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The '''gambler's fallacy''' is a [[logical fallacy]] based on a misunderstanding of [[statistics]]. The fallacy holds that because a fair [[gambling]] device has produced a run, the next trial of the device is less likely than normal to continue that run.<ref>[http://www.fallacyfiles.org/gamblers.html Gambler's Fallacy]</ref> For example, if nineteen flips of a [[coin]] have produced heads, the gambler's fallacy holds that the next flip will more likely than not yield tails. The gambler's fallacy is a fallacy because statistically independent effects by definition do not affect one another; for example, if a coin flipped 19 times has produced heads, the probability that the next flip will produce tails is still 1 in 2.
The fallacy takes the form:
*Several statistically independent occurrences of ''X'' have produced outcome ''Y''.
*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==
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 indpendent 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.
==References==
<references/>
[[Category:Logical Fallacies]]
[[Category:Statistics]]
The fallacy takes the form:
*Several statistically independent occurrences of ''X'' have produced outcome ''Y''.
*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==
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 indpendent 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.
==References==
<references/>
[[Category:Logical Fallacies]]
[[Category:Statistics]]