[fbartos]

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) 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 And the open data, code, and video recordings at OSF: 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

[noobermin]

>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]

The 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]

Make 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.

[kqr]

...from where does the money come that you pay back? Even if you have an unlimited stake, your winnings will be constrained by the counterparty eventually.

[mewpmewp2]

Right, I forgot, you also need someone willing to take those bets. So I think what you should do is make bets against multiple casinos/institutions where you can develop an algorithm that will find you an optimal method of betting for reasonable 50/50 results, if it makes easier to think. So for example at some point you might want to go to a casino and put the max bet on a single number in roulette, but do it enough times that you would have 50% odds of winning.

Once that is exhausted, you would have to become more creative, like trying powerball enough times, but I'm not sure how good the odds are there vs the reward. Maybe that wouldn't ever work.

Actually, I forgot. You can just play with highly leveraged options. It's not infinite yet, but come back to me until you've multiplied enough times that even options are not enough.

Forget everything I said before, just play with options and automated algorithm to buy more. And post here once you can't buy any higher cost options, and we'll figure something out together.

[andrewinardeer]

The Federal Reserve. They can print money arbitrarily.

[cortesoft]

You also have to factor in your time cost. Your hourly rate is going to be really low, better off just getting a job.

[thih9]

> in order to get just the $19, assuming you played a modest 20 times a day, it'd take 10 weeks (not including weekends)

What if one play session consisted of 10 coin tosses (each an independent $1 bet)? I guess 20 games like this per day would still be doable. Would that mean $19 per week?

Next we up the bet to $10 per throw.

[nomilk]

It becomes clear why there's a same-side bias when watching the video: 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.

[fbartos]

There 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]

How 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?

[glandium]

With 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.

[gowld]

Bouncing erases the bias due to flipping.

[Semaphor]

> 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.

[jonahx]

From 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?

[fouronnes3]

What is the physical explanation for this bias?

[bradrn]

It’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]

After 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.

[aspenmayer]

Your comment reminded me of two-up.

> Two-up is a traditional Australian gambling game, involving a designated "spinner" throwing two coins, usually Australian pennies, into the air. Players bet on whether the coins will both fall with heads (obverse) up, both with tails (reverse) up, or with a head and one a tail (known as "Ewan"). The game is traditionally played in pubs and clubs throughout Australia on Anzac Day, in part to mark a shared experience with diggers (soldiers).

https://en.wikipedia.org/wiki/Two-up

[pirates]

Two-up sounds pretty fun. Your comment in turn made me think of Chō-han, which is somewhat similar but involves rolling dice instead of flipping coins. https://en.m.wikipedia.org/wiki/Ch%C5%8D-han

[aspenmayer]

It’s wild and raucous and usually takes place outside pubs and local workers’ clubs, lawn bowl clubs etc. The spinner puts the coins on a flat stick made for the game, but basically a popsicle stick but wider like a tongue depressor. They toss the coins up and flick the stick to tumble the coins. If you called the two matching coins correctly, you double your bet in winnings. Each game takes like 30s-1m and they go on from mid-morning til early afternoon ish.

The losers indirectly pay the winners based on your call of two heads or two tails and if there’s a split, the house wins that game.

Most of the time is spent drinking your beverage of choice, yelling and cursing your own luck and talking smack to the coins, the spinner being booed or cheered for a well run game or a bad string of luck, but the bets are typically fairly low in my experience, although it’s up to each individual player what they bet, but the spinner or venue sets the bet amount per section or per spinner. Each spinner usually takes a set amount, like $5/$10/$20/$50, even $100 in some cases. It’s all cash and pretty much the honor system in that they aren’t handing out receipts or tickets with your bet, so that sets an upper limit on how many bets each spinner can keep straight. No one wants to see someone lose their shirt, so most folks play against their friends/mates for fun, and ultimately you’re all playing your own game because you decide your bet and it’s actually fairly unpredictable due to the crowds milling around the spinners, of which there will be many, all taking bets and running games independently and simultaneously, and players can place bets in any or all of the games around them if they want.

It’s a wild affair. Highly recommended.

[rstarast]

I 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]

Yeah, 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.

[alkonaut]

> 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").

[gala8y]

This 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]

Yeah, 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]

I 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]

Noticed 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).

[gala8y]

This. When I tried to comment briefly yesterday that the coin falls on the same side, only then did I remember that I actually did that and that it has nothing to do with physics but rather neuroscience (and innocently bent morality - which was also the object of my internal observations). then I remembered that I had actually considered different coin sizes, but I was never as thorough in my attempts to bend the results as you were. oh... the playgrounds of childhoods...

[JKCalhoun]

Fair coin, unfair flip.

[kzrdude]

Fun that you're showing your own work. Since this is your work, it could have Show HN in the title; I guess.

[gala8y]

No, not really.

https://news.ycombinator.com/showhn.html

[input_sh]

If there isn't a product one can try, it's not a Show HN material.

[kzrdude]

I see, it's a bit more narrow than I knew