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. 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 independent occurrences of X happen, the probability that none of them will yield Y is (1-z)n. Thus, the probability that at least one occurrence of X will yield Y is 1-(1-z)n = 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.