Is a Coin Flip Really 50/50? What 350,757 Flips Found

The player who kept winning the toss

A five-a-side game I played in for a couple of years started with a coin toss, always the same battered coin out of somebody's kit bag. One of the regulars never called heads or tails. He called whichever side was already facing up when the coin left your hand, and across a season he won noticeably more than his share. It became a running joke about his luck.

It was not luck, and it was not sleight of hand either. He had read a paper. The rest of us had not.

Heads or tails really is 50/50

Start with the part of the folklore that survives, because most of it does. Flip an ordinary coin, ask somebody to call heads or tails, and that is a genuine even bet. The same 2023 study that found the effect below measured this directly: with the starting side randomised, the probability of heads came out at 0.500, with an interval running from 0.498 to 0.502. Nothing. And the absence of a lean did not vary from one currency to the next.

So a modern coin is not weighted toward one face in any way you could use. Call heads every time and you will win about half. The bias is not in which face. It is in which face you started with.

The 2023 study: 350,757 flips

A team led by Frantisek Bartos at the University of Amsterdam recruited 48 people and had them flip coins from 46 currencies, 350,757 times in total, recording which side started face up and which side landed. Coins came down matching their starting side 50.8 per cent of the time, with an interval from 50.6 to 50.9 per cent. Small, but a long way from zero: about one flip in 125 nudged back toward where it began.

This was not a surprise result. In 2007 Persi Diaconis, Susan Holmes and Richard Montgomery published a physics model of a hand-thrown coin that predicted close to it, around 51 per cent. A thumb does not impart a clean rotation. The coin precesses as well as spins, wobbling about its axis like a plate slowing down on a table, and that wobble means it spends slightly more of its flight with the original face pointing up. Slightly more time up, slightly better odds of coming down that way. The 2023 flips were the large-scale test of a prediction that was already sitting there, and it held.

The detail most write-ups leave out is how unevenly the effect was spread. It was not a constant across the 48 flippers: some showed a clear same-side lean, others showed none at all, which the authors put down to differences in technique. That is why this looks like one person on a lucky streak rather than a law of physics. Some people really do put more wobble into a toss than others, and my five-a-side regular was calling against whoever happened to be flipping.

If Diaconis's name looks familiar, he is also behind the seven-shuffles result for a deck of cards, which has since needed updating too. He has spent a career measuring randomness that everybody takes for granted and finding it is not quite what they assumed.

Spinning a coin is a worse bet than flipping one

These are two different experiments and they get different answers. A coin spun on a table balances on its edge as it slows, and an edge is exquisitely sensitive to where the metal sits. Any asymmetry in the design gets amplified while it rattles down, and the coin tends to fall onto its heavier face. The stock example is an older US penny, which people who have tried it will tell you comes up tails well over half the time when spun.

Treat that particular number as folklore rather than as a finding from the 2023 study, which tested flipping and not spinning. The mechanism is sound regardless. If somebody offers to spin a coin instead of flipping it, you are being offered a shakier proposition, and it is fair to ask for a flip.

When 50.8 per cent is worth caring about

Almost never. If the toss decides who does the washing up, an edge of eight parts in a thousand is noise, and you would need hundreds of tosses before you could even see it.

It becomes a real advantage only when three things line up at once: the toss decides something worth having, the caller can see which side starts face up, and the caller gets to choose. That is the combination my five-a-side regular had wandered into. Break any one of the three and it disappears.

• Hide the starting side. Shake the coin between two cupped hands before the throw, so not even the flipper knows which face was up.
• Let the other person call. If the call is made while the coin is in the air, nobody can feed the caller a starting side.
• Catch and turn. Catching the coin and slapping it onto the back of the other hand, the way most sports do it, inverts the starting face and cancels the effect outright.

That last one is why a properly run sporting toss was never really exposed to this in the first place.

What a digital flip does instead

A virtual coin has no starting side, no wobble and no heavier face, because there is no coin. The result is a number before it is ever a picture of one.

Here is exactly what happens on this one. Each flip asks the browser's Web Crypto generator, crypto.getRandomValues(), for a 32-bit random number and checks whether it came out even or odd. Even is heads, odd is tails. There are precisely as many even values as odd ones in that range, so the split is exact rather than close, and there is no technique, no starting face and nothing for anybody to read.

That is a cleaner job than the same generator can do for a six-sided die, which is not the obvious way round. Six does not divide evenly into the range of a 32-bit number, so a fair die has to throw away the draws that land in the uneven tail and take them again. Two divides that range perfectly, so a coin never has to discard anything at all.

The animation is decoration. It runs for a fixed length of time and carries no information: the number is drawn when the spin finishes, and the coin is then simply told which face to show. There is no physics underneath that could drift one way, which is the same arrangement the wheel uses. If you would rather check the generator than take my word for it, the randomness page runs a live chi-squared test on it across up to a hundred thousand draws.

The flip counter is the part teachers want

The tool keeps your last twenty flips and shows a running tally underneath: total flips, how many heads, how many tails, each as a percentage. The history survives a page reload, so a lesson can pick up where it left off rather than starting from nothing.

That tally is a law of large numbers demonstration that costs nothing to set up. Flip ten times in front of a class and the split is usually lopsided: seven to three, eight to two, sometimes worse. Students read that as a rigged coin, because "random" sounds like it ought to mean "evens out quickly" and it does not. Keep going and the percentage crawls toward fifty. The convincing part is not where it ends up, it is watching an early run that looked obviously unfair turn out to be nothing at all.

The same idea applied to dice, including why a result being "due" is a fallacy, is in the random number generator versus dice post.

What it will not do

• One coin per tap. There is no flip-ten-at-once mode. You tap repeatedly and read the history.
• No weighted coin. You cannot set it to 60/40 to model a biased coin for a statistics lesson. It is a true even split and nothing else.
• No best-of-three button. Run three flips and count them yourself.
• The history stops at twenty. Older flips drop off the list and there is no export, so it is a live demonstration rather than a way to log a long experiment. For a run of hundreds, write the counts down as you go.

So which should you use

A real coin is fine for a casual call, provided nobody in the group is quietly working the starting side. It needs no screen and no battery, and the ritual of it is half the appeal.

Reach for the digital version when the toss has to be above suspicion, when it is happening over a video call where nobody can see anybody's hands, or when you want the counter as a teaching aid. And when the real question is yes or no rather than heads or tails, the yes or no wheel is the more honest instrument, because it does not dress a plain binary choice up as a coin. Whether to hand the decision to a toss at all is a different question, and the guide to deciding without overthinking is about that one.