Is a coin toss really 50/50?

A coin toss is physics. Once the coin leaves the thumb, its upward speed, its spin and the height at which it is caught decide how it lands. It behaves like a random event only because those starting conditions vary from toss to toss in ways that are too small to control. The question is whether that variation evens out to exactly 50/50. For a normal toss caught in the hand, it very nearly does, with one measurable exception.

The same-side bias

In 2007 the statisticians Persi Diaconis and Susan Holmes and the mathematician Richard Montgomery published an analysis of the coin toss in SIAM Review. Earlier models assumed the coin turns end over end about an axis lying in its own plane. Real coins also wobble, or precess: the axis they spin about changes direction during the flight. The paper shows that this wobble means a vigorously flipped coin spends slightly more of its flight with its starting face up, so it tends to be caught the same way up.

They first built a coin-tossing machine. With careful adjustment it made a coin that started heads land heads every time, which makes the point that the outcome is determined by how the coin is launched. They then filmed 50 human flips with a high-speed camera at about 600 frames a second, recovered the wobble angle from the 27 flips that gave usable measurements, and calculated each flip’s chance of landing as it started. The mean was 0.508, which they rounded to the 0.51 in the abstract.

The model applies to coins caught in the hand. A coin that bounces on a table or floor is a different, less predictable system.

350,757 flips

Diaconis and colleagues noted that checking a 51% bias with a standard error of 1 in 1,000 would need about 250,000 tosses. A study led by František Bartoš, first posted in 2023, went further. Forty-eight people flipped coins from 44 currencies a total of 350,757 times in sequences of 100. The first flip of each sequence started on a randomly chosen side and each later flip started on the side the previous one had landed on; the flippers caught each coin in the hand and recorded the result.

That is the 51% the physics predicted, and no sign of any bias towards heads or tails as such. People differed: individual same-side rates ranged from about 0.487 to 0.601, and the bias fell as people practised, to the point that the authors suggest around 10,000 flips might virtually remove it. The paper was posted on arXiv in October 2023, revised to a fourth version in April 2025, and published in the Journal of the American Statistical Association in 2025.

What 50.8% means in practice

For one toss, very little. If you know which side starts up and call it, you win 50.8% of the time instead of 50%. Over many even-money bets that edge adds up slowly: 1,000 one-pound bets at 50.8% gain £16 on average, but the ordinary scatter of results over 1,000 bets has a standard deviation of about £32, and about 3 runs in 10 still end in a loss.

Detecting the bias takes far more tosses than anyone makes casually. To have the difference between 50.8% and 50% stand out at two standard errors you need about 15,625 tosses; at three standard errors, about 35,000.

The easy fix is to keep the starting side from mattering. If the person calling can’t see which side was up before the toss, or calls while the coin is in the air without having seen it, their call has exactly a 50% chance whatever the coin’s tendency.

How ordinary 50/50 results look

Fair coins produce more streaks and lopsided runs than intuition expects, and none of that is evidence of bias.

Spinning is not tossing

Spinning a coin on its edge on a table is a different experiment, and a much less fair one. Diaconis, Holmes and Montgomery describe data from a class of 103 Berkeley students who each tossed a penny 100 times and then spun the same penny 100 times. The tossed results clustered around 50 heads, as fair coins should. The spun results leaned clearly towards tails, and several students’ coins came up heads less than 10% of the time. A spinning coin’s result depends on the exact shape of its edge and on where its centre of gravity lies, and the authors note that magicians use coins with slightly shaved edges, invisible to the eye, that always come up heads.

The same paper suggests that a tossed coin allowed to land and bounce on the floor may spin on its edge before settling, and so may be less fair than one caught in the hand.

Landing on the edge

A coin can land and stay on its edge. Daniel Murray and Scott Teare, writing in Physical Review E in 1993, combined a dynamical model with experiments in which objects were dropped from a height onto a flat surface, and estimated that an American nickel lands on its edge about once in 6,000 tosses.

At 1 in 6,000, the chance of seeing it at least once is about 1.7% in 100 tosses, 15.4% in 1,000 and 63.2% in 6,000. The figure is for a nickel landing on a flat surface; other coins and surfaces give other rates, so treat it as an order of magnitude rather than a figure for your pocket change.

When it has to be exactly 50/50

For a call between two people in good faith, a tossed and caught coin is fair to within about a percentage point, and hiding the starting side removes even that. For anything where the result has to be beyond dispute, or where you need many tosses at once, a generator is simpler. The coin flip on this site draws each toss from the browser’s cryptographic random number generator, which has no starting side, no edge and no wobble, and the random number generator set to 1 to 2 does the same for a yes-or-no choice.