Gambler’s Fallacy and the Law of Small Numbers




“The idea that beliefs about probability show systematic biases is somewhat older than experimental psychology.
Throughout his “Essai Philosophique sur les Probabilités,” Laplace (1796) was concerned with errors of judgment  and even included a chapter concerning “illusions in the estimation of probabilities.” It is here that we find  the first published account of what is now widely known as the gambler’s fallacy—the belief that, for random  events, runs of a particular outcome (e.g., heads on the toss of a coin) will be balanced by a tendency for the opposite outcome (e.g., tails). (Peter Ayton and Ilan Fisher, below)

For a simple description of the phenomenon see also  wikipedia :

<<Gambler’s fallacy arises out of a belief in the law of small numbers, or the erroneous belief that small samples must be representative of the larger population. According to the fallacy, “streaks” must eventually even out in order to be representative.   Amos Tversky and Daniel Kahneman first proposed that the gambler’s fallacy is a cognitive bias produced by a psychological heuristic called the representativeness heuristic, which states that people evaluate the probability of a certain event by assessing how similar it is to events they have experienced before, and how similar the events surrounding those two processes are. According to this view, “after observing a long run of red on the roulette wheel, for example, most people erroneously believe that black will result in a more representative sequence than the occurrence of an additional red”,  so people expect that a short run of random outcomes should share properties of a longer run, specifically in that deviations from average should balance out. When people are asked to make up a random-looking sequence of coin tosses, they tend to make sequences where the proportion of heads to tails stays closer to 0.5 in any short segment than would be predicted by chance (insensitivity to sample size);Kahneman and Tversky interpret this to mean that people believe short sequences of random events should be representative of longer ones.The representativeness heuristic is also cited behind the related phenomenon of the clustering illusion, according to which people see streaks of random events as being non-random when such streaks are actually much more likely to occur in small samples than people expect.


Tversky and Kahnemann Belief in the law of small numbers  (advanced)

Rabin Rabin-Small Numbers  (advanced)



Charles T. Clotfelter, Philip J. Cook , in their paper “Gambler’s Fallacy in Lottery Play”  (NBER Working Paper No. 3769, Issued in July 1991) provide evidence on the time pattern of lottery participation to see whether actual behavior is consistent with this fallacy. Using data from the Maryland daily numbers game, we find a clear and consistent tendency for the amount of money bet on a particular number to fall sharply immediately after it is drawn, and then gradually to recover to its former level over the course of several months. This pattern is consistent with the hypothesis that lottery players are in fact subject to the gambler’s fallacy.

One further empirical studio confirming the fallacy is presented in

The Gambler’s Fallacy and the Hot Hand: Empirical Data from Casinos” by Rachel Croson and James Sundali

The Gambler’s Fallacy by Charles T. Clotfelter and  Philip J. Cook

Gambler’s Fallacy and Hot Hand  by Hayton and Fisher

see also Scott Plous, “The Law of Small Numbers”,  The Psychology of Judgement and decision making , page 112-115


see also more papers at the page on Expected Utility  (back)


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