Definition
The gambler's fallacy is the belief that, after a streak of one outcome, the opposite outcome becomes more likely in independent trials — that a coin which has just shown five heads is somehow 'due' for tails. The fallacy assumes a mechanism (a balancing force, a memory of past outcomes) that does not exist when trials are genuinely independent.
The mistake is so universal that it has its own name in psychology, gambling literature, and behavioural economics. The most famous historical instance: at Monte Carlo in 1913, black came up 26 times in a row, with players losing millions betting that red was 'due'.
Why it matters
How it works
When trials are independent (the formal condition: P(A and B) = P(A) × P(B)), the past gives no information about the future. The fallacy substitutes a different mental model: the trials are imagined to be sampling from a small bag of fixed outcomes that gets 'depleted' as you go. That model fits sampling-without-replacement experiments — but the wrong setting for coin flips, roulette spins, or die rolls.
The corrective is to ask explicitly whether outcomes are independent, and if so, to refuse all reasoning that depends on past streaks. The dice do not remember; the coin does not balance.