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The gambler's fallacy , also known as the Monte Carlo fallacy or the fallacy of the maturity of chances , is the belief that, if an event whose occurrences are independent and identically distributed has occurred less frequently than expected, it is more likely to happen again in the future or vice versa.
The fallacy is commonly associated with gambling , where it may be believed, for example, that the next dice roll is more than usually likely to be six because there have recently been fewer than the expected number of sixes. The term "Monte Carlo fallacy" originates from an example of the phenomenon, in which the roulette wheel spun black 26 times in succession at the Monte Carlo Casino in The gambler's fallacy can be illustrated by considering the repeated toss of a fair coin.
In general, if A i is the event where toss i of a fair coin comes up heads, then:. If after tossing four heads in a row, the next coin toss also came up heads, it would complete a run of five successive heads. This is incorrect and is an example of the gambler's fallacy.
Since the first four tosses turn up heads, the probability that the next toss is a head is:. The probability of a run of coin tosses of any length continuing for one more toss is always 0. The reasoning that a fifth toss is more likely to be tails because the previous four tosses were heads, with a run of luck in the past influencing the odds in the future, forms the basis of the fallacy.
