The Gambler's Fallacy: Is Tails Ever "Due"?

A fair coin has landed heads five times in a row. What’s the chance the next flip is tails?

It feels like it should be more than 50%. Surely tails is due. But it’s exactly 50%, just as it always was. Believing otherwise is the gambler’s fallacy.

Why it’s wrong

Each flip of a fair coin is independent: the coin has no memory of what it did before, and nothing in the world is keeping score. The chance of heads on the next flip is 1 in 2, whatever came before.

Five heads in a row is unusual: it happens about 1 time in 32. But once it has happened, it’s done. The question is only about the next flip, and that’s a fresh 50/50.

The night at Monte Carlo

On 18 August 1913, at the casino in Monte Carlo, the ball on a roulette wheel landed on black 26 times in a row. As the streak grew, gamblers bet more and more heavily on red, sure it was overdue. Each spin was still close to 50/50, and the casino made millions of francs. The gambler’s fallacy is sometimes called the Monte Carlo fallacy because of it.

Why it feels so right

We expect small samples to look like big ones. We know that over thousands of flips, heads and tails come out about even. So we expect even a short run to balance itself out. But the balancing happens by dilution, not correction: the early streak gets swamped by thousands of later flips, not cancelled by extra tails. See the law of large numbers.

Real randomness is streakier than we think. Ask people to write down a “random” sequence of 100 coin flips, and they alternate far too often and avoid long runs. In 100 real flips, a run of six or more in a row is more likely than not.

The opposite mistake

Some people make the reverse error: believing a streak will continue. A basketball player on a “hot hand”, or a roulette colour that’s “running hot”. For a truly random process, like a coin or a fair wheel, that’s wrong too: streaks don’t predict anything.

(Human performance is different from coins. Whether real players get genuinely hot streaks has been hotly debated by researchers, and recent studies suggest a small hot-hand effect may be real. Coins and roulette wheels, though, have no confidence to gain.)

Where it matters

• Gambling. Betting systems that raise stakes after losses assume a win is “due”. It isn’t, and the losses can grow fast.
• Family planning. After several boys, many parents feel a girl is more likely next time. The odds are about the same as ever.
• Judging decisions. Studies have found that judges, loan officers and umpires are slightly less likely to rule the same way several times in a row, as if balancing out.

Try it

The coin above flips for real. Look only at what comes straight after five heads in a row, and see how often tails shows up.

It’s the first question of the free probability test.