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The Gambler's fallacy, also known as the Monte Carlo fallacy (because its most famous example happened in a Monte Carlo Casino in 1913)[1] or the fallacy of the maturity of chances, is the belief that if deviations from expected behaviour are observed in repeated independent trials of some random process, future deviations in the opposite direction are then more likely.

For example, if a fair coin is tossed repeatedly and tails comes up a larger number of times than is expected, a gambler may incorrectly believe that this means that heads is more likely in future tosses.[2] Such an expectation could be mistakenly referred to as being due, and it probably arises from one's experience with nonrandom events (e.g., when a scheduled train is late, we expect that it has a greater chance of arriving the later it gets). This is an informal fallacy. It is also known colloquially as the law of averages.

The gambler's fallacy implicitly involves an assertion of negative correlation between trials of the random process and therefore involves a denial of the exchangeability of outcomes of the random process. In other words, one implicitly assigns a higher chance of occurrence to an event even though from the point of view of "nature" or the "experiment", all such events are equally probable (or distributed in a known way).

The reversal is also a fallacy, in which a gambler may instead decide that tails are more likely out of some mystical preconception that fate has thus far allowed for consistent results of tails; the false conclusion being: Why change if odds favor tails? Again, the fallacy is the belief that the "universe" somehow carries a memory of past results which tend to favor or disfavor future outcomes.