The Hot Hand Fallacy in Gambling: What the Research Shows

The hot hand fallacy is the belief that a run of wins makes the next win more likely when the outcomes are in fact independent, as they are on a roulette wheel, a fair shuffle or a slot machine's random number generator. Casino data show gamblers acting on that belief, raising bets and staying at the table after wins. The research history is more complicated than the label suggests: the 1985 basketball study that made the "fallacy" famous was later found to contain a statistical bias, but the correction applies to human performance, not to games of pure chance.

Two opposite errors with one root

Gamblers make two mirror-image mistakes about streaks. The gambler's fallacy expects a streak to end: after five reds, black feels "due." The hot hand belief expects a streak to continue: after three winning spins, the next one feels more likely to win too. They predict opposite things, but they share a starting point, which Amos Tversky and Daniel Kahneman described in "Belief in the Law of Small Numbers," published in Psychological Bulletin in 1971.

Their argument was that people expect short sequences to look like the long-run average. The "law of small numbers," as they put it, "asserts that the law of large numbers applies to small numbers as well." On the gambler's side of the error they wrote: "The heart of the gambler's fallacy is a misconception of the fairness of the laws of chance." A wheel has no memory and no way to correct a run of reds. In their phrase, deviations "are not canceled as sampling proceeds, they are merely diluted." Ten extra reds in the first hundred spins are still ten extra reds after ten thousand spins; they just matter less as a share of the total.

The hot hand version was named and tested in 1985 by Thomas Gilovich, Robert Vallone and Amos Tversky in "The Hot Hand in Basketball: On the Misperception of Random Sequences," Cognitive Psychology, volume 17. They surveyed 100 basketball fans recruited from Cornell and Stanford. Ninety-one percent believed that a player has "a better chance of making a shot after having just made his last two or three shots than he does after having just missed his last two or three shots," and 68 percent held essentially the same belief about free throws.

The researchers then checked the belief against shooting records of the Philadelphia 76ers, free-throw records of the Boston Celtics and a controlled shooting experiment with the men's and women's varsity teams at Cornell. Their abstract summarizes the experiment: "The outcomes of previous shots influenced Cornell players' predictions but not their performance." The paper concluded that people see streaks in random sequences because they misjudge what randomness looks like.

What roulette players actually did

Basketball is a test of skill. Roulette is not, which makes it a cleaner place to watch both errors at work. Rachel Croson and James Sundali did that in "The Gambler's Fallacy and the Hot Hand: Empirical Data from Casinos," Journal of Risk and Uncertainty, volume 30 (2005) (Croson and Sundali, 2005).

They analyzed 18 hours of security video from a single roulette table at a large casino in Reno, Nevada, recorded in July 1998 in three six-hour blocks. The data covered 904 spins, 139 players and 24,131 individual bets on a double-zero wheel, where the house edge is 5.26 percent.

The gambler's fallacy appeared after long streaks. Of 531 even-money bets placed after at least one prior spin, 255, or 48 percent, went against the previous outcome, which is close to an even split. After streaks of one to four of the same outcome, the researchers found no significant tendency to bet against the streak. After streaks of five, the tendency became significant, and after streaks of six or more, 85 percent of bets went against the streak.

Hot hand behavior appeared after wins. Players who had just won a spin behaved as though their luck was running:

Quitting after a loss and betting more after a win is consistent with a belief that wins come in runs. The authors were careful to note other explanations. A player may be spending "house money" more freely, may simply have more chips after a win, or may be betting a fixed fraction of a stack that just grew. The data show the behavior; they cannot fully separate the reasons for it.

What the data cannot show is any payoff. Each spin of the wheel is independent, so betting more after wins only raises the amount exposed to the same 5.26 percent edge.

A coin-flip worked example: the streak selection bias

In 2018, Joshua Miller and Adam Sanjurjo published "Surprised by the Hot Hand Fallacy? A Truth in the Law of Small Numbers" in Econometrica, volume 86 (Miller and Sanjurjo, Econometrica). They identified a bias in the method the 1985 study used to test for streaks, and they explained it with coins.

The question is simple: in a short sequence of fair coin flips, what share of flips that come right after a heads are themselves heads? Intuition says one half. The worked example below shows why the expected answer, measured sequence by sequence, is lower.

Step 1: List every possible sequence of three flips. There are eight, each equally likely: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

Step 2: In each sequence, find the flips that follow a heads. Only flips 2 and 3 can follow anything. A flip counts if the flip just before it is H.

Step 3: Drop sequences where no flip follows a heads. In TTT and TTH, neither flip 1 nor flip 2 is a heads, so there is nothing to measure. Six sequences remain.

Step 4: In each remaining sequence, compute the share of those flips that are heads.

Step 5: Average the shares across sequences. The sum is 1 + 1/2 + 0 + 0 + 1 + 0 = 2.5. Dividing by six sequences gives 2.5/6 = 5/12, or about 0.417.

The coin is fair, yet the expected proportion of heads after a heads, computed this way, is 5/12, not 1/2. The bias comes from selection: in a finite sequence, a streak of heads uses up heads, so the flips that follow a heads are drawn from what remains, which tilts toward tails. The longer the streak being conditioned on and the shorter the sequence, the larger the effect.

Step 6: Apply it to the 1985 experiment. Miller and Sanjurjo reanalyzed the Cornell shooting experiment: 26 players, 14 men and 12 women, each taking 100 shots from a distance chosen so they would hit about half. The players hit .49 after three straight hits and .45 after three straight misses, a 4-point difference the original study treated as no effect. For a shooter with a 50 percent hit rate and 100 shots, the authors calculated that the bias in this measurement is about minus 8 percentage points. A shooter with no hot hand at all would be expected to show a difference of minus 8. The observed plus 4 is therefore 12 points above that baseline, and after their full bias correction the average difference was 13 points. In the authors' word, the original conclusions are "reversed."

What the correction does and does not change for casino games

The Miller and Sanjurjo result says that skilled performers may, in fact, have streaks. It applies to activities where the outcome depends on a person's state: a shooter's confidence, fatigue or rhythm. It does not apply to a roulette wheel, a properly shuffled deck or a regulated random number generator, because none of them has a state that a streak could reflect.

The coin-flip example makes that distinction concrete. The 5/12 figure is not evidence that coins are streaky. It is evidence that a particular way of measuring streaks understates the true conditional probability, which for a fair coin is exactly 1/2 on every flip. Correcting the measurement on a coin brings the answer back to 1/2. Correcting it on a basketball shooter revealed a positive effect. For a casino game, the corrected answer is the same as the uncorrected truth: the next outcome has the same probability it always had.

The Croson and Sundali data show why the distinction matters at the table. The roulette players in Reno bet more after wins and walked away after losses, behavior that fits a belief in hot hands, but the wheel gave them the same odds and the same 5.26 percent edge on every spin. Whatever happened in the previous spin, the cost per dollar wagered on the next one was unchanged. Larger bets after a win simply mean more dollars at that fixed price.

The research leaves two practical points. First, a streak in a game of chance is information about the past and nothing else. Second, the instinct that streaks mean something is not a sign of poor reasoning in general. It is the same instinct that, applied to human skill, turns out to have some basis. The error lies in carrying it from the basketball court to the roulette table.