When an event with equally probable outcomes comes out the same way several times in succession, the other outcome is not more likely next time. For example, if a roulette ball ends up on black many times in a row, and not once on red (as reportedly happened 26 times on August 18, 1913, in the Monte Carlo Casino), the next ball is not more likely to land on red; red is not "due." For a fair wheel, neither is red less likely. This misconception is known as the gambler's fallacy; in reality statistical independence holds, and red is just as likely or unlikely on the next spin as always - sometimes expressed as "the system has no memory." If the event is physically determined, and not perfectly random, the repeated outcome may be more likely. For example, a die that has rolled a six ten consecutive times might be loaded or controlled by hidden magnets, and would be more likely to roll another six.

The gambler's fallacy, also known as the Monte Carlo fallacy or the fallacy of the maturity of chances, is the mistaken belief that if something happens more frequently than normal during some period, then it will happen less frequently in the future (presumably as a means of balancing nature). In situations where what is being observed is truly random (i.e. independent trials of a random process), this belief, though appealing to the human mind, is false. This fallacy can arise in many practical situations although it is most strongly associated with gambling where such mistakes are common among players.

Remembering that coins have no memory can be useful. Many people falsely believe that if, for example, we got three tails in a row, then head is, in some sense, "overdue," and has become more likely. That is not true. If one meets a person with such false beliefs, one might as well take advantage and make some money. If the person resents having been taken advantage of, call it a tuition fee.