Gambler's Fallacy

The gambler's fallacy is a common invalid inductive inference. It involves the mistaken intuition or belief that the likelihood of a particular outcome of a process that generates independent random events increases as a function of the length of a run of consecutive non-occurrences of that outcome.

For example, a person playing a casino roulette wheel would commit the gambler's fallacy if he or she had a greater tendency to gamble on red than on black after four consecutive black outcomes, than after a shorter run of black outcomes. Such a tendency, or belief that red is more likely to occur as a function of its nonoccurrence, is erroneous, because the outcomes of the spins of a properly calibrated roulette wheel are independent, and the probabilities of red and black are equal and remain constant from spin to spin of the wheel. Similarly, a flip of a fair coin is not more likely to produce tails after a run of heads; nor is a pregnant woman more likely to give birth to a girl if she has, in the past, given birth to three boys consecutively.

The most widely cited explanation of the gambler's fallacy effect involves the hypothesis that people judge the randomness of an observed series of outcomes in terms of the extent to which it represents the output that would be expected of a prototypical random process—one that contains few orderly sequences such as long runs, symmetries, or strict alternations of one outcome, and few over- or under-representations of possible outcomes. Perhaps the gambler's fallacy arises because the occurrence of a locally less frequent outcome would produce a sample that would better represent randomness than the alternative sample would. For example, given five flips of a fair coin, heads might seem more likely after a series such as THTTT, because THTTTH has a shorter run of tails, and overrepresents tails less, than THTTTT does. People may also believe that a random device is somehow capable of correcting for the local scarcity of one outcome by overproducing instances of that outcome. Such thinking is faulty. A random device has no memory or means by which to correct its output, or to prevent patterns from appearing in a sample of outcomes.

Generalization from frequently encountered cases involving finite populations sampled without replacement could also explain this fallacy. For example, [Page 385]a motorist who is stopped at a railroad crossing waiting for a freight train to pass would be using sound reasoning if he or she counted freight cars that have passed the crossing and compared this number to his or her knowledge of the finite distribution of train lengths to determine when the crossing will clear. However, such reasoning is invalid when applied to large populations sampled without replacement.