What is the gambler's fallacy, and why is it a mistake?

The gambler's fallacy is the belief that an independent event which has happened more or less often than expected will become less or more likely in the future. It is also called the Monte Carlo fallacy or the fallacy of the maturity of chances. A standard example is thinking that after several heads in a row, a tail is "due." Because each flip is independent, the chance of heads on the next toss stays at one in two. [1] [2]

For a fair coin, the chance of two heads in two tosses is 1/4, and of five heads in five tosses is 1/32, a little over 3 percent. A person who has seen four heads may reason that a fifth is unlikely and expect tails. The error is that the 1/32 figure applies only before the first toss. Once four tosses have happened, their results are known, and the chance of heads on the fifth toss is still 1/2. The sequences "five heads" and "four heads, then a tail" are equally likely, each at 1/32. [1]

A Duke University mathematics page makes the same point with roulette. On a wheel where 18 of 38 numbers are black, six blacks in a row has a probability of about 0.0113. But after five blacks, the chance of a sixth is still 18/38. [3]

Believers in the fallacy often make a correct observation: in the long run, the proportion of heads approaches one half as the number of flips grows. They then infer that some natural adjustment must compensate for a run of heads with extra tails. The Statistics LibreTexts text presents this as the intuition behind the error, and it treats it as a mistaken inference about independent events. [2]

The name Monte Carlo fallacy comes from an episode in 1913, when the roulette wheel at the Monte Carlo Casino came up black 26 times in succession. [1] [4]

The fallacy runs in both directions. It can mean expecting a reversal after a streak, or expecting an event that has been rare to turn up more often. It applies only to independent events. When events are not independent, as when cards are drawn from a deck without replacement, earlier results can change later probabilities. [1]

Concluding from a long run of tails that a coin may be biased is a different matter. Wikipedia describes this as a rational, Bayesian inference and not a fallacy, since the coin may not be fair. The fallacy lies in believing that a sequence of trials carries a memory of past results. Most illustrations also assume the trial is fair. [1]

The error is discussed in gambling and in investing, where people may predict outcomes from previous results that do not influence them. [5]