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*Therefore, the probability that the next statistically independent occurrence of ''X'' will produce ''Y'' is less than it normally would be.
*Therefore, the probability that the next statistically independent occurrence of ''X'' will produce ''Y'' is less than it normally would be.
==What the fallacy is not==
==What the fallacy is not==
It is not the gambler's fallacy to say that the more often you do ''X,'' the more likely it will be that at least one occurrence of ''X'' will yield ''Y''. This can plainly be seen through a simple statistical analysis. Suppose that the probability that one occurrence of ''X'' will yield ''Y'' is ''z''. Then the probability that one occurrence of ''X'' will not yield ''Y'' is 1-''z''. If ''n'' statistically independent occurrences of ''X'' happen, the probability that none of them will yield ''Y'' is (1-''z'')<sup>''n''</sup>. Thus, the probability that at least one occurrence of ''X'' will yield ''Y'' is 1-(1-''z'')<sup>''n''</sup> = 1-(1-''nz''+...), which is some quantity less than ''nz''. A popular misconception is that the probability is ''nz'', which cannot be true because it could easily lead to probabilities greater than one. Also, our understanding of the gambler's fallacy must be adjusted in cases in which the events are not statistically independent, e.g., dealing cards from a deck that is not replenished.
It is not the gambler's fallacy to say that the more often you do ''X,'' the more likely it will be that at least one occurrence of ''X'' will yield ''Y''. This can plainly be seen through a simple statistical analysis. Suppose that the probability that one occurrence of ''X'' will yield ''Y'' is ''z''. Then the probability that one occurrence of ''X'' will not yield ''Y'' is 1-''z''. If ''n'' statistically independent occurrences of ''X'' happen, the probability that none of them will yield ''Y'' is (1-''z'')<sup>''n''</sup>. Thus, the probability that at least one occurrence of ''X'' will yield ''Y'' is 1-(1-''z'')<sup>''n''</sup> = 1-(1-''nz''+...), which is some quantity less than ''nz''. A popular misconception is that the probability is ''nz'', which cannot be true because it could easily lead to probabilities greater than one. Also, the gambler's fallacy does not apply when the events are not statistically independent, such as dealing cards from a deck that is not replenished.
==References==
==References==