.
-A result with a given p-value is -_-some updated to your belief about the likelihood.-_- The p-value tells you how much to update it by (so we got from 1 in 10000 odds of the null-hypothesis being false to 1 in 10 odds, a 1000x improvement from our p-value of 0.001), but if the thing being tested is astronomically unlikely to begin with, this hardly lets you be confident.-
-Of course, in this case, we know exactly the prior probability of the null-hypothesis, which isn't usually true, making interpreting p-values in the real world much harder. But in general, when you see a study with a p-value of 0.05, the correct interpretation is not "this is true with probability 95%", it's -*-"this is 20 times more likely to be true than I thought it was."-*- If the study is "some random food cures cancer"... well, there's a whole lot of foods and most of them probably don't cure cancer.-
Footnote: the easy way to do the math here is in terms of odds. We started with 1:9999 odds of rigged:fair. The likelihood of 1/1024 tells us that we have an odds-ratio of 1024. This tells us that we have 1024:9999 posterior odds of a rigged coin. (This is exactly equivalent to Bayes theorem, where you would have P(rigged | 10 heads) = P(10 heads | rigged) * P(rigged) / P(10 heads) = P(10 heads | rigged) * P(rigged) / (P(10 heads | rigged) * P(rigged) + P(10 heads | fair) * P(fair)) = 1 * 0.0001 / (1 * 0.0001 + 1/1024 * 0.9999) = 1024/(9999 + 1024)
Useful comment, but I was initially confused what statement you were responding to, since you didn't quote it in your reply. (Maybe timestamps can be a bit finnicky.)
This is mistaking p-values with Bayes Factors - a p-value is always relative to some base (null) hypothesis, and is defined as the likelihood that the observed results are obtained if the null hypothesis is true. You’re correct about the odds calculations, and that a common and incorrect interpretation of a .05 p-value is as a likelihood of untruth.
The idea that you can use the p-value to weight your belief in a particular model also assumes that the space of possible models is both finite and known, which is an ongoing question in statistics.
Source: I’m a predominately Bayesian biostatistician who taught mixed research methods and statistics.
@@ForsakenDAemon trying to get my head around all of this - Mr Roof has mistakenly said that we can use the measured result to calculate the likelihood that the null hypothesis is true/false.
Yaksher is unknowningly describing a Bayesian approach which says that p-values are used to update our beliefs.
So both are mistaken. What is the correct interpretation?
@@parkerstroh6586 The first paragraph from Forsaken's comment tells you how to interpret a p-value. I'm not following exactly what people are responding to otherwise. (Seems I missed the Bayesian part of the original comment as I was focused on figuring out the original mistake.) Spitballing, it seems people have added in extra context to try to explain a concept, but that context isn't exactly right. (Which I've noticed happens with statistics decently often since a lot of terms are very particular in how they work.)
@@P0rtalGunn3r ohhhh super nice, cheers!
If I'm not mistaken, the slightly greater probability of an odd numbered house only holds in places where houses are numbered sequentially starting on one side of the street, that is where houses 1 and 2 are adjacent. In places like Canada and the US, odd numbered houses are on one side of the street, and even numbered houses are on the opposite side, so we would except the probabilities to be equal for a sufficiently large number of houses.
And in Australia (or, Queensland at least; not sure about other states), houses are numbered by their theoretical standard allotment facing, so if a block is wider than that the road numbers will skip (this is to future-proof road numbers from potential subdivisions), and if your block is _narrower_ you get A, B, C, D, etc (because the future-proofing doesn't always work). If you have multiple residences within the same building on a single allotment, then it's 'Unit 1', 'Unit 2', etc.
So you can have a road where the numbers go 1, 2, 3, 4, 6, 7, 8A, 8B, 9 (Units 1-20), 10, 12, 14, 15. Odds and evens are on opposite sides of the road here, so in this example you've got a very dense road frontage on the even side, and a sparse development on the odd side that has a large block of units built on one of the properties. It's not at all uncommon to see this; it can make finding people's houses a pain if they've told you the street number and haven't given any landmarks to find the place by.
14:00
They said 250,000 = 500² since.
With that many flips the standard deviation is 50%/500 = 0.1% which is sufficient to differentiate 50% and 50.3%
Hmm, perhaps it really is as simple as that. Thanks!
When I flip a coin, I can make it land 'starting side up' about 80% of the time.
For years I just though I had some weird talent which allowed me to win any coin toss.... but no... science.
Thanks science (grumble...)
Well TBF if you're both calling and flipping then the other person just deserved to lose. The sneaky way to do it is tell them to call it as you're showing the side facing up. A lot of people simply pick that side then you flip it onto the back of your other hand. If they call the bottom, it's on you to decide to just go for it or if they'll complain about not flipping it over after the catch.
the talent of being a wobbly tosser
Nothing like last call to bring out the wobbly tossers 🍸
How to say your British without saying your British? 😅
14:14 Here's my guess as a statistical practitioner. When I have to do this kind of "power analysis" I usually just program a simulation and try different values until I find the sample size needed to reject the null at some given effect size.
great video, great explanation, great everything! i expected you to have millions of subs from the quality of this one, here’s to getting you one step closer 🙏
As a data scientist, my insight into why they chose that number of flips is that nobody does power analyses before studies like they're supposed to do.
Data science is a job field not an academic field.
@@alessandrorossi1294Data science is also an academic field, at least in the US.
@@SoftBreadSoft No it's not, it's just a hot-job-field-degree. Actual academic disciplines are math, computer science, physics, statistics. Data Science is just "here's the bare minimum of t-tests you need to know so you can import sklearn and plug things into it wildly".
it varies for each university.
no formal criteria exist for defining an academic discipline
@@alessandrorossi1294 It is. There are researchers producing theories and models, and professors teaching, i.e. it is an academic field.
like the other fellow said it does depend on the college whether they focus on the theory/modeling side of things or the applied engineering side of things, which yeah those curriculums could/should be a specialization of computer/software engineering.
"Rosencratz and Guildenstern are dead" opening scene sums it up
16:15 I was expecting a joke about the term “wobbly tossers” and I was not disappointed 😂
You need a mechanical flipper to test this. Build it with a randomised starting energy and only count flips which had a minimum number of flips before landing.
i've done 10 flips with each of the six types of australian coin, including the dodecagon that is the 50 cent coin! this is the real research that needed to be done
1:23 minutes in an my mind is already blown
Wow, didn't expect to see you here, great videos!
14:07 Besides deciding the p value you want to consider a result significant, you also have to decide on the power of the test. The power is the ability of the test to find a difference using your data, if in fact one exists. The closer your two values are together, the more samples you need to be able to find that difference. Statistics programs can give you want sample size you need to have a particular power. In this case, if your significance level is at 3 sigma (~p of 0.001), and you want about a 99% power to find a difference between 0.5 and 0.51, you need about 250,000 coin flips
3:26 "in twenty o seven" stealing this
2016 = twenty sixteen
2024 = twenty twenty-four
2007 = twenty oh seven
There is still always the chance a coin flip ends up on it's edge. I witnessed it once on a hard floor. The coin landed and bounced onto it's side and slowly rolled into a wall where it stopped and stayed on edge. The odds are extremely low, but I can say they aren't zero.
The most funny thing about stochastics is, that it will never will tell you anything about individuals, other than how atypical they are. This includes individuals rolling dice or flipping coins. So technically the distribution of how well specific individuals fit the expectation in their results is in itself a bell curve. Which means that people exist that can't roll dice for shit. Their whole life. It's not particularly LIKELY, but that doesn't mean they don't.
Before watching this I’m going to guess that the actual physical construction of the coin will matter quite a bit. A US Quarter may be weighted slightly to one side as compared to a Pound Sterling or Penny
Edit: not even ten minutes in and I was wrong lol
Respect for the edit instead of deleting - this guy sciences 🤜🤛
Problem: The researchers all operated by flipping and CATCHING the coins, ending the rotation immediately and confirming that the coin spends more time on its starting side than the other side. However, coin flips are not always caught - they are often dropped, or allowed to land naturally, where they bounce and further randomize.
I have no idea why this ended up on my recommended, but I'm actually learning about hypothesis tests right now, so that's neat!
Before watching the video, the issue I have with studying actual coin flips is that they technically aren't random at all.
If we were capable of understanding every single thing about the coin flip, from the force applied to the coin the angle of the flip, distance from the ground, any interference in the room etc. then we can calculate where the coin would land 100% of the time. It's a bit like how a roulette wheel isn't technically random. For all intents and purposes it is since people can't predict where it will land, but there absolutely is a way to determine it, it's just not something we're capable of doing at the moment.
So the issue with having several hundred people flip a bunch of coins is that all these minute factors that impact a coin flip will change very slightly from person to person. They each aren't in control of the flips enough to dictate exactly where it will land, but the flip itself is still impacted by these things. This is why I don't like the term "fair coin flip", because it's not really fair, it's just that we're incapable of accurately measuring all the things that impact the flip from the outset.
@@RAGEAlanBun Without reading your whole comment, the problem with pointing out that coin flips aren't random is that this is addressed in the video and the study is very much about what happens when people flip coins.
Take-away: calculating the probability of heads/tails requires defining the path from initial orientation through rotational vectors, and accounting for resistance or modifiers like temperature, pressure, air currents, etc.
In a standard game of Two-up, you would always use a kip (a sort of paddle) to flip the coins, which might do something about that wobbling effect.
15:53 I can't believe they're not funding this of all things
Your point on “what is a fair coin toss?” made me think about a possible answer - is it 50/50 to spin a coin on a table?
If the coin and the spin were perfectly balanced then the coin would just stay on edge.
@ that is true, but consider the perception of the coin - it balances out and you can see how “fair” is the spin by the radius of the circle it makes. Then the chance is down to when it loses the energy and drops on the table.
Spinned coins should then be biased to weight I’d assume. Probably the heavier face of the coin would end up pulled down by gravity more times that the other?
technically speaking, even a flip isnt a perfect 50/50 without human intervention. It is incredibly rare outside of the contexts where the outcome is orchestrated, but landing on the edge is always a remote possibility. It comes from the nature of being a 3d object: 3 dimensions means a minimum of 3 "faces", attributed directly to the x, y, and z coordinates. cylinders might just be the only 3d shape with exactly 3 faces simply because the rounded shape is a bit difficult to properly define, so i am just simplifying it into a single face. By that same logic, a coin already balanced on its edge to produce a spin is even more likely to remain on that edge depending on the energy added into the mechanical system of spinning the coin. Just like flipping a coin, it is human error that forces the coin to pick a side. The difference is that it seems much easier to rig a spin in your own favor compared to a coin flip. Even acknowledged in this study is the idea that isolating variables for a coin flip to properly test is practically impossible, simply by nature of the comparisons to "perfect flipping" machines that can orchestrate the desired outcome with 100% accuracy. But spinning a coin? Comparatively, the action requires a lot more attention, all while being much easier to hide an intentional tilt in the coin in the initial action to start the spin. Add in the direction of spin and rigging the action seems much more plausible than the mechanisms out of a coin flipper's control like involuntary micro spasms or the force of the initial liftoff. Just look at the overemphasized example of a wobbly coin flip, it starts to look an awful lot like the last few moments of spinning a coin on a table, where the outcome is already determined it is just a matter of waiting for it to slow enough to identify it.
18:04 Definitely in keeping with the null expectation of 50/50 odds. X-squared = 0.016, df = 1, p-value = 0.8993.
Randomness often can come from outside. An electronic dice doesn't need to be internally random. It can simply be fast so that a human can't reliably stop it on any particular number.
The fun thing is that statistically, a coin has three possible results. Because it can also land on the rim.
Now the chance to land on that side is extremely small, but not zero.
But fun to see that coin flips are just like toast falling off the table. It's all a result of time, distance, and speed.
Worth mentioning that coins are not symmetrical. The embossing on the heads is not the same pattern, nor does it have the same amount or weight of embossed material as the tails side.
So why would you expect the results to be 50/50:in the first place?
I thought it was worth mentioning too, hence 2:27
1:28 loved the sheet!
I once read a paper which concluded that with American coins, heads is slightly more probably than tails. They concluded that the results are due to the different style of markings that the heads side of the coin has from the tails side of the coin.
@@edwardsong7628 Can you share the paper?
The next Nobel award will be given to the scientist who walked 350,757 steps to prove that by walking on legs, legs become stronger than arms.
For 13:20:
Sq.rt.(250,000)/2 = 250 per stdev
Fair coin = 125,000
125,000 + 250 = 127,500
125,250 / 250,000 = 0.501
So at n = 250,000, you can claim precision to +-0.001 with 68% confidence.
It's kind of arbitrary, but I think it makes sense to suggest 250,000 flips to get to 0.001 precision with ~68% confidence.
bro...
12:30 The last line is wrong: a 10% chance that the null hypothesis leads to some result does not mean that there's a 10% chance that the null hypothesis is true given this result. This is a simple case of conditional probabilities. Let A be the event that we get a result this significant, and let B be the event that the null hypothesis is true. What we have worked out is that P(A|B) = 0.1 (the probability of this result given the null hypothesis is true is 0.1), but you're saying that this means that P(B|A) = 0.1 (the probability of the null hypothesis being true given this result is 0.1) as well. This is not the case.
Bayes' theorem states that P(B|A) = P(A|B) P(B)/P(A). So the actual value also depends on the unconditional probabilities of the null hypothesis being true, and of getting a result of this significance (specifically, the ratio between these). Of course, we don't really know these values, so that's why we don't usually claim anything about the chance of the null hypothesis being true given this result (we basically just don't know with this information).
This sort of mistake with switching round conditional probabilities can lead to real problems. For example, it's the difference between the probability that some medical test comes back positive given that you have a disease (which we certainly want to be very close to a 100% chance), and the probability that you have this disease given that the test comes back positive (which is what you actually want to know, but could be very small if the unconditional probability of having the disease is tiny, and there is any chance of a false positive).
I'm not sure exactly how the authors calculated a sample size of 250,000. One thing you need to take into account is also the power of the test (i.e., the probability of rejecting the null given that the alternative of .5027 is correct). A common value for power is .80. Using power = .80, alpha = .05 and alternative proportion of .5027, the number I obtain is 212,019. See the R code below:
library(pwr)
power
Interesting. What is the power of test measuring in terms of the probability that the null hypothesis would be rejected in favor of the alternative? (Statistics isn’t my strong suit, have never heard of this before)
@@photoniccannon2117, good question. Power is a probability that you will reject the null hypothesis (e.g., the coin is fair) given that the alternative hypothesis (e.g., the probability of heads is .55). Power depends on three quantities: the assumed effect size (e.g., how different is the assumed probability of heads from fair?), the number of samples, and the statistical significance level (typically set to .05, which is a 5% chance of rejecting the null hypothesis given that it is true). The basic relationships are intuitive. An effect that is large (e.g., 80% heads) is easier to identify than one that is small (e.g., 51% heads), the more samples you collect, the easier it is to find a difference, and increasing alpha makes it easier to identify an effect (but also find a false effect).
RUclips sometimes removes comments with links. So I recommend seaching "rpsychologist statistical power" to find a visual illustration using a continuous variable as opposed to a discrete count. If you select "solve for Power", you can change the significance level, sample size, and effect size to see how it changes power .
8:12 I see what you did there…
I'm not a rocket surgeon, so take this with a corn of pepper, but, isn't the reason it doesn't matter what side it started on have a lot to do with the fact that at the very start the first side is already "half flipped"... ie- a flipped coin (usually) starts parallel to the ground. So, it only takes a quarter-rotation for it not to be on top any more, and then the second side gets a full half-rotation before giving up the top. From then on they trade full half-rotations on top. So, in your chalk diagram at 0:50 the very first pink bar at the top left should be half the height of all the other bars that follow (pink and blue).
Which begs the question... if you start the coin standing straight up and down so the first side gets a full-half rotation before not being on top, THEN does the first side have a slightly higher probability? It must, right?
You know it's convoluted when it all sounds like Vizzini explaining his logic in The Princess Bride
Methinks the protocol may entail a possibly-cumulative but easily-removed bias, namely, start half the tosses on heads.
I used a 1922 Peace dollar weighing 26.63 grams which is easy for me to handle and, being silver, provides a pleasant "ching" upon launch and some Doppler effect from the spinning flight, with a muffled thud when caught.
10 tosses starting heads -- 2 heads and 8 tails, surprising
10 tosses starting tails -- 6 heads and 4 tails
Bonus: 100 tosses starting 50 heads and 50 tails, results exactly 50/50.
Keeping track of the individual trials might be interesting as a follow-on.
Good vid, quirky edutainment at its best.
P.S. I should've read the description first about the survey. I'll go do the survey but leaving my comment anyway.
I have a biased for opposite of start when catching, mostly due to I trained some time ago to do that. I am more 50/50 when landing on the ground or table.
For the number I got out of 10 landing in hand: it was 9 opposite 1 starting.
Congrats on the Ig-Nobel :-) I have to also let you know that I live on a strange road (I cannot use the word 'odd') where all eight house numbers are even!
I did 10 flips with my right hand using a US penny from 2022 with a shield on the tails side. I started with heads and maintained the same face from previous flips.
Started with H, then THTTHHHHHH.
3 tails, 7 heads
6 same-sides results out of 10
The
Sides each Have very small different mass due to different images. The one with the larger mass will theoretically.
End up on the down side and the smaller mass on top, which is side that wins.
As discussed, this isn't true! Surprised me, too.
Interesting bit about those tossers you highlighted in blue...
Great video too!
Biostats instructor here... I am just itching to see the heads and tails tallies... Gonna plug those shits into a Chi-square test so quick...
Edit( 18:16 ): X-squared = 83.104, df = 1, p-value < 2.2e-16, reject the null of 50/50 chance.
12:13, typo: “after *1,000 flips”. Hope it helps. Great video!
You can't start with a cat and then later dispense with it as unnecessary, that's just bad science.
Euro 20 cent: 8 heads, 2 tails. All started as heads.
Started with flipping 5 heads in a row. Low height flips, but two did roll to floor, both heads.
Used a quite large coin:
(same, other) = (13, 7), but (8, 2) for the first ten flips!
50 flips: (27, 23)
80 flips: (39, 41)
100 flips: (47, 53)
I did the survey on an Australian 20 cent and got HTHTHTHTHT starting on heads, so that’s only 1 of ten flips that ended on the same side as they started. Interesting. This clearly means that either there is actually a less likely chance that you flip a coin and it lands on the same side and all the maths is wrong, or that my Australian money is skewed. Clearly with such a high sample size of ten one of these is the case and not simply luck.
Seriously though cool video!
Got to explain the angle, if it is as I think, then if force coin flip to hit way more closer to the edge would make way less biased. And yes I might later but not now look at it myself. And what did the paper say about this? This should be a huge point in the paper. As well as tilt hand rotated from from thumb as well.
Tails never fails
Heads never yeads
definitely a tails guy. in Italian we say "cross or head", I always go for cross
Its because of flippers. Wr have a tendancy to try and do things the same way. The energy put in dictates rotation. I can consistently flip a coin to either heads or tails after just a few flips to calibrate.
I can say it's harder to get good at not dropping the coin than it is encouraging the outcome. I can convincingly flip a quarter 10 times and be pretty confident in it landing heads nearly every time. I'm much less confident that it won't land awkwardly then roll over or simply bounce off.
For whatever reason, a lot of people calling the side get fooled into picking the one they can see and then lose when the person flips it onto their other hand. If anyone volunteers to flip for something, the safer bet is to just tell them to let it fall to the floor.
Man taught statistics better than any of my professors
Looked to me like it wasn't measuring whether coins are biased, but how good people are at throwing a coin properly.
@@DjVortex-w That's exactly the point. Although in doing so we can be pretty certain there is no heads / tails bias, as discussed
Did they consider the average force applied to flip the coin and the height at which the coin drops relative to the height of the hand that flipped it?
Is it 50-50 whether a street ends in an odd or even house? Under some assumptions is this the same thing as how a random number is more likely to start in a 1 than a 9?
Two possibilities: 1. The flipper getting one result and inadvertently putting the wrong answer down. I know I would be a bad candidate for the coin test for that reason. 2. The roughness of one side vs a smother side may affect the aerodynamics of the coin.
Doing my part for science: Used a 2 Belgian euro coin. Started with heads side up, keeping the result for the next flip as discussed in the video.
Sequence: HHHHTHHHTH
Same side landings: 6, different side landings: 4
Self-report on flipping technique: very poor. Dropped coin multiple times, on one occasion (flip 3 or 4), I believe the coin didn't actually flip at all.
EDIT: I forgot there was a form, and have now filled that in as well with these results.
I see what you did there at the end of the video. Nice one.
Okay, since this contains trig functiona I bet on Matt Parker recruiting 1000 volunteers for coin tossing to determine the value of π.
Looking forward to the Premier but can predict if you did an odd number of flips its not 50/50 😉
@Kounomura that's true. Landing on its edge would also make it not 50/50 as there are now 3 states: head, tails and edge. So the paradigm 50/50 is actually invalid. Good point.
Well the more coins you flip, the less likely it becomes for the results to be exactly 50/50 (only considering even numbers of flips because it’s impossible for odd numbers). But as it gets less likely for it to be exactly 50/50, it becomes more likely for it to be arbitrarily close to 50/50 as a proportion of the total number of coins thrown. Not close to 50/50 in terms of by how many flips off it is. I’m pretty sure. I might be wrong.
Was thinking the exact same thing! 😂
And the distribution isn’t 50/50 because a coin landing on its edge is a possibility for most coin designs… A rounded edge can bias the results by shifting the edge possibly to either heads or tails…
Interesting fact, in the Pokémon TCG players prefer using dice instead of coin flips
I noticed this as a child. Heads had a slight bias and people start with heads. I just never knew why. I was told it was a random distribution. This is one of the world's greatest lies.
The more I see about coin flipping the more I get convinced that there is a significant chance that the outcome of a coin flip is not random. For every coin flip there are at least four possible outcomes, it land heads up, it land tails up, it lands on its side or it does not land at all. The fifth possible outcome we can ignore, that the coin had not flipped at all.
I predict that even if a totally impartial coin flipping machine would be made, that the outcome can be manipulated. The cumber of flips can be regulated by the controlling the force, the distance is also of influence to the number of flips. True randomness will only appear if and when it is done manually and if the starting position is unknown and the force of the flip is unknown and also the distance between the flipper and the catching hand is unknown. I must be done by two individuals both with their eyes closed. If one coin flipping machine would be made, the outcome could be ordered, every coin would be after one flip at exactly the opposite side up as the original position was. The number of flips can be controlled and therefore it is not random.
Football referees that flip a coin at the start of the game are just clever bastards who are in favour of one of the two teams. So clever that they van hide their preferences.
Great video! Also you have nice hand writing!
I laughed out loud when you said you could make neither heads nor tails of statistics. That was a good one
I went from flipping houses to flipping burgers, and now I'm ready to move on to flipping coins.
8:40 Are they actually independent though? If someone picks up a coin to flip it again, depending on the way they do that they may favour keeping it same side up (or flipping it over) before the next flip, which would mean that for any one person flipping a coin multiple times in sequence you can't assume the flips are independent right? Since the previous result affects which side starts up on the next flip. Am I missing something?
As long as there is no heads-tails bias (and we have extremely strong evidence to support this) then starting from the same side as the last flip doesn't influence the outcome.
Flip the coin with your non dominant hand, while standing. The coin must reach a height of 2 meters, and fall to the ground. The side up is then recorded. A rotating laser will determine if the coin breached the 2 meter mark if needed.
The ground will be a standard Astro turf as used in most football stadiums. This way, the experiment can be conducted with on going results for decades to come.
it just depends on all the angles of forces working on the coin and the way you are holding it. So you could say it is undefined by just randomly flipping it without precision
I would love to participate in this experiment, but I haven't even seen a coin in nearly a decade.
If you look carefully. Most of the flip video the main side of the coin is tossed and when catched. The person uses the same side and flip again. So due to the fact the main side is over 50%. And when they flip second time they used the catched side as main side. It reinforces this bias. And repeat this 27 times. Most likely the start side is the same and hence easier to be over 50%. They should control the starting side with equal heads and tails for this study.
You don't have to look carefully -- this was the protocol as explained in the video. Since the results show 50/50 heads vs tails we can conclude there isn't a bias there.
This is why the best way to flip a coin is to flip it without showing the person guessing, and having them guess with the coin in midair.
interesting note, the paper says Canadian quarters are biased towards tails. 52% or something like that. I just flipped one 10 times from tails and got heads 8 times. Statistics are fun. Changed sides to the side it's biased against.
my coin flips were: 4:6
Is there a number of times you can flip a coin to be 99.999% random, knowing the starting position, if you only count the last flip? For example, if you need to make a random decision, you flip a coin 10 times in a row, with the starting position for each flip the same as how the previous flip landed, then you count the 10th flip as the one result .
I'm a magician. I can flip a coin on the side I want it almost all the time and have it look perfectly legit. I do f up sometimes, cos of being a meatbag.
Cats are never unnecessary.
Would spinning rather than tossing, eliminate the bias due to precession?
I figured this out in highschool minus the science. I won so many bets due to guessing the probability was in my favor knowing how it started.
"Every coin flip is deterministic." Mostly true, but there are instances where a true random result is possible due to quantum effects, like if the coin bounced perfectly balanced off of a corner of its edge, but these circumstances would be rare.
It seems to me that the only way to get procession is to have an object with the center of mass which is NOT at its physical center. Thus, a coin should follow procession.
Also, IF the coin were biasly flipped, and IF that coin were perfectly balanced, center of mass is the same point as the physical center, then the coin should flip concentrically.
@@lourias Precession occurs in objects whose centre of mass is the physical centre
Would spinning it on its side be more random? Since it is not starting on a particular side. Y’all know the flick spinning of quarters game where everyone at the table tries to keep it spinning by flicking it while it spins.
The quantum test would be to have 10 people in a room and all just focusing on a particular side of the coin. Flip the coin 100 times where everyone is thinking heads. Then another 100 flips where everyone thinks tails. Maybe what the people are thinking about changes the chance of landing on a certain side.
No, leaning the coin very slightly can guarantee which side it will stop on when you spin it
the 5% significance level is a holdover from when these calculations had to be done by hand, which lead to just selecting a few values to use depending of the sensibility of the data. For instance you want to be more sure about medical data than general population data, so you may use 1% when dealing with that.
With the ease of calculation of the probability of having the experimental results or more extreme under null hypothesis (we call that the p-value), these fixed levels are not really needed anymore, unless you happen to get a borderline p-value like 0.035 or something, when you would need to think if 3.5% chance of error is acceptable based on the kind of data you're working with.
Bro says his probability skills are subpar and immediately after effortlessly writes out a probability equation that 99.99% of humans couldn’t possibly create
There is a way of determining the outcome of a coin toss which magicians use.
After the toss that the magician does they run their fingers very quickly on the underside of the tossed coin and turn it over and call it out correctly.
The secret? Use a coin with a rougher tails side (which is usually the case). If it feels rough on the finger scrape then it is tails otherwise heads.
When I do it with a UK £2 coin I can call it correctly > 90% of the time.
I haven't edited watched the full video yet, but if tails is facing up before the flip it will usally land on tails, same with heads
If you do a flip then whatever the result you turn it upside down then do another flip would that get rid of the bias?
@@dexterdixon2000 No, but it lessens it!
Assume the same-side bias is 51% and say we start on heads. What's the probability of getting heads at the end of both flips?
There are two options for getting heads. Either we get TH or HH.
To get TH, that means we got a different side landing (0.49), then we turn it upside down onto heads, then a same-side landing (0.51), so P(TH)=0.49x0.51.
To get HH we get a same side landing (0.51), then turn it upside down onto tails, followed by a different side landing (0.49), so P(HH)=0.51x0.49.
All in all, 0.49x0.51 + 0.51+0.49 = 0.4998. this is less biased than 0.51 but still biased.
Interesting idea though, got me thinking!
Next Ignoble prize is for proving that toast always lands buttered side down 😂
Genuinely, a guy called Robert Matthews investigated this in 1996 and showed that butter-side down is more likely.
And yes, he won the Ig Nobel prize 😆
@AnotherRoof Thanks, I have a factoid for the next dinner party lol.