Coin Flips Are Biased

Experimental result:

Many people have flipped coins but few have stopped to ponder the statistical and physical intricacies of the process. In a preregistered study we collected 350,757 coin flips to test the counterintuitive prediction from a physics model of human coin tossing developed by Persi Diaconis. The model asserts that when people flip an ordinary coin, it tends to land on the same side it started—Diaconis estimated the probability of a same-side outcome to be about 51%.

And the final paragraph:

Could future coin tossers use the same-side bias to their advantage? The magnitude of the observed bias can be illustrated using a betting scenario. If you bet a dollar on the outcome of a coin toss (i.e., paying 1 dollar to enter, and winning either 0 or 2 dollars depending on the outcome) and repeat the bet 1,000 times, knowing the starting position of the coin toss would earn you 19 dollars on average. This is more than the casino advantage for 6 deck blackjack against an optimal-strategy player, where the casino would make 5 dollars on a comparable bet, but less than the casino advantage for single-zero roulette, where the casino would make 27 dollars on average. These considerations lead us to suggest that when coin flips are used for high-stakes decision-making, the starting position of the coin is best concealed.

Boing Boing post.

Posted on October 16, 2023 at 7:06 AM17 Comments


GregW October 16, 2023 8:35 AM

The study authors interest and observation about people-specific variances in frequency of whether their coin flips land on the same side caught my eye. Was that deemed truly statistically significant?

I ask because this people-specific phenomena really bedeviled academic investigations into measuring mind control over attempted-to-be-random events done by tenured (ie couldn’t be fired) engineering faculty at Princeton Engineering Anomalies Research lab in the 80s/90s/00s. They setup many experiments with people trying to influence various classic random phenomena with their minds to obtain non-random outcomes. My perception was that they found small effect sizes but they kept finding effect sizes that were statistically significant particularly when looking at certain individuals. What seemed to intrigue those researchers most was that some people seemed to have more impact than others and that too was statistically significant. Over time the critics whittled down the findings:

I would wait to see if others replicate these coin flip findings before making too much of them.

Mexaly October 16, 2023 11:11 AM

It takes little practice to learn to “flip” a coin so it lands without reversing. I tried it long enough to realize a person could get good at it.
Coin tosses are very, very rarely fair.

Clive Robinson October 16, 2023 12:07 PM

@ Bruce, ALL,

First off, notice how many contributors the paper has, apparently most were “contributing thumbs” for the coin flips 😉

This is not the first time the physics of coin tossing has come up on this blog,

As has been noted in the past nearly all coins are in fact “heads-tails biased” very slightly due to the fact they have been “struck” / “embossed” thus the density on either side of the coin is not the same and so the center of gravity about which it spins is off-set. However the last time I tried calculating it, it came out,

“below one part in ten to the five”

which is for all but a few, well below what their available test instruments could measure. So under normal circumstances it would be considered negligable against other variables in a coin toss made by humans. Who shall we say lack the fineness of control that machines can have 😉

But, look at the lower left pannel in fig 2 of the paper, and you will see that despite using many different coins, they found a small heads-tails bias…

That said the right hand pannel is rather more of interest…

Which is about,

“The model asserts that when people flip an ordinary coin, it tends to land on the same side it started”

I’ve noted this in the past and given a reason why it might be and it was not due to “wobble” in flight but human issues.

I’ve also described how I’ve honed that and used it to advantage to win coin tosses. I got it down to a fine art such that wining ten out of ten tosses was shall we say uninteresting except to the observers who could be incredulous. Especially when I made it so that they would call correctly ten times out of ten.

But this is not what the paper is about. As such it is a data gathering and analysis, that shows that there is an effect… But does not test/eliminate what might be a cause of the observation.

Q October 16, 2023 12:43 PM

This problem was solved long ago.

Flip the coin twice. If the results are the same then discard and go again. If the results differ use the first result.

Same results (discard):
p * p
(1-p) * (1-p)

Different results (use the first):
p * (1-p)
(1-p) * p

The coin and flip can be biased to p=0.99 or 0.01 and anything non-zero and non-1.

Anonymous October 16, 2023 12:57 PM

For the casino odds in the closing paragraph: Maybe the margin is comparable, but I don’t think it is worth the effort of 1000 tosses (and the croupier could cheat quite easily)…

Still a fun experiment.

Clive Robinson October 16, 2023 7:28 PM

@ Q,

“This problem was solved long ago.”

It’s actually inefficient because you throw away so many coin flips without sufficient reason.

But as I’ve mentioned before you can easily implement the idea in logic chips using a couple of latches (74LS74) and either nand gates (74LS00) to make an xor or XOR gates directly (74LS86). Back when I was designing and building TRNG’s using the “Roulette Wheel” method of generation.

The algorithm was devised by John Von Neumann and you can read a copy of what he wrote,

anon October 16, 2023 10:13 PM

What about 75 heads in a row?

Thanks, Tom Stoppard.

The question is can you use a robot to reliably get a heads or tails? I bet Shane Wighton could build a robot to test this.

Clive Robinson October 17, 2023 3:04 AM

@ Anon,

Re : Coin Tossing reliably.

“The question is can you use a robot to reliably get a heads or tails?”

As indicated above humans can with practice do as many heads or tails in a row as you like.

Further have a look in the paper @bl5q sw5N above provides a link to, you will see pictures of a simple mechanical device that reliably flips coins.

Adding a simple “robot arm” or similar is just simple electronics and mechanics I would expect most engineering or physics under grads to do as a small group project.

The fact nobody has done it and attracted news attention is because I suspect it’s a,

“So what, that’s dull dull dull”

But there is another aspect…

There is “Online Gambling” where idiots bet on the outcome of dice being thrown, and those running it make a big show of having the dice thrown/rolled by a mechanical device that has a video feed in real time.

Ask yourself the question,

“As we know a mechanical coin flipper can be made to flip a coin to a desired outcome, how much more effort would be required to do the same with a dice?”

Before you start waving your arms around consider this.

“What is the difference between a coin and a dice?”

The answer is from the point of view of physics “thickness”…

Because, if you take a bar/rod of material of uniform density and cut a thin slice off the rod you get a “coin blank” increase the thickness to the point where it’s the same as the cross section of the bar then you have a “dice blank”. The rest is just “finishing”.

Such rods and blanks are made with “milk proteins” turned into the oldest plastic known “casein” which is made in a very similar way to cheese.

It’s still made into rods and bars and similar industrially for making “artificial bone” etc for making buttons for the clothing industry. Also handles on cutlery and keys for instruments, as it’s considered safer than nitro-cellulose that has a habit of going “high-order” when used for things like billiard balls…

Yes it realy happened and was actually accepted as a price to pay back in the Victorian era for an upper middle class life style. You can blaim a man by the name of John Wesley Hyatt who read about a $10,000 prize for finding a replacment for elephant and walrus tusks (ivory) that were in increasingly short supply in the 1860’s as the new wealthy middle class wanted the niceties of life formally enjoyed only by the landed gentry and barrons etc. John began experimenting and came up with the new wonder chemical of the age “celluloid” (which by the way is still used for making ping-pong balls). Chemically it’s based on cellulose nitrate what might politely be called a pre-cursor to gun-cotton a very powerful explosive, for the time. The thing is whilst celluloid does burn extreamly vigoursly it generallt burns well below the speed of sound. However change the nitrates around and you get TNC… This can and does happen with celluloid if you don’t take certain precautions during manufacture that were not realised at the time… So yes John was inadvertently the inventor of the “Exploding Billiard Balls”, but… He and his brother first used celluloid for making another Victorian necessity “dentures” and no I do not know if any of them ever exploded in use, it’s the sort of thing that would ruin Xmas dinner if gramps used his new nashers to chomp down on a walnut or similar…

denton scratch October 17, 2023 5:40 AM

Apparently the effect is only observed when the coin is allowed to land on the ground; it disappears if you toss coins the way I do – I catch the spinning coin, and invert it onto the back of my other hand.

This seems counter-intuitive. The more the tosser handles the coin, the more opportunities he has to influence the outcome, and you’d expect a coin landing on the ground to tumble, in an unpredictable way. A caught coin is predictable from the moment it’s caught.

When I toss a coin, I balance it on my index-finger, then flick the edge sharply with my thumb. That’s not a “flip” in the sense that the coin turns over once or twice; it spins fast, so that it becomes a blur. I wonder if the phenomenon of predictable/controllable coin-tosses is restricted to cases where the toss really is a “flip”, and the coin is only inverted a few times.

John Levine October 17, 2023 9:25 PM

A few days ago Matt Levine’s Bloomberg “Money Stuff” newsletter had a segment on the way finance types in general and Sam Bankman-Fried in particular bet on coin flips. He referenced that paper with all the authors but as a throwaway at the end mentioned an acquaintance at a large financial firm who said “we have a guy who can flip 55% heads.”

Bob Paddock November 6, 2023 8:27 AM

[Splitting due to the auto moderator]


When the Princeton Engineering Anomalies Research lab – or PEAR Lab was closed in 2007 its work was transferred to International Consciousness Research Laboratories (ICRL). The PEAR Lab equipment was stored in Brenda Dunne’s basement. Brenda was the lab manager for its 28 year run.

Three years ago Brenda donated all the equipment to Broughton Sanctuary, Skipton UK, where a museum has been setup to house the equipment. My friend Rachel has been getting things up and running there.

Broughton is now housing these Random Event Generators (REG) equipment from PEAR in what they call the ‘Wyrd Experience’. Wyrd being an old English name for magic and the exploration into interconnectedness.

The late Brenda, having died early last year, explains how some of the equipment works in these videos. She states that the while the effects are small, they are statistically significant and repeatable.

Bob Paddock November 6, 2023 8:35 AM


My longer post explaining PEAR history and results won’t make it past the auto moderator, has anyone figured out exactly what triggers this thing?

The 3rd annual Science & Consciousness Summit organized by Ubiquity University, starts next Sunday Nov 19th. This years theme is Non-Linear Time. It would be great to see @Clive Robinson there to explain to people how Random Event Generators REGs could be done better. This is where PEAR equipment is now.


Related would be the Global Consciousness Project 2.0 (GCP 2.0) is a newly envisioned and expanded version of the global network of physical electronic devices which detect and quantify important aspects of human consciousness (i.e. attention and emotions). As these effects are observed on a global scale, they indicate a deep interconnectedness between human consciousness and the physical world. GCP 2.0 was developed and is maintained by the donor-funded HeartMath Institute (HMI), a nonprofit research and education organization in collaboration with other scientists and engineers from around the world.

Scott November 15, 2023 9:13 AM

denton scratch wrote “A caught coin is predictable from the moment it’s caught.”. This is true; as am amateur magician I’ve taught myself to flip quarters and produce whatever side I want with the toss – catch – slap-on-back-of-hand procedure, even if the toss is “called in the air”. I’ve found that newer quarters are better, and I think the Connecticut state quarters are the easiest.

Clive Robinson November 15, 2023 11:00 AM

@ denton scratch, ALL,

Re : The rate of spin fails in the service of magic.

“I wonder if the phenomenon of predictable/controllable coin-tosses is restricted to cases where the toss really is a “flip”, and the coin is only inverted a few times.”

The laws of physics apply when no agency is involved.

However as I explained above,

“I’ve also described how I’ve honed that and used it to advantage to win coin tosses. I got it down to a fine art such that wining ten out of ten tosses was shall we say uninteresting except to the observers who could be incredulous. Especially when I made it so that they would call correctly ten times out of ten.”

The secret is not in how many times the coin spins in the air, it’s in the catching it and what you do next.

When you catch the coin out of the air in your cupped hand you can see which side is facing you. As you bring your hand down so the coin is on the back of the hand then the other side of the coin would be up. UNLESS in the proces knowing what is up you slight of hand it over.

For those starting out and honing this apparently magic trick, remember catching the coin high gives you more time to see it and flip it. The fact you flip it so high in the air it goes over so many times, incorrectly makes the person calling think you can not put the fix in.

I used to tease my son with it when he was young, so that he would learn one of lifes most important lessons,

It is an important and popular fact that things are not always what they seem.

(Douglas Adams version 😉

Scott November 15, 2023 2:31 PM

@Clive Robinson You don’t need to see the coin to know which side is up in your hand. Once you catch the coin, use your thumb to feel which side is up as you start the slapping motion. If the “called” side is up, curl your fingers to turn the coin over. My record with a random quarter is 17 correct in a row. For some reason people watching me always think I control the flip, but that would be doing it the hard way.

Dave November 16, 2023 8:23 PM

This is only my opinion, But after a bit of critical considerations I find this assumption that coin tosses are biased to be flawed. So I will discount the results as a poor experiment.

I can find it believable that a person can toss a coin and get consistent results.

Consider only one variable… Let’s consider that if the distance to landing was reduced by 1/2 a coin spin, the results would be the opposite.

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