Gambler’s fallacy
Five heads in a row. Surely tails is overdue — the coin needs to “balance out.” This intuition is wrong, and it is wrong in a precise way: a fair coin has no memory.
Each flip is independent. The probability of tails on the next flip is exactly , regardless of what came before. The coin cannot “know” it owes you tails.
What to notice
- All sequences converge. Every run eventually drifts toward the true probability p, not because the coin corrects itself, but simply because new flips dilute old history. With 1000 flips, one outlier streak is a tiny fraction of the total.
- Short runs diverge wildly. In the first 20 flips, the running proportion swings dramatically. This is where gamblers are tempted to “see patterns.”
- Biased coins (p ≠ 0.5). Drag p away from 0.5. All sequences still converge — now to the true p, not 0.5. The gambler’s error remains the same.
Law of Large Numbers vs the fallacy
The Law of Large Numbers says the running average converges to the true mean. It does not say past outcomes must be corrected. Long-run balance happens through dilution, not compensation.
The gambler’s fallacy is the belief that the correction mechanism is active even on individual flips. Casinos have made fortunes from this belief. The same fallacy appears in investing (“this stock is down three years running, it must recover”), sports (“the shooter is cold, he’s due”), and everyday decision-making.