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The gambler's fallacy is a formal fallacy. It is the incorrect belief that the likelihood of a random event can be affected by or predicted from other, independent events.
The gambler's fallacy gets its name from the fact that, where the random event is the throw of a die or the spin of a roulette wheel, gamblers will risk money on their belief in "a run of luck" or a mistaken understanding of "the law of averages". It often arises because a similarity between random processes is mistakenly interpreted as a predictive relationship between them. (For instance, two fair dice are similar in that they each have the same chances of yielding each number - but they are independent in that they do not actually influence one another.)
The gambler's fallacy often takes one of these forms:
A particular outcome of a random event is more likely to occur because it has happened recently ("run of good luck");
A particular outcome is more likely to occur because it has not happened recently ("law of averages" or "it's my turn now").
Similarly
A particular outcome is less likely to occur because it has happened recently ("law of averages" or "exhausted its luck");
A particular outcome is less likely to occur because it has not happened recently ("run of bad luck"). A more subtle version of the fallacy is that an "interesting" (non-random looking) outcome is "unlikely" (eg that a sequence of "1,2,3,4,5,6" in a lottery result is less likely than any other individual outcome). Even apart from the debate about what constitutes an "interesting" result, this can be seen as a version of the gambler's fallacy because it is saying that a random event is less likely to occur if the result, taken in conjunction with recent events, will produce an "interesting" pattern.