What is Gambler's Fallacy?

The gambler's fallacy is the mistaken belief that independent random events are influenced by what came before. After a run of red on the roulette wheel, it whispers that black is now due — yet each spin is wholly independent, and on a single-zero wheel the chance of red stays at 18/37, about 48.6%, however long the streak. Worked example: before any spins, the probability of ten reds in a row is (18/37) to the tenth power, roughly 0.07%, or about 1 in 1,350 — genuinely rare. But once ten reds have landed, that history is spent: the next spin is still 18/37 red. The wheel has no memory, and no outcome is owed. This matters because the fallacy drives dangerous behaviour. Believing a Cold Streak must end, players raise stakes to recoup losses — chasing — which is exactly how a Martingale system blows up a bankroll. Casinos lean into it with scoreboards displaying recent results, nudging players toward the colour that is supposedly due. The core mistake is treating independent trials as if they self-correct in the short run. They do not; only the long-run average drifts toward expectation, with no guarantee about the next bet. Understanding this is central to responsible gambling and to seeing why a negative Expected Value cannot be reversed by reading patterns. See also the Cold Streak, Variance and Expected Value.