8 ms·
About 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 i
by fbartos 3y ago
About 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.
- bradrn 3y agoIt’s in the paper: > The standard model of coin flipping was extended by Persi Diaconis [12] 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 (i.e., ‘same-side bias’). [12] Diaconis P, Holmes S, Montgomery R. Dynamical bias in the coin toss. SIAM Review 2007; 49(2): 211–235.
- Philip-J-Fry 3y agoAfter reading this the first thought I had was how do you stop people flipping the same way? Like, give me a baton and I could throw it at varying heights and control which side I caught it on. In theory the same applies to coin flipping. You can get quite consistent with your positioning and power. You could probably control for it by making people alternate which side was face up before the flip. That's my intuition anyway.
- gala8y 3y agoThis was my intuition in childhood. If you choose tails to be yours and start with tails then catch it, it is most likely to be tails. I came up with this observation myself. Weird.
- kqr 3y agoYeah, I think you were just lucky that your superstition happened to be true. Trying not to be disrespectful but I don't believe you intuited a 50.8 % bias. So nothing weird going on at all.
- gala8y 3y agoI know it's not on par with any real stats, but still... I just had this strong conviction that worked this way. The really interesting stuff is why it is so. I would think along lines of brain timing tossing and catching, eye-brain-timing rather than gravity and coin itself. Added: Yeah, now I remember actually manipulating timing of catch to achieve this. > Trying not to be disrespectful No worries, it is just me using high context communication style, where I assume that you know that I know this and I just share what was my experience in childhood (it was not like a single thought).
- bryanbuckley 3y agoNoticed as a kid I could flip a quarter with a certain consistency, so I experimented a bit and quickly got to be >90% accurate with an ordinary (controlled) flip. Pretty simple. In fact I just picked up a quarter and practiced (20+ years out of practice) and have some observations: 1) harder than when I was a kid, my fingers are lot bigger + stronger so it's not as precise from the start. A bigger and heavier coin would help. 2) the timing factor is bigger than I recalled.. essentially you can watch the coin flipping and get a subconscious/automatic/predictable sort of count/feedback to it. You can bring your hand up to the coin in the air at a precise moment pretty easily and "tell" (>90% accuracy today of the flips I just did that I considered successful before looking at the result) if the flip was predictable. Hand eye coordination, spatial awareness is very correlated to this skill, I suppose. 3) it really is the same side that comes up.. again I think because of the automatic watching/count/completion of full rotations, i.e. catching the coin at the end of a full rotation instead of a partial. Came in handy occasionally.. if I knew I was going to be wrong (other person usually waits to call mid-flip) I could catch the coin a little lower to give myself a chance, or punk them by not putting it on the back of my hand as is more standard (they might demand a re-flip.. kind of like if you are playing rock paper scissors and one person goes on 3 and the other on 4).
- Semaphor 3y ago> but less than that for single-zero roulette (27$). This reminds me of the casino I was at with a fair roulette. Betting Black/Red and getting a zero meant all bets stayed for the next spin ;) My winning strategy was to always bet the opposite color of a friend of mine.
- noobermin 3y ago>If you bet a dollar on the outcome of a coin toss 1000 times, knowing the starting position of the coin toss would earn you 19$ on average. This is more than the casino advantage for 6-deck blackjack against an optimal player (5$) but less than that for single-zero roulette (27$). This sounds like the plot of a western where a man travels from town to town and gleans a little cash from the local waterhole a little every time. I did the math though, in order to get just the $19, assuming you played a modest 20 times a day, it'd take 10 weeks (not including weekends), and by that point people would definitely figure out your trick. In order to make any profit quickly, you'd have to distribute the strategy, after which your secret would explicitly be out there. Even assuming perfectly honest colleagues, having that many parallel people using the same strategy in the open means that before you turn any real profit, people will find out. It's a fun idea to fantasize about though. Anyway, cheers on the paper! Pretty cool result that you guys put the effort in in implementing.
- kqr 3y agoThe upshot is that as long as you only stake $1 at a time, you're unlikely to lose more than $50. On the other hand, /if/ you do, you'll have to play for 6000 more flips until you can be fairly certain that you're even again. What's worse is if, after having lost $50, you're down to your last $50, there's almost a 1/5 chance you'll blow all of it trying to recover if you wager $1 each time. If you grow wise and start Kelly betting you'll get back to your starting $100 on average in 5000 flips, though. If you can take out a loan of $500 first, you can Kelly bet your way to even much faster, in an expected 700 flips. Whether this is worth the interest on the loan depends on how quickly you can find challengers to bet with.
- mewpmewp2 3y agoMake a deal with all the banks or systems that can print money that would allow you to take an infinite loan from them. Then just double the bet every time you lose. If they can print money, why not infinitely as you will always pay it back anyway, so you don't have to worry about introducing inflation. There will always be a point when you can just burn the money that you temporarily introduced.
- nomilk 3y agoIt becomes clear why there's a same-side bias when watching the video: https://www.youtube.com/watch?v=3xNg51mv-fk https://www.youtube.com/watch?v=3xNg51mv-fk These are fairly gentle coin tosses; barely going a foot into the air! When I think of a coin toss, I think high and spinning fast (like the ones before sports games, where the coin goes into the air and lands on the ground, usually rolls a short way, and is collected on whatever side it landed). I would guess the 50.8% same-side bias would be much closer to 50% if the coins were tossed this way in the experiment.
- glandium 3y agoWith a 1 foot to 2 feet toss, landing in the hand, I can get the same side as the starting side more than 90% of the time, without even trying. I wouldn't trust such a toss to be fair. Landing on the floor would change the game.
- fbartos 3y agoThere was indeed a lot of variation in the height of the tosses. I however disagree with the conclussion: two of my friends at the video had the most different height of tosses (one tossed thrice as hight as the other one), yet both of them had exactly the same bias (0.505). The amount of spin is unfortunatelly very misleading from the 30fps videos--the coins often seem like not spinning at all but that's just a result of the poor video quality.
- sixstringtheory 3y agoHow do you control for bias coming from the same coin flipper? Do they usually flip their coin from the same starting height (whatever comfortable arm positioning they have, which I assume would also introduce bias by how they catch it as well) and to the same arc peak height? Or were they encouraged to try a different body position, strength and angle of launch for each flip?
- gowld 3y agoBouncing erases the bias due to flipping.
- JKCalhoun 3y agoFair coin, unfair flip.
- alkonaut 3y ago> This is more than the casino advantage for 6-deck blackjack against an optimal player (5$) I have seen that figure (roughly 0.5% edge) but that has to depend on how deep the shoe is dealt? I remember playing only the last hands with dealers playing down to between 1.0 and 0.5 decks left. That meant you could play hands where you knew almost all remaining cards were suited. I guess the average edge assumes constant bet and doesn't include betting strategies based on counting at all? (And those strategies obviously wouldn't work in any real casino because it's "frowned upon").
- jonahx 3y agoFrom the Method section: "In each sequence, people randomly (or according to an algorithm) selected a starting position (heads-up or tails-up) of the first coin flip, flipped the coin, caught it in their hand, recorded the landing position of the coin" Presumably if you instead allow the coin to land and bounce on a hard surface, the bias would disappear?