The Gamblers’ Paradox – Adventures With 3 Coin Flips, Part 1.

Many gambling activities involve betting on events for which the outcomes obey rigid, specified odds. When there is no mechanical bias, roulette wheels have fixed odds, including some that are binary (50:50), such as red vs. black. Betting on the flip of a coin is likewise a binary 50:50 proposition: heads or tails. Why is it, then, that there is a propensity for some gamblers to place wagers in a pattern conflicting with the known 50:50 odds? For example, after a string of blacks on a roulette wheel, why do some gamblers keep increasing the amounts bet on red with each succeeding black?

Such behavior is related to the concept that if some process deviates from a known probability for a period of time, future events will counter that deviation in what is called “reversion to the mean”. However, the expectation that such a process will start with the next process event cannot have a probability different from the known odds. If someone flips 10 heads in a row, the next coin flip still has even odds of heads or tails.

Australian economist Jason Collins has considered a coin-flipping problem involving 3 flips per sequence. He shows that by choosing the sample size that he has here, a bias is introduced such that the probability of a head being followed by another head is only 42%, not 50% expected for random coin flips.

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