The gambler's fallacy, also known as the Monte Carlo fallacy or the fallacy of the maturity of chances, is the mistaken belief that, if something happens more frequently than normal during a given period, it will happen less frequently in the future (or vice versa). In situations where the outcome being observed is truly random and consists of independent trials of a random process, this belief is false. The fallacy can arise in many situations, but is most strongly associated with gambling, where it is common among players.
After a consistent tendency towards tails, a gambler may also decide that tails has become a more likely outcome. This is a rational and Bayesian conclusion, bearing in mind the possibility that the coin may not be fair; it is not a fallacy. Believing the odds to favour tails, the gambler sees no reason to change to heads. However, it is a fallacy that a sequence of trials carries a memory of past results which tend to favour or disfavour future outcomes.
Retrospective gambler's fallacy
Researchers have examined whether a similar bias exists for inferences about unknown past events based upon known subsequent events, calling this the "retrospective gambler's fallacy".
An example of a retrospective gambler's fallacy would be to observe multiple successive "heads" on a coin toss and conclude from this that the previously unknown flip was "tails". Three studies involving Stanford University students tested the existence of a retrospective gambler's fallacy. All three studies concluded that people have a gamblers' fallacy retrospectively as well as to future events. The authors of all three studies concluded their findings have significant "methodological implications" but may also have "important theoretical implications" that need investigation and research, saying "[a] thorough understanding of such reasoning processes requires that we not only examine how they influence our predictions of the future, but also our perceptions of the past."
Monte Carlo Casino
Perhaps the most famous example of the gambler's fallacy occurred in a game of roulette at the Monte Carlo Casino on August 18, 1913, when the ball fell in black 26 times in a row. This was an extremely uncommon occurrence: the probability of a sequence of red or black occurring 26 times in a row is (18/37) or around 1 in 66.6 million, assuming the mechanism is unbiased. Gamblers lost millions of francs betting against black, reasoning incorrectly that the streak was causing an imbalance in the randomness of the wheel, and that it had to be followed by a long streak of red.