The gambler’s fallacy is a mistaken belief that recent occurrences are indicative of odds, or that the odds increase or decrease based on recent occurrences when they are actually based on fixed probabilities. The probability of certain outcomes when the dice are rolled are believed to increase or decrease, when they are actually the same for every roll.

The gambler’s fallacy is sometimes called the Monte Carlo Fallacy in relation to an event at a casino there where the roulette wheel came up with black 26 times in succession on August 18, 1913.

Statistical independence is the operative concept. One occurrence has no statistical effect on another occurrence. Randomness, where one set of numbers has no importance in predicting the outcome involving another set of numbers, is the other operative concept.

Another concept is “clustering illusion”, the idea of Daniel Kahneman and Amos Tversky, where people have a cognitive bias toward seeing “streaks” that occur in small random samples as either “winning” or “losing”. According to Kahneman and Tversky, the cause of “clustering illusion” is the Representativeness Heuristic, which is another very complex topic.

Gambling thrives on statistical independence and randomness of outcomes. Recording or observing the number of outcomes of one sort or another is of no value whatsoever when the system, coin, machine or roulette wheel is fair. But people have biased cognitive reasoning that favors expecting short runs to be predictive of longer runs of independent and random occurrences.

The “Hot Hand” fallacy is a related fallacy that is based on a lucky streak that is used to predict that the next bet will have a positive outcome. More money is bet based on the prediction that past occurrences are indicators of future occurrences, when they are not. The same goes for losing streaks, where less money is put at risk.

Whether lottery players choose numbers based on whether they have come up more frequently, represent personal milestones, or simply choose quick picks that are randomly generated, the odds will still be the same: poor. The mathematical odds are in no way affected by past draws.

Betting in roulette involves a myriad of choices and combinations of choices: black, red, corners, specific numbers, the green zeroes and so on. Yet the gambler’s fallacy leads people to bet on numbers that have come up several times or to bet on red when black has come up several times in a row. The fact is that the odds of any choice being right or wrong are the same every time: Yes or no, 36 to 1.

A committed slot machine player will be glad to announce the win, but will never admit to the greater losses over the years before that win happened. The slot machine fallacy is that a string of wins that double or even triple a player’s initial stake means that a big payoff is imminent, or that certain “signs” mean that a big payoff is imminent. The player is far more likely to eventually walk away empty handed after playing back any gains either on the same machine or at a new machine.

Conversely, when a slot machine gambler sees evidence of a sizable recent win, the reasoning is that the machine will not pay off in the near future and the gambler moves on.

Skeptic’s Dictionary, “Gambler’s Fallacy”

Skeptics Dictionary, “Hot Hand Fallacy”

Wikipedia, “Gamblers Fallacy”

Wikepedia, “Clustering Illusion”