Gambler's Fallacy: Why Five Reds in a Row Make Black Feel Due

A fair wheel has no memory. The probability of red on the sixth spin is the same as on the first, whatever the previous five showed, because nothing about the wheel, the ball or the croupier is changed by the earlier results. Each spin is independent. The fallacy is to treat the sequence as if it were a single draw from a fixed supply, so that five reds have used up red's share and black must now come up to restore the balance.

Tversky and Kahneman traced the error to a belief they called the law of small numbers: the intuition that a short sequence should look like the long-run proportion it comes from. Over a million spins, red and black will each be close to half. People expect the same of six spins, so a run of five reds looks like a deviation that the process is obliged to correct. The correction is real, but it works by dilution rather than by compensation: the five reds become a smaller and smaller share of a growing total, and no individual spin leans the other way.

The same intuition produces the opposite error. When people believe that the process is not random, for example that a basketball player can be hot, the same streak is read as a signal that the next outcome will match it. Croson and Sundali found both patterns in one casino: roulette bettors, who know the wheel is random, bet against streaks, while the same players' betting on their own luck showed the hot hand pattern, placing more bets after a win.

In the casino data the fallacy grew with the length of the streak. It is also easier to fall into when past outcomes are displayed in a row, which is why the demo above shows the five spins one after another and why casinos put recent results on a board above the wheel. Knowing that the spins are independent does not switch the intuition off; it has to be managed, which is what the counter-measures below are for.

The name is older than the research. The usual origin story is a long run of black at the Monte Carlo casino, usually dated to 1913, with players betting ever larger sums on red as it went on; the account comes from later retellings rather than from casino records. Laplace had described the error in his Philosophical Essay on Probabilities, cited below in its 1825 edition, in men who, wanting a son, believed that the boys already born that month made a girl more likely next. Controlled experiments on how people read random sequences were under way by the 1950s and 1960s. The most influential account of the fallacy is Tversky and Kahneman's 1971 paper "Belief in the law of small numbers" in Psychological Bulletin, which showed that even research psychologists expected small samples to mirror the population they came from, the intuition that makes a run look like a deviation in need of a correction.

The cleanest field evidence is Rachel Croson and James Sundali's 2005 paper in the Journal of Risk and Uncertainty, which analyzed security camera footage of 18 hours of play at a Nevada casino, covering 139 roulette players and 24,131 recorded bets. For each spin they classified the bets placed on even-money outcomes (red or black, odd or even, high or low) as with the streak or against it, as a function of how long the streak had run.

After a streak of one or two the bets were evenly split. As the streak lengthened, bets moved against it: after five of one outcome roughly two-thirds of the bets were on the other, and after six or more, 85 percent of the bets went against the streak. The same players also bet more numbers after winning a spin and fewer after losing, the hot hand belief about their own luck sitting alongside the gambler's fallacy about the wheel.

A 2016 study of asylum judges, loan officers and baseball umpires by Chen, Moskowitz and Shue found a related pattern outside gambling: a decision in one direction made the next decision somewhat less likely to go the same way, even when the cases were unrelated. The authors read this as the gambler's fallacy at work, while allowing that a contrast between consecutive cases could play a part.

The fallacy appears wherever independent events are watched in sequence.

The fallacy is an intuition, and intuitions are not reliably removed by knowing better. They are managed.

• Ask the memory question. Does anything about this process carry information from the last outcome to the next? A wheel, a coin, a lottery machine: no. A tired goalkeeper, a depleted inventory, a pitcher's arm: perhaps. Only in the second case can a streak mean anything.
• Think in single events rather than sequences. The probability of black on this spin is a fact about this spin. The run of reds is a fact about the past.
• Expect streaks. In 100 fair coin flips, a run of five or more of one side is very likely to occur somewhere. A streak is what randomness looks like, not a departure from it.
• Fix the decision rule in advance. A loan officer with a written scoring sheet, or an investor with a rebalancing rule, has nothing to correct for.
• Be suspicious of the word due. It almost always marks the fallacy, in your own thinking or in someone else's pitch.

Several biases involve streaks and patterns in random data. They differ in what the person expects next and in whether the process is believed to be random.

• It is not the hot-hand fallacy, which is the mirror image: expecting a streak to continue because the person or process is on a run. The two coexist, and Croson and Sundali found both in the same players, one about the wheel and the other about their own luck.
• It is not the clustering illusion, which is seeing a pattern in random data that is not there at all, rather than expecting a correction to a pattern that is there by chance.
• It is not the illusion of control, the belief that one's own actions influence a chance outcome. A gambler who blows on the dice shows illusion of control; one who switches to red after five blacks shows the gambler's fallacy.