Gambler’s Fallacy

What Is Gambler’s Fallacy?

The Gambler’s Fallacy is an intuition that was discussed by Laplace and refers to playing the roulette wheel. The intuition is that after a series of n “reds,” the probability of another “red” will decrease (and that of a “black” will increase).

As with the hot-hand fallacy Opens in new window, the gambler’s fallacy was also attributed to the representativeness heuristic Opens in new window. The reason given was that “the occurrence of black will result in a more representative sequence than the occurrence of an additional red” (Tversky and Kahneman 1974, 1125). People do not predict another “red” because “black” will make the resulting series of n + 1 events look more representative to a series with the true probability of .5.

With the gambler’s fallacy, people expect outcomes in a random series to reverse systematically. For example, if you flip heads on a coin three times in a row, subjects assess the probability of flipping a tails next at 70 percent. The reason is that people expect a short sequence to resemble a larger population, so that heads and tails roughly balance out.

The gambler’s fallacy is the most extreme version of the hot-hand fallacy. Think again about coin tosses, and suppose that there has been a run of five heads. It is quite common for people to believe that there is therefore a high probability of tails on the next throw, but, as the saying goes, the coin has no memory. The logical possibility of tails is still .5; this gives an expected frequency of 50/50, which is what it will approach in the long run. People who fall for the gambler’s fallacy or the hot hand hypothesis are confusing the one for the other.

Alter and Oppenheimer (2006) review numerous studies of the hot hand fallacy and show how it and the gambler’s fallacy can be seen as two sides of the same coin (forgive me).

The hot hand idea incorporates the notion of skill as a causal mechanism, so that when there is a long streak of one particular outcome, such as hits, people expect the streak to continue. However, when the streak comes from a random, skill-free process such as coin tossing, people expect the streak to end, so that the general sequence balances out.

An exception occurs with gamblers playing roulette or dice games in casinos: these are random processes and yet people do often believe in hot hands when gambling. This shows that gamblers have mythical beliefs about the processes that generate outcomes at the tables—a very dangerous state of affairs for the gambler, but a very happy one for the house.

Note that these two phenomena are exactly opposite. In the hot-hand fallacy fallacy, the intuition is that after a series of n equal outcomes, the same outcome will occur again; in the gambler’s fallacy, the intuition is that after a series of n equal outcomes, the opposite outcome will occur. Nevertheless, the notion of the representativeness heuristic is flexible enough to account for both logical possibilities (Ayton & Fischer, 2004).

In the hot-hand fallacy, the similarity is taken to be between the new outcome and the series of n outcomes; and in the gambler’s fallacy, the meaning of similarity is switched to that between the series of n + 1 outcomes and the underlying probability of the outcome. If any time a basketball coach can be found who exhibits a cold-hand fallacy, then the version of the representativeness heuristic used for the gambler’s fallacy will also be able to explain this phenomenon.