Gambler's fallacy
After eight heads in a row, surely tails is due? It is not. A coin has no memory, and the chance of heads on the next flip is still exactly 50%. This simulator flips a coin by itself, 100 times a second, keeps track of every streak, and shows what came right after each one. A fair coin flips by itself, 100 times a second. Set a streak length and watch what comes right after every streak that long: the same face again about half the time, however long the streak was. The coin has no memory. Turn on Only the flips after a streak to see each of those moments. Click the coin to flip it once.
How it works
- Press Flip and let the coin run. The tape shows the latest flips with the streaks marked.
- Set a streak length, such as 5, and turn on Only the flips after a streak to see each of those moments.
- Read the chart: after a streak of any length, the next flip is the same about half the time.
A coin has no memory
Each flip of a fair coin is independent of all the flips before it. The coin does not know that it just came up heads eight times, and the chance of heads on the ninth flip is ½, as on any other flip. Believing that tails is now due is the gambler's fallacy. Its mirror image, believing that a streak will go on because it is hot, is just as wrong for a coin.
Why streaks feel wrong
People expect random sequences to switch between heads and tails more often than they really do. In 100 flips a run of at least five heads or five tails in a row turns up almost every time, and runs of seven or more in about half of all sequences. When we see such a run we suspect a pattern, and our sense of fairness wants it corrected. The coin has no such urge.
Monte Carlo, 1913
The most famous case happened at the Monte Carlo Casino on 18 August 1913, when the ball in a roulette game fell on black 26 times in a row. Gamblers lost fortunes betting on red, sure that the streak had to end. The chance of red was the same on every spin: 18 pockets out of 37, just under one half.
Then why does it even out?
Over many flips the share of heads does approach 50%, but not because tails catches up. An early lead of a few heads is simply drowned out by the thousands of flips that follow. The difference between the numbers of heads and tails even tends to grow, roughly like √N, while the share settles. That is the law of large numbers, and it needs no memory.
Frequently asked questions
What is the gambler's fallacy?
The mistaken belief that after a streak of one outcome the other outcome becomes more likely, for example that tails is due after many heads. For independent events like coin flips the chances stay the same.
After 8 heads in a row, what is the chance of heads?
Exactly 50% for a fair coin. The chance of eight heads in a row, seen from before the first flip, is 1/256, but once they have happened the next flip is a fresh 50/50.
Why is the chart not exactly 50% for long streaks?
Long streaks are rare, so those bars rest on few cases and wobble. The error bars show the range that would contain the true chance in 95% of runs. With more flips they shrink around 50%.
Is the gambler's fallacy the same as the hot hand?
They are opposites. The gambler's fallacy expects a streak to end; the hot-hand belief expects it to go on. For a fair coin both are wrong. Whether people, such as basketball players, can really have a hot hand is still debated.
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