18 ms·
Fair coins tend to land on the same side they started
- avipars 3y agoThey actually flipped a coin >350,000 times and the paper has 49 authors. Based on the paper https://arxiv.org/pdf/2310.04153.pdf https://arxiv.org/pdf/2310.04153.pdf, they recorded video footage of each coin flip I thought they would have run a computer simulation instead.
- fbartos 3y agoYes, I mentioned that in the original paper but it got eddited for some reason.
- fbartos 3y agoAbout a year ago, we embarked on a quest to answer one of the most intriguing questions: If you flip a fair coin and catch it in hand, what's the probability it lands on the same side it started? Today, we are finally ready to share the results. Thanks to my friends, collaborators, and even strangers from the internet, we collected flippin 350,757 coin flips. We ran several "Coin Tossing Marathons" (e.g., https://youtu.be/3xNg51mv-fk?si=o2E3hKa-ReXodOmc https://youtu.be/3xNg51mv-fk?si=o2E3hKa-ReXodOmc) and spent countless hours flipping coins. In short, we found overwhelming evidence for a "same-side" bias predicted by Diaconis, Holmes, and Montgomery 2007: If you start heads-up, the coin is more likely to land heads-up and vice versa. How large is the bias? In our sample, the mean estimate is 50.8%, CI [50.6%, 50.9%]. We also found considerable variance in the same-side bias between our 48 tossers. The bias varied with a standard deviation of 1.6%, CI [1.2%, 2.0%], in our sample. The variation could be explained by a different degree of "wobbliness" between our tossers. If you bet a dollar on the outcome of a coin toss 1000 times, knowing the starting position of the coin toss would earn you 19$ on average. This is more than the casino advantage for 6-deck blackjack against an optimal player (5$) but less than that for single-zero roulette (27$). The manuscript is at arXiv: https://arxiv.org/abs/2310.04153 https://arxiv.org/abs/2310.04153 And the open data, code, and video recordings at OSF: https://osf.io/pxu6r/ https://osf.io/pxu6r/. Diaconis, P., Holmes, S., & Montgomery, R. (2007). Dynamical bias in the coin toss. SIAM Review, 49(2), 211-235. https://doi.org/10.1137/S0036144504446436 https://doi.org/10.1137/S0036144504446436
- kzrdude 3y agoFun that you're showing your own work. Since this is your work, it could have Show HN in the title; I guess.
- fouronnes3 3y agoWhat is the physical explanation for this bias?
- deleted 3y ago[deleted]
- rstarast 3y agoI would guess it's rather mathematical. Each coinflip has some number of half-flips. Now analyze the distribution of that number. If this distribution were to start at its maximum with 0 half-flips and decay as it increases, summing over the even values (same side up) clearly gives more than summing over the odd values. Now the distribution isn't going to be like that, but I expect that it's generally "front-loaded" in a way that causes a similar effect.
- mewpmewp2 3y agoYeah, and seeing that the bias occurs only in some people, perhaps it occurs in people who do as little rotation as possible. Not sure if this study has a graph including amount of rotations occurred in general. E.g. you could take all coin flips where 0-5 rotations occurred and compare them to 6-11 rotations.
- ss1996 3y agoThank you for this. I'll make sure to request a best of 350,757 next time I'm deciding anything by coin toss.
- LadyCailin 3y agoIt’s 50.8% bias, so you only need to do best of 1015!
- iamflimflam1 3y agoI would have expected some variance between different currencies and denominations - but maybe that would need a lot more data.
- morelisp 3y agoIf you read the original Diaconis, Holmes, and Montgomery paper you'll see why it shouldn't depend on this. There's also the classic Gelman & Nolan https://www.tandfonline.com/doi/abs/10.1198/000313002605 https://www.tandfonline.com/doi/abs/10.1198/000313002605 "You Can Load a Die, But You Can't Bias a Coin", in case you're imagining more complex dynamical behavior.
- deleted 3y ago[deleted]
- goindeep 3y ago[dead]
- jbandela1 3y agoVon Neumann described a very elegant way to get fair results from a biased coin. 1. Flip the coin twice 2. If you get the same result both times, goto 1 3. Now that you have different results for your pair of flips, use the first element of the pair of flips as your result. https://en.wikipedia.org/wiki/Fair_coin#Fair_results_from_a_biased_coin https://en.wikipedia.org/wiki/Fair_coin#Fair_results_from_a_...
- extraduder_ire 3y agoI take it a double headed coin doesn't count as biased for this algorithm. (because it's too biased to still be considered a "coin", for probability's sake.) Otherwise it never halts.
- garfieldnate 3y agoCouldn't you still game that by choosing the starting side? If you want heads, and you start on heads and get a tails for the first round, you could start on tails on the second round to try to get another tails and reset to step 1.
- dentalperson 3y agoI love the math/stats history around gambling-related things, thanks for mentioning this. This method assumes the flipper can't introduce bias. ET Jaynes in his book Probability Theory also mentions that it is easy to learn to flip a fair coin in such a way that the result can be predetermined. I searched a tiny bit for this but couldn't find what he was referring to though.
- ryanar 3y agoLots of practice, use your thumb to hit the edge of the coin to give you more control, aim for a specific spot so you use the same amount of force and control the amount of times it flips. You can also use a surface that absorbs more so the coin is less likely to flip after hitting it.
- aspenmayer 3y agoI’ve heard that, with many hours of practice, dedicated amateurs and many famous magicians are able to do this kind of thing. I wouldn’t call it sleight of hand, but it is similar, although it may fall under that category broadly. I’m not a domain expert but I was taught some simple coin tricks as a child by my artist mom’s artist friend who ran the local frame shop. I never tried or thought to try to favor the coin flip or introduce bias, but it’s definitely a skill that can be acquired.
- dmarchand90 3y agoExcellent ig nobel prize contender here
- hurtuvac78 3y agoThis is incredibly puzzling. Is there a minimum number of rotations per coin flip to consider it valid? If looks like the bias was not evenly distributed across people. How did you protect your experiment from skilled bad actors who could influence the data with a few bad/skilled flips? Did strangers on the internet fare any differently than in-person attempts from trusted people?
- morelisp 3y agoWhy do you think this is puzzling? This bias has been analytically and dynamically predicted for years.
- hurtuvac78 3y agoBecause I find it counter-intuitive. And because I am not aware of scientific development in this field. @bradrn on this thread kindly extracted the description of the proposed physical model from inside the paper, it is helpful: https://news.ycombinator.com/item?id=37830265 https://news.ycombinator.com/item?id=37830265
- fbartos 3y agoWe told people that the coin has to flip at least once (which would bias it for the opposite site). Whenever instructing people, I tried to explaining that the coin flip should look like you were trying to determine an outcome of a bet. You can find the complete experimental protocol here: https://osf.io/hkv8p https://osf.io/hkv8p Also, I wish I had (any) budget to hire proffesional skilled tossers haha.
- cantrevealname 3y agoI assumed that the 1% bias was entirely due to coins that did not undergo any rotation at all. However, reading that you told people that the coin has to flip at least once, I think I assumed wrongly. It sounds like the bias is due to coins that have undergone an integral number of 360-degree rotations (not zero rotations). But what exactly is the physical mechanism causing this bias? It's easy to understand why zero rotations would introduce a bias, but I can't easily picture a reason for a bias toward an integral number of 360-degree rotations. Is there a simple and intuitive way you can explain the physical reason?
- brettwall 3y agoBased my back-testing of stock market, I found a similar conclusion: if a stock rise yesterday, then today the probability of raise > the probability of fall. P(raise) is about 50.1%, P(fall) is about 49.9%. Vice versa. Having this theory means you can't rely on a single bet, you have to bets many many times to make profit from stock market. Even though I knew that, I am still working as a developer, I wish one day I have enough money to start stock career.
- hmate9 3y agoYou have to take into account how much it rises and how much it falls too. It might be you win more often but when you lose you lose more.
- mucle6 3y agoI don't know enough, but I bet with options you could make something close to a double or nothing boolean bet
- mewpmewp2 3y agoAlert: It's not going to be as simple as this to make money even with large numbers. So don't fret about not having the money and missing some golden opportunity. There are enough actors out there trying to develop complex algorithms to find an edge, and so there wouldn't be any simple edges like this left anywhere as they would be arbitraged away. If there is a simple pattern, it is noticed and traded until the pattern disappears.
- kqr 3y agoHas hmate9 pointed out, a simple pattern like the one described can persist indefinitely – but the risk of exploiting it is priced in already.
- mewpmewp2 3y agoYeah, you could find a pattern that wins 99% of the time to yield 1% of what you risk, but you don't consider that there's always 1% odds of losing it all and it's priced in, it just hasn't happened to happen yet, but systems with more precise data have accounted for it. E.g. something like selling 0dte options sufficiently out of money might seem like a free money hack, but once something unexpected happens you are down to just "who could have possibly foreseen that event to occur, my decision making was solid.".
- shapefrog 3y agoCan someone who has access to a precision robot and controlled environment please check; if one applies the same force on the coin in flipping, with it landing on a (soft) surface at the same height it - will it land the same side every time.
- morelisp 3y agoYou could read the Diaconis paper?
- shapefrog 3y agoIt is paywalled so I can not. However, I was unaware that they already conducted said experiment. I am now left confused as to why they would not mention such in the abstract, and refer only to natural experiments and second order measurements. I shall aquire the paper and learn why.
- tgv 3y agoUsing Google Scholar (1), you can find PDF versions of the paper (1) https://scholar.google.com/scholar?cluster=16877003867074896522&hl=en&as_sdt=0,5 https://scholar.google.com/scholar?cluster=16877003867074896...
- tiagod 3y agoYour comment sniped me into thinking of some over-engineered systems to measure this with the human factor included. Some sort of RFID coin balanced on all axis with an IMU chip that measures the force applied to the force (through acceleration) and the mode and number of rotations of the coin. Maybe a computer vision solution would also work
- shapefrog 3y agoMoving hoop won't let you miss - https://www.youtube.com/watch?v=myO8fxhDRW0 https://www.youtube.com/watch?v=myO8fxhDRW0 I guess in theory you could "catch" with the correct call by varying the height of the catching platfrom.
- cesaref 3y agoI think the sporting approach of letting the coin drop to the ground rather than catching it is the answer to avoid bias in coin tossing.
- fbartos 3y agoNot neccessarily because spinning and bouncing coins are often much more biased then flipped coins. (Unequal weight distribution on the side can bias a spinned coin while it doesn not bias a flipped coin. There are a couple of studies on it too.)
- mihaic 3y agoInterestint, my intuition was inverse to the result: landing on the opposite side takes an odd number of flips and on the same side it would be an even number of flips. Since for every number of flips either the number of odds is the same as the number of evens, or higher by 1, the chances of getting an odd number of flips is higher. There seems to be more at work here, and only testing validates a model!
- 1f60c 3y agoHow is that possible when both outcomes have a probability of exactly 50%?
- nottorp 3y agoAn ideal coin flipped by an unbiased super person would have that 50% probability. A real coin flipped by a real human no, as the study shows. Let's consider a spherical cow...
- tgv 3y agoIt's simpler: an ideal coin flip is simply assumed to be uniformly distributed, on the basis of there being two possible outcomes and no influence. Where the bias in reality comes from, doesn't matter. This also happens to be the great divide between frequentists and Bayesians.
- kqr 3y agoEven simpler than that, actually. There's no requirement for any distribution at all. (And I would argue strongly against a uniform prior, but that is a separate discussion.) What's necessary to guess 50 % on the first toss is simply (a) complete ignorance about the bias, whatever it is, and (b) the hypothesis that the bias is just as likely to be negative as positive (i.e. a symmetric prior.)
- kqr 3y agoWait, didn't Ed Thorp argue for the exact opposite – if you have a super-person that can guarantee lack of bias, then you also have absolute Newtonian predictability on virtue of the mechanical perfectitude. The randomness must come from somewhere, and it comes from imperfections which also incidentally introduce bias.
- shawabawa3 3y agobecause they don't have a probability of exactly 50%
- saalweachter 3y ago
- havnagiggle 3y agoI've always caught the coin and flipped it onto the back of my hand to display the result. So I guess I have an opposite-side bias.
- quietbritishjim 3y agoThat's the usual way of doing it.
- rootusrootus 3y agoHuh. I thought everyone did it my way. Flip it onto the floor, hunt it down, and see which side came up. I'm mostly serious. I know that's not ideal. But I've never been able to master the art of flipping a coin onto the back of my hand, and even catching it mid-air is hit-or-miss for me. The vast majority of time I or anyone I'm with has flipped a coin it's ended up on the floor.
- quietbritishjim 3y ago> I've never been able to master the art of flipping a coin onto the back of my hand That's not the usual way of doing it. Usually, you catch it in the palm of your throwing hand, and only then put it on the back of your other hand, flipping it once more in the process.
- a_c 3y agoI always tell people that result of coin flip is highly start state dependent. Imagine a sequence of H(ead), T(ail), H, T, H, T, ... if the sequence starts with H first, in no way can the number of T exceed that of H, but the number of H might be 1 greater that that of T. I never tested my self, but I hypothesize that the propability will be more skewed if the number of revolutions is less, i.e. having a shorter Head-Tail sequence. Edit: The sequence was meant to represent the sequence of head and tail facing up during the rotation. A sequence of [H, T] denotes one full rotation, starting with head. [T, H, T, H] denotes two full rotation, starting with a tail, ends with a head. I didn't mean the result of a flip. So the result of a flip is the final element of the sequence.
- deleted 3y ago[deleted]
- cabirum 3y agoCan you toss a coin without it flipping at least once (changing state)? I find it quite unlikely to happen, so if the starting state was H, your sequence will be THTH...
- a_c 3y agoI can't quite define what counts as start of sequence. Maybe a sequence should always have at least one element, and toss straight up is allowed. But if some flipping is mandatory, then the start of sequence would mean the other side of the coin. All I could deduce before this paper is the probabilty of coin flip is skewed.
- roflmaostc 3y agoOf course that is true, if you wait not significantly long enough. Imagine the situation that you flip the coin (starting at H) and you grab it in air immediately. Of course, you will get H as result. But let's say, the time to stop the coin can be a relatively long time T. Then, I think the probability is some kind of sum. Let's choose \Delta T= 10ms as time discretization: P(H) = 1 / T * (10ms-0ms) + (30ms-20ms) + (50ms-40ms) + ... = 1/T \sum_{i=0}{floor(T / (2 * \Delta T))} \Delta T P(T) = 1 / T ((20ms-10ms) + (40ms-30ms) + (60ms-50ms) + ... = 1/T \sum_{i=0}{floor(T / (2 * \Delta T)) - 1} \Delta T For T -> \infty P(H) and P(T) getting more similar. But, in practice you wouldn't wait equally distributed in time but more like a Gaussian distributed time period. Hence, each term of the sum would get weighted differently. And the variance and the offset of the Gaussian distribution can shift the probability in favor of H or T. It's really dependent of the concrete parameters. If you grab always after 35ms, then you'll always get T for example.
- cabirum 3y ago50 authors in a paper about flipping coins?
- jb1991 3y agoSounds fair to me. Better than if it was only 49, or 51.
- boomboomsubban 3y agoBlind guess, it's cheaper and easier to recruit volunteers to flip a coin 7000 times if you promise them credit on the paper.
- fbartos 3y agoYou are absolutely right :)
- blantonl 3y ago[flagged]
- shusaku 3y agoI guess that’s one way to prevent p-hacking: you can’t exclude/include someone’s flips without them getting mad!
- bunderbunder 3y agoMake me sit around flipping coins 10,000 times and recording the results, and you damn well better at least put my name on the paper.
- slingnow 3y agoThen I guess we should include every study participant as an author across all disciplines. 200 participant study in psychology? 200 authors on the paper.
- bunderbunder 3y ago
- harimau777 3y agoSupposedly dice tend to role higher if you roll them starting with the largest side up. I use that when I'm creating my character in D&D and it seems to work.
- helsinkiandrew 3y agoI think you can argue that the experiment wasn't representative of 'normal' coin flips. On average, each flipper in that experiment flipped a coin over 7000 times, after that amount many people will have learned to flip in a comfortable way with less variance between physical action and force they use. I'd imagine that in that case the coin would more likely land with the same orientation. I don't think this would be true if someone flipped without practice.
- misja111 3y agoI bet that with enough practice, you can learn to toss the coin so that you get a much bigger than 51% chance of landing it on the same side.
- rootusrootus 3y agoI think you'd need more than practice. Most people would need a teacher. Just doing something over and over isn't automatically going to make you better. E.g. The old "10000 hours of practice makes you an expert" rule assumes deliberate practice. And even then, it's incorrect.
- SamBam 3y agoI think this is a great point. One flipper 7000 times is quite different than 7000 flippers one time, if the aim is to see whether there is an underlying bias.
- fbartos 3y agoI personally did 20,100 flips and I can assure you I have no clue how to control the flip. I centrally got much better at flipping and catching the coin in hand without dropping it---which takes some practice on its own. (I know that there are techniques for adding the wobble to the toss, but I didn't study them and I have no clue how to do them. I think it is safe to say you don't discover them intuitevelly.)
- deleted 3y ago[deleted]
- rax0m 3y agoI watched one of the 12-hour coin tossing marathons. They were sitting with their laptops and pressed a button for every result. I wonder if human error can explain (at least part of) the deviation from 50/50: * locations of the buttons they pressed on the laptops (they only pressed once per toss before enter, meaning the button represented same-side or other-side) * remembering what the coin started out as may be harder (or easier, but probably harder) when the result is other-side * other?? Need to repeat this amount of tosses but with a higher degree of supervision to be sure of the result.
- fbartos 3y agoThat's actually not completelly accurate. The study protocol (https://osf.io/hkv8p https://osf.io/hkv8p) describes the procedure in greater detail. People were pressing one button for heads and another button for heads (which we deemend less error prone and less likely to be subcontiously influenced). The trick was that the next coin flip started the same side-up as the previous landed. Therefore there was no need to record the start (and we randomized the starting position of every 100th flip) We also did some auditing of the video recordings (trying to decode the outcomes from the videos) and they showed quite consistent degree of bias as the original responses.
- alexmolas 3y agoI would like to see how TianqiP flips the coin. This user have a ratio of 0.601 [0.582, 0.619] heads, which is a lot. This is the type of skill that can get you a couple of bucks if played strategically.
- jcoder 3y agoOr perhaps their coin was biased
- m3kw9 3y agoWhat about if they were flipped by computer random functions
- fowlie 3y agoVery interesting! Is there reason to believe that the outcome would be different if the experiment was re-run but replacing the humans with coin-flipping machines?
- kortex 3y agoFolks have made flipping machines than can achieve near perfect reliability. Same guy mentioned in the paper. https://www.npr.org/2004/02/24/1697475/the-not-so-random-coin-toss https://www.npr.org/2004/02/24/1697475/the-not-so-random-coi...
- qup 3y agoSorry, does this mean perfect 50/50, or 100% controllable?
- magicalhippo 3y agoThis reminds me of the opening scene of Rosencrantz & Guildenstern Are Dead[1]. Didn't get to see the play, but loved the movie. [1]: https://www.youtube.com/watch?v=gOwLEVQGbrM https://www.youtube.com/watch?v=gOwLEVQGbrM
- omneity 3y agoThis got me thinking, is it physically feasible to slightly bias the coin against the starting side to get closer to fairness?
- sjducb 3y agoHow do I bias a coin flip? Based on the paper it looks like 55% chance that it will land on the same side it started is possible. This was the most extreme subject. The bias is caused by procession so I want my flip to process as much as possible. Maybe I offset my finger as far away from the center of the coin as possible. Also putting as much force into it as possible is probably a good idea. Finally I have to catch it in a way that the top side is facing up when I reveal it. Any thoughts?
- hddqsb 3y ago@robocat linked a great video showing how to do this: https://m.youtube.com/watch?v=A-L7KOjyDrE https://m.youtube.com/watch?v=A-L7KOjyDrE
- curriculum 3y agoGive it some spin with your index finger. Imagine putting a coin between your index finger and thumb, heads side up, resting on your middle finger below — not too different than how most people start a coin flip. With your index finger, rotate the coin in its plane, so that it stays “heads up” but the head is rotating. Now try doing this and simultaneously flipping the coin by flicking it with your thumb at a point on the bottom, close to the edge. If you do it right, you’ll impart a spin. If you impart a modest spin, the coin will never actually flip over, but will just wobble, and will therefore land heads up. An observer will likely not know what you did, because it is hard for the eye to tell the difference between a flip and a wobble at high speeds.
- rootusrootus 3y ago> procession [...] process Nitpick: Precession. This is one case where a minor mispelling really does throw off the meaning of the sentence. At least to me.
- derbOac 3y agoPlaying games with my family, I've often wondered if there's something similar with dice.
- cantrevealname 3y agoI'm still looking for an intuitive or ELI5 explanation of the mechanism for this bias. The original paper says: The standard model of coin flipping was extended by Persi Diaconis who proposed that when people flip a ordinary coin, they introduce a small degree of precession’ or wobble—a change in the direction of the axis of rotation throughout the coin’s trajectory. According to the Diaconis model, precession causes the coin to spend more time in the air with the initial side facing up. Consequently, the coin has a higher chance of landing on the same side as it started. Another coin toss experiment[1] site says this: The basic reason is that, instead of rotating around a horizontal axis as one might imagine, a typical tossed coin is rotating around a tilted axis which is precessing in 3-space, and this entails a certain degree of "memory" of the initial parameters. The Diaconis paper[2] has the definitive explanation but it's hardly intuitive. I got a feel for why it is, but I can't do an ELI5. The best I'm able to write is this: A human being is likely to introduce some precession in the coin toss. If there is precession, then the angular momentum vector is going to spend more time in the heads direction if starting from heads, and that accounts for the bias. What I think would work well for an ELI5 is an animation of a coin toss showing the angular momentum vector sweeping out a region during its flight, and showing it spends slightly more time pointing toward heads. [1] https://www.stat.berkeley.edu/~aldous/Real-World/coin_tosses.html https://www.stat.berkeley.edu/~aldous/Real-World/coin_tosses... [2] http://epubs.siam.org/doi/10.1137/S0036144504446436 http://epubs.siam.org/doi/10.1137/S0036144504446436
- Q_is_4_Quantum 3y agoPerhaps it helps to imagine someone had a "screwy thumb" and the coin only precesses when they "flip" it (in fact people can train themselves to do this, and its very difficult for you, the sucker, to see in the air that the coin is not rotating but just precessing!). Hopefully its obvious that whatever side is initially facing up will be the same one facing up when its caught? The next step is not at all intuitive to me, namely that even someone trying to do a fair flip causes some precession, and that this isn't decoupled from the rotation.
- robocat 3y agoVideo of how to train yourself to precess the coin and cheat: https://m.youtube.com/watch?v=A-L7KOjyDrE https://m.youtube.com/watch?v=A-L7KOjyDrE
- p00dles 3y agoCould it have anything to do with contact with the hand? Moisture/oil from the hand making the “same side they started on” a tiny bit heavier than the other side? Or, and I’m no physicist, static electricity or something?
- Q_is_4_Quantum 3y agoGiven access to repeated uses of a coin of unknown bias "p" (which is not 0 or 1) you can (eventually) always generate a new coin flip with bias given (exactly) by: 1. 1/2 (i.e fair - von Neumann) 2. p^2 3. p^2/(p^2+(1-p)^2) 4. sqrt(p) Number 4 really surprised me, I learned it from this paper: http://www.math.chalmers.se/~wastlund/coinFlip.pdf http://www.math.chalmers.se/~wastlund/coinFlip.pdf But you can never generate the biases: 5. 2p 6. 4p(1-p) Although... if you change the game to allow a quantum coin then 5. and 6. are possible (a paper of mine: https://arxiv.org/abs/1509.06183 https://arxiv.org/abs/1509.06183)
- matteoraso 3y agoI actually used to be really good at manipulating this as a kid. Basically, if you toss a large coin with a stiff arm, you can get it to flip exactly 1 and a half times before you catch it. I would always use this to win bets with my friends.
- gowld 3y agoIf you are allowed to cheat, you can also spin the coin to not flip at all.