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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 then these deviations are likely to be evened out by opposite deviations in the future. 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. 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.
The reversal is also a fallacy, the inverse gambler's 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.
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An example: cointossing
The gambler's fallacy can be illustrated by considering the repeated toss of a fair coin. With a fair coin, the outcomes in different tosses are statistically independent and the probability of getting heads on a single toss is exactly ^{1}⁄_{2} (one in two). It follows that the probability of getting two heads in two tosses is ^{1}⁄_{4} (one in four) and the probability of getting three heads in three tosses is ^{1}⁄_{8} (one in eight). In general, if we let A_{i} be the event that toss i of a fair coin comes up heads, then we have,
Now suppose that we have just tossed four heads in a row, so that if the next coin toss were also to come up heads, it would complete a run of five successive heads. Since the probability of a run of five successive heads is only ^{1}⁄_{32} (one in thirtytwo), a believer in the gambler's fallacy might believe that this next flip is less likely to be heads than to be tails. However, this is not correct, and is a manifestation of the gambler's fallacy; the event of 5 heads in a row and the event of "first 4 heads, then a tails" are equally likely, each having probability ^{1}⁄_{32}. Given the first four rolls turn up heads, the probability that the next toss is a head is in fact,
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