The gambler's fallacy is the belief that, in a game of chance, an outcome becomes more likely because it has not come up for a while, or less likely because it has come up several times in a row. After a run of reds in roulette, black seems "about to come"; a number that has not been drawn in a lottery for a long time seems "due". The intuition is very widespread and very strong, but in games with independent outcomes it is wrong.
Independence
Two events are independent when the occurrence of one does not change the probability of the other. A roulette wheel, dice, and lottery balls returned to the drum for every draw have no memory: they do not "know" what happened before and have no mechanism for making up for the past. At every spin of a European roulette wheel, the chance of red is 48.65%, however many times red has come up before. The same holds for black.
Modern gaming machines work on the same principle: the result of each spin is set by a random number generator, independently of earlier spins. A machine that "hasn't paid out for a long time" is not, for that reason, any closer to paying out.
The probability of a streak and the probability of the next result
The fallacy often comes from confusing two different questions:
- Before the first spin: what is the probability that the next ten spins all come up red? The answer is 0.0743%, a very small value.
- After nine reds have already come up: what is the probability that the tenth spin comes up red? The answer is 48.65%, as on any other spin.
The first nine results are no longer uncertain, so they no longer enter the calculation. What remains is a single spin, with its usual probabilities.
What is more, ten reds in a row are no less likely than any other specific sequence of ten colours. The sequence red, black, red, black and so on, alternating ten times, has exactly the same probability, 0.0743%, because red and black have the same chance at every spin. The run of reds stands out only because it looks orderly. And in a casino where many tables spin for hours on end, long runs occur regularly, precisely because the number of spins is very large.
The law of large numbers does not correct the past
A common justification of the fallacy is the law of large numbers: if the proportion of reds must approach its probability over time, then many reds should be followed by blacks. The law says something else. Proportions approach probabilities because a very large number of new results dilutes the old deviations, not because the deviations are cancelled by results going the other way. The absolute difference between the number of reds and the number of blacks can even grow while the proportion settles down. The full explanation is in Variance and the law of large numbers.
Amos Tversky and Daniel Kahneman described the tendency to expect small samples to resemble the whole already as "belief in the law of small numbers" ("Belief in the law of small numbers", Psychological Bulletin, 1971). The gambler's fallacy is a particular case of that tendency.
The hot hand
The opposite belief is that a streak will continue: a player "is on a roll", a table or a machine "is hot", so the luck should be followed. In games with independent outcomes, this belief is as mistaken as the gambler's fallacy: a run of wins does not make the next win more likely, just as a run of losses does not make it "due".
The term comes from the study of the "hot hand" in sport. Thomas Gilovich, Robert Vallone and Amos Tversky analysed basketball shooting and found no evidence that successful shots come in streaks more often than chance would produce ("The hot hand in basketball: On the misperception of random sequences", Cognitive Psychology, 1985). Joshua Miller and Adam Sanjurjo later showed that the method used had a subtle statistical bias and that, once it is corrected, the data are consistent with a real but modest effect ("Surprised by the hot hand fallacy? A truth in the law of small numbers", Econometrica, 2018). That debate, however, concerns activities in which skill and the athlete's condition matter. A roulette wheel or a random number generator has no form, confidence or fatigue, and therefore no "hot hand".
Numbers that are "due"
In lotteries, statistics always show numbers drawn less often, often called "cold", and numbers drawn more often, called "hot". The differences are normal fluctuations of chance. If the draws are fair, no number has a greater or smaller chance than the others at the next draw, as the Lotteries and keno page shows.
There is one real exception to independence: card games in which the cards dealt do not go back into the pack until it is shuffled. There, every card played changes the make-up of the cards that remain, so the past does matter. But it matters as information about what is left in the pack, not as a "debt" owed by luck, and it does not remove the house edge under ordinary conditions of play.
Near misses
A near miss is a result that looks very close to a win: two jackpot symbols on the line and the third just above it, or a lottery ticket with numbers next to those drawn. Mathematically, a near miss is a loss like any other. It does not show that a win is "getting closer" and does not change the chance of the next result.
On gaming machines, the symbols displayed are determined by the result of the random number generator, and the way the symbols are arranged on the reels determines how often such combinations appear on the screen, with no connection to the next spin. Psychological studies have shown that near misses are experienced differently from ordinary losses and can strengthen the urge to keep gambling (for example, Luke Clark and colleagues, "Gambling near-misses enhance motivation to gamble and recruit win-related brain circuitry", Neuron, 2009).
See also
- Probability basics
- Betting systems
- House edge and RTP
- Variance in the glossary