Gambler's fallacy

The gambler's fallacy is the mistaken belief that an "overdue" result becomes more likely after a run of opposite results. After several "reds" at roulette, you convince yourself that "black" is now "due". This intuition, very widespread, is false.

The error comes from a confusion about the notion of independence. Draws are independent when the result of one in no way influences the next. A coin, a die, a wheel have no memory: they do not "know" what just came up and seek to compensate for no imbalance. On every turn, the device starts afresh.

The numbers show it clearly. In a coin toss, the probability of getting heads is 1/2 on every toss, whatever the past tosses. After four consecutive heads, the probability of a fifth heads stays 1/2, neither more nor less. What is rare is the whole sequence seen in advance — five heads in a row have a probability of 1/2 to the power of 5, that is 1 in 32 — but once the first four heads have fallen, the fifth toss knows nothing of what came before.

The trap is reinforced by a confusion with the law of large numbers. This law says that the share of heads approaches half over a very large number of tosses, but it works through no toss-by-toss "catch-up": the deviations are not erased, they are diluted in the mass of draws. No force actively restores the balance.

This fallacy feeds many losing strategies, where one bets on the "overdue" outcome believing one can force fate. On this site, the drawing tools are designed to be independent on each use. Understanding that chance has no memory is probably the most useful key to a responsible reading of chance, because it dissolves the illusion of control.