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The Gambler's Fallacy and Other Tricks of the Mind

After five heads in a row, tails feels due. It is not. Understanding why is one of the most useful pieces of everyday probability.

The fallacy

The gambler's fallacy is the belief that if something happens more often than usual for a while, it will happen less often in future, or the reverse. A fair coin has no memory, so after a run of heads the next toss is still an even chance.

Independence

Two events are independent when the outcome of one does not change the probability of the other. Coin tosses, dice rolls and draws from a well-run lottery are designed to be independent. Card games are different: a dealt card is no longer in the deck, so the odds of what remains really do change.

Its mirror image

The "hot hand" feeling runs the other way: a streak seems likely to continue. Whether streaks are real depends on the activity. In skilled play, confidence and form may matter; in a machine driven by a random number generator, they cannot.

Why the brain falls for it

People expect short sequences to look like long-run averages, so a run of identical results feels like a correction is coming. Real randomness is lumpier than intuition expects, which is why genuinely random playlists and coin tosses look suspicious.

Machines that cannot remember

Electronic games of chance generate each result independently with a random number generator. That makes "signals", "patterns" and "it has not paid out for a while, so it is about to" claims a textbook case of the fallacy. One clear debunking, written in Traditional Chinese about a particular slot game, walks through the mechanics and explains why so-called winning signals in slot games do not hold up (老虎機「必勝訊號」為何站不住). The argument applies to any game whose outcomes are generated independently: no past result tells you anything about the next one.

filed 2026-10-22