Is a coin flip really 50/50?

The share settles. The streaks don’t.

Two things happen as the pile of flips grows, and they feel like they contradict each other. The share of heads drifts towards 50% — not because the coin is correcting itself, but because each new flip is a smaller and smaller slice of the total. Ten flips can come out 7–3 and nobody should blink; a thousand flips coming out 700–300 would be astonishing.

Meanwhile the streaks never stop. In a hundred fair flips, the longest run of one side is most often six or seven; about four sequences in five contain a run of at least six. Your housemate’s four heads in a row were not a sign of anything except that four is a small number.

Jacob Bernoulli proved the first half of this in a book published in 1713 — the law of large numbers — and schoolteachers have been using the second half ever since to catch students who fake their coin-flip homework. Invented sequences almost never contain the long runs real ones do.

So is a real coin fair? Almost.

Here the story takes a small twist. In 2007 the statistician Persi Diaconis and colleagues filmed coins with a high-speed camera and worked out the physics of a wobbling flip. Their prediction: a coin flipped and caught in the hand lands the same way up it started slightly more often than not — about 51%.

In 2023 a team led by František Bartoš put that to the test the only way that works: 48 people flipped coins 350,757 times. The coins landed on the same side they started 50.8% of the time. Heads versus tails, though, was exactly even — 50.0% — because the bias is about the starting side, not the face. It varied a lot from flipper to flipper and shrank as people practised. The paper won an Ig Nobel Prize, and then a place in the Journal of the American Statistical Association.

Two more things physics will tell you. A tossed US nickel has been estimated to land on its edge about once in 6,000 flips. And never settle anything by spinning a coin: in one Berkeley class, pennies spun on their edge came up heads far less than half the time, some students’ coins less than 10%.

Back to the bins.

Four heads in a row proves nothing about the coin. Best of five is as fair as best of three, which is as fair as one toss; the extra rounds just delay the chore. If you want to be precise about it, agree on the number of rounds before the first flip, and let the other person call.

Or use a coin that has no starting side, no weight and no feelings. Ours is one tap away, and it has never once hated anybody.