The Gambler’s Fallacy refers to an incorrect belief about a sequence of independent random events. Someone falling prey to the fallacy believes that if a sequence of observed random events deviates from expected behavior, subsequent events will be biased in the opposite direction in order to move his observed sample towards a known population mean. A commonly overheard example is the expectation that something is “due”; a roulette ball is “due” to fall on a red number after a long series of black numbers.

Such beliefs have been demonstrated experimentally by Estes (1964) , who asked subjects to observe a sequence of coin flips and predict outcomes. Subjects behave as if every segment of the sequence must reflect the true proportion of 50 percent, and expect corrective biases in the opposite direction of the observed sequence. This can also be seen by asking experimental subjects to generate a random sequence of hypothetical coin flips. Tune (1964) showed that subjects maintain the proportion in any short segment at much closer to 50 percent than the laws of chance would predict.

Tversky and Kahneman (1971) suggest that the Gambler’s Fallacy is driven by belief in the Law of Small Numbers (the Law of Large Numbers applied to small samples). This can be described as the belief that a sample randomly drawn from a population is highly representative; thus each sample must be similar to each other sample and to the population. In surveys of professional psychology researchers, they demonstrate fundamental misunderstandings of interpreting replication studies. This includes overestimating the likelihood of replicating study findings, and underestimating the significance of successful replication with differing magnitudes. They suggest that across naïve subjects and trained scientists alike, there exist strong intuitions about random sampling that are fundamentally wrong. They argue that belief in the Law of Small Numbers (and thus the Gambler’s Fallacy) is a result of the representativeness heuristic, a cognitive bias that operates regardless of motivational factors. This has been replicated in field experiments by Croson and Sundali (2005) .

A related fallacy is known as the Hot-Hand Fallacy, where future outcomes are believed to be biased in the same direction as a previous sequence (this is sometimes also called the Gambler’s Fallacy, as both fall into a general category of inference from previous independent results). There may be some validity to such beliefs in a human-generated outcome like the shot of a basketball, but it is just as misguided as the Gambler’s Fallacy with respect to independent, randomly generated outcomes.

Correcting or avoiding the Gambler’s Fallacy has proven to be difficult, as education about the nature of random events has been ineffective at reducing the prevalence of picking “with” the fallacy. Beach and Swensson (1967) tested how people predict Page 170  |  Top of Articledraws of reshuffled cards with and without prior education about the Gambler’s Fallacy and found that both groups made similar predictions that relied on the fallacy.

Roney and Trick (2003) have demonstrated that the effect of the Gambler’s Fallacy can be reduced by “grouping” observations to make the next outcome appear as though it were the beginning of a sequence. Participants were shown the results of a sequence of six coin flips, with the last three all coming up heads. Those who were asked to predict the outcome of the seventh coin flip relied more heavily on the Gambler’s Fallacy (by choosing tails), more so than those that were asked to predict the first flip for the next sequence of six. They thus argue for encouraging people to view each event as a beginning and not a continuation of events.

Kevin Laughren and Robert Oxoby

See also: Errors and Biases ; Heuristics ; Ignorance of Base Rates ; Kahneman, Daniel ; Tversky, Amos