Gambler's Fallacy Demo

The gambler's fallacy is the mistaken belief that if an event has occurred more frequently than expected in the past, it is less likely to happen in the future (and vice versa). After seeing 10 heads in a row, many people feel that tails is "due", but the coin doesn't know what happened before.

P(Heads on flip 11 | 10 heads in a row) = P(Heads) = 0.50

The conditional probability is exactly the same as the unconditional probability. Prior outcomes provide zero information about the next independent trial.

People confuse the Law of Large Numbers with the gambler's fallacy. The law states that as the number of trials increases, the proportion of outcomes converges to the expected probability. It does not say that short-term deviations will be "corrected."

If you flip 100 heads in 200 flips (50%), then flip 100 more getting 55 heads, the ratio becomes 155/300 = 51.7%. The extra 5 heads aren't erased, they're diluted by more data. Convergence happens by overwhelming deviations, not by reversing them.

Two events are independent when the outcome of one has no influence on the other. Fair coin flips, roulette spins, and dice rolls are all independent. The physical mechanism resets completely between trials, the coin has no muscles, memory, or conscience.

P(A and B) = P(A) × P(B), the definition of independence

This is exactly why the probability of 10 heads followed by tails is the same as the probability of 10 heads followed by heads: (0.5)^10 × 0.5 = (0.5)^11 in both cases.

The hot hand fallacy is the mirror image: believing that a winning streak will continue. In casino games with fixed odds, neither hot hands nor cold streaks have any predictive power. The roulette wheel doesn't know it just landed on red 8 times.

Casinos exploit both fallacies. They display previous roulette results on electronic boards precisely because players use that information to make irrational bets. Pattern-seeking in random data is a feature of human cognition, not a feature of the data.