Five Reds in a Row

Question

A fair roulette-style wheel has just landed red five spins running. A colleague offers you even money that the next spin is black, arguing red is “overdue to break.” A second colleague offers you even money on red, arguing the wheel is “hot.” The spins are independent and the wheel is fair (ignore green for this question). Which bet do you take, and what's the trap each colleague fell into?

Accepted Answer

Take neither bet at even money: both are fair coins dressed up as edges. The core fact. The spins are independent, so the next spin is red with probability frac12 and black with probability frac12, regardless of the five reds behind it. The wheel has no memory of its own history. An even-money bet on either color has expected value exactly zero, you are indifferent before costs and a loser after any spread or commission. Colleague one fell for the gambler's fallacy: the belief that an outcome is "due" because it has been absent, as if independent trials must self-correct in the short run. They will not. Nothing forces the proportions to balance over the next spin; balancing happens only in the limit, and the limit does not reach back to nudge spin number six. Colleague two fell for the hot-hand fallacy: the belief that a streak signals a propensity to continue, that the wheel is "running hot." On an independent fair process there is no such propensity to detect. The one caveat a sharp candidate adds: the math above assumes the wheel is fair. If you did not know that, five reds is mild evidence the wheel might be biased toward red, in which case the hot-hand colleague accidentally p

Five Reds in a Row

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