Is a Coin Flip Really 50/50?

Short answer: 50/50 for heads vs tails

Yes, a fair coin flip is 50/50 for heads versus tails. Each side has the same chance on every flip, and the largest real experiment so far backs that up: across 350,757 recorded flips, heads came up 50.0% of the time. There is one small catch, and it has nothing to do with heads or tails. A coin flipped by hand tends to land on the side that was facing up when it left your thumb, slightly more often than half the time. If nobody knows which side started up, that effect cancels out and the toss is as close to 50/50 as anything in daily life gets. The sections below explain the numbers.

The same-side effect: 50.8% (Bartoš et al. 2023)

A flipped coin lands on the same side it started on 50.8% of the time, according to a 2023 study by Bartoš and colleagues. The team recorded 350,757 real flips on video and logged which face was up before each toss and which face ended up on top. The 95% confidence interval was 50.6% to 50.9%, so the effect is small but real. In plain terms: if you can see heads facing up before the flip, calling heads gives you roughly 51 chances in 100 instead of 50. Over a handful of tosses you would never notice it. The full paper is free to read on arXiv under the number 2310.04153.

Why it happens: wobble/precession (Diaconis et al. 2007)

The same side effect comes from wobble. A real coin does not just turn end over end; it also tilts and precesses as it spins, a bit like a spinning top. In 2007, Persi Diaconis, Susan Holmes and Richard Montgomery published a physics model in SIAM Review showing that this wobble keeps the starting face pointing up for slightly more of the flight. Their model predicted about 51% same side results for a coin that is caught. That was a prediction, not a measurement. The 2023 video study by Bartoš and colleagues later measured 50.8%, very close to what the model said it would be.

Is heads more likely? No

No, heads is not more likely than tails. In the study of 350,757 flips by Bartoš and colleagues, heads came up 50.0% of the time, so there was no difference between the two faces. Any edge comes from the starting side, not from the picture on the coin. A portrait on one face does not pull a flipped coin toward that side in any way the data can detect. So if you are wondering whether to call heads or tails, either is fine. If you happen to see which side is up before the toss, calling that side gives a tiny edge, which is exactly why a fair flip keeps the starting side hidden.

Can a coin be weighted? (Gelman & Nolan 2002)

You cannot meaningfully weight a coin for a flip that is caught in the air. Statisticians Andrew Gelman and Deborah Nolan made this point in a 2002 paper in The American Statistician, titled You Can Load a Die, But You Can't Bias a Coin. A flipped coin turns over in the air and lands in your hand, so where its weight sits barely affects which face ends up on top. Dice are a different story, because they bounce and roll on a surface. The same goes for a coin that is spun on a table: that is where an uneven coin can lean toward one side, which is the next topic.

Spinning vs flipping

Spinning a coin on a table is much less fair than flipping it. When a coin spins on its edge, the rim and the weight of each face affect how it slows down and topples, so some coins lean noticeably toward one side. Gelman and Nolan point to spinning as the case where bias is possible, and the Wikipedia article on coin flipping notes that a spun coin is much more likely to be biased than a flipped one. A flip spends its time in the air, where the coin turns many times and those small differences barely count. If the result matters, flip and catch, or flip and let it land on a soft surface. Don't spin.

How a digital coin avoids physical bias

A digital coin avoids physical bias because there is no physical coin: no thumb, no wobble, no starting side and no table. On FlipACoin, each result comes from our server, which publishes a hash of the secret seed (an HMAC SHA256 commitment) before the flip. After the flip the seed is revealed, so anyone can recompute the result on the Verify page and see that it was not changed. Global Flip rounds also mix in the public drand randomness beacon. The 50.8% same side effect simply does not apply here. What matters for a digital coin is whether you can check it, and with FlipACoin you can.

What studies found