D'Alembert system
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 some period, it will happen less frequently in the future, or that, if something happens less frequently than normal during some period, it will happen more frequently in the future (presumably as a means of balancing nature). In situations where what is being observed is truly random (i.e., independent trials of a random process), this belief, though appealing to the human mind, is false. This fallacy can arise in many practical situations, but is most strongly associated with gambling, where such mistakes are common among players.
The use of the term Monte Carlo fallacy originates from the most famous example of this phenomenon, which occurred in a Monte Carlo Casino in 1913.
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 Ai be the event that toss i of a fair coin comes up heads, then we have,
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