Hot hand fallacy

Numbers on a roll

A lottery player looks up the last month of draws and picks the numbers that came up most often. “Those ones are hot right now. Why fight the trend?”

Lottery balls don’t warm up. Each draw is independent, so a number that has come up three times this month is exactly as likely next week as one that hasn’t come up at all. This is the clear-cut form, because the independence is known for certain. It isn’t only a hypothetical: in Danish national lottery records, players bet more on a number the longer its run of recent wins (see Evidence).

A coin that looks streaky

A teacher has a class flip a coin 100 times and write down the results. A student points to a run of six tails in a row. “That coin’s not fair. It gets stuck on one side.”

A fair coin flipped 100 times almost always produces a run of five or more of the same side somewhere in the sequence, and usually a run of six or more. The run feels too long to be chance because people expect random sequences to switch back and forth more often than they do. When Gilovich, Vallone and Tversky showed basketball fans a sequence of hits and misses with exactly the amount of switching chance produces, 62% called it “streak shooting”. Nothing about sports is needed for the error.

Feed the hot shooter

In a pickup basketball game, a player who usually makes about half her shots has just made four in a row. “She’s on fire,” a teammate says. “Give her the ball every time down the court.”

This is the contested edge. A shooter who makes half her shots will often make four in a row somewhere in a game by chance alone, so four makes are weak evidence that she’s hotter than usual. But shooting isn’t a lottery: confidence, rhythm and fatigue could make one shot depend on the last, and some research now finds a real, if uncertain, hot hand in basketball. The teammate may be right that she’s hot and wrong about how much, or about whether “every time” is worth letting the defense know where the ball is going.

• Seeing streaks in independent outcomes: betting on “hot” lottery numbers or roulette colors, or judging a random sequence to be streaky, as in the first two examples.
• Overreacting to a real hot hand: the streak does carry information, but the response is too big. Benjamin’s review describes studies finding a real hot hand in baseball and darts, alongside pitchers and darts players who reacted to it more strongly than it justified.
• Hot places and hot people: expecting luck to cling to whatever was nearby when it happened. Benjamin’s review describes stores that sold a winning lottery ticket selling substantially more tickets in the weeks that followed.

• When the streak tells you about ability you didn’t know. If you don’t know how good someone is, a run of successes is evidence that they’re better than you assumed. Raising your estimate of their next attempt is correct even if each attempt is independent of the last.
• When something really carries over from one attempt to the next. Confidence, practice within a session, fatigue or a machine drifting out of adjustment can make outcomes depend on each other. Expecting a streak to continue then points the right way; the open question is by how much.
• When the adjustment matches the evidence. A small update after a short streak, in a process shown to have some carry-over, is proportionate. The error is treating a few successes as proof of a lasting “hot” state.

The test: would this streak be surprising if nothing carried over from one try to the next, and am I expecting more continuation than the process has actually shown?

The newcomer

A new player joins a darts league and wins his first eight matches. The captain moves him up to play against the strongest opponents: “He’s clearly better than we thought.”

This looks like trusting a hot streak, but the captain isn’t predicting that the streak will carry him through the next match. He’s using eight wins to estimate a skill level nobody knew. A player who was only average would rarely win eight straight, so the run is good evidence about his ability, whatever happens in any one match. That’s the ability you didn’t know condition.

A machine that keeps making bad parts

A quality inspector finds three defective parts in a row from the same machine. She pulls the machine for inspection and expects the next parts to be faulty too.

A run of three defects in a row, followed by an expectation of more, has the shape of the fallacy. But defects from a machine aren’t like lottery draws: a tool that has worn down or slipped out of alignment stays that way until someone fixes it, so each defect raises the chance of the next. That’s the something carries over condition.

Gilovich, Vallone and Tversky traced the belief to the same misconception of chance behind the Gambler’s fallacy: people expect even short random sequences to look like the process that produced them, so they expect far more alternation than chance delivers. A genuinely random sequence then looks streaky, and “she’s hot” is a ready explanation for the streaks. They also suggested that the many plausible reasons a player could run hot, such as confidence or fatigue, may make the belief easier to hold. On this account the two fallacies are linked rather than separate: a person who expects too much alternation ends up seeing too many streaks. Benjamin’s review describes formal models of this idea, and lottery data in which players who bet more on numbers with winning streaks also tended to be the ones who avoided a number that had just won once.

Ayton and Fischer proposed a different account based on experience. Human performance, they argued, often really does show runs of success and failure, while many sequences of natural events tend to reverse. People carry those expectations into random sequences: a streak produced by a person suggests a hot hand, and a streak produced by nature suggests a reversal. In their experiment, participants expected reversals in the outcomes of a roulette game while expecting streaks to continue in their own record of correct and incorrect predictions about it, even though the two sequences were statistically identical.

Either way, the hot hand fallacy and the gambler’s fallacy are opposite expectations about the same kind of streak: one expects it to continue, the other expects it to end. Ayton and Fischer describe them as positive and negative recency, two sides of how people perceive randomness.

• Ask whether one outcome can affect the next at all. For lotteries, dice and coins it can’t, and the streak can be set aside entirely.
• Remember how streaky chance is. A 50% shooter taking 20 shots has close to even odds of making four in a row somewhere. Before reading a streak as “hot”, ask how often chance alone would produce one that long.
• Separate “is there a hot hand?” from “how big is it?” Where carry-over is plausible, treat a streak as modest evidence, not a guarantee.
• Don’t cry fallacy at every belief in streaks. After the research described below, believing that basketball players sometimes run hot is not an established error. Overestimating how hot is still a live possibility.

Status: contested. The label turns on what the fallacy claims. Expecting streaks of known independent outcomes to continue is a real error, seen in lottery betting and in how people judge random sequences. But the entry’s namesake claim, that the widespread belief in a basketball hot hand is an illusion, rests on an analysis later shown to be biased. Researchers now disagree about how large a real hot hand is, and whether people’s beliefs exceed it has not been settled. In reviewing this entry, no preregistered or multi-lab replication of the bias was found. Where the evidence falls between labels, the site chooses the less confident one.

The original study. Gilovich, Vallone and Tversky (1985) combined four kinds of evidence.

• A survey of 100 basketball fans at Cornell and Stanford found that 91% believed a player has “a better chance of making a shot after having just made his last two or three shots”. Asked about a player who makes 50% of his shots, fans estimated 61% after a make and 42% after a miss.
• Shooting records of the Philadelphia 76ers (1980–81), free throws by Boston Celtics players, and a controlled experiment with 26 Cornell varsity players showed no reliable tendency to hit more often after hits, with one Cornell player as the exception. The Cornell players’ bets on their own shots followed their recent hits and misses, but didn’t predict them.
• Shown sequences of hits and misses, fans rated sequences with more switching than chance produces as the best examples of “chance shooting”, and 62% called a sequence with chance-level switching “streak shooting”.

They concluded that the data showed “a powerful and widely shared cognitive illusion”. In the years that followed, Miller and Sanjurjo write, “a consensus has emerged that the hot hand is a ‘myth’”.

The statistical flaw. Miller and Sanjurjo (2018) proved that the standard way of measuring streakiness is biased in any finite sequence. Flip a fair coin three times and, in each sequence, look at the flips that come right after a heads. Averaged over the possible sequences, the proportion of heads among those flips is 5/12, not 1/2. The same selection effect means that a shooter with no hot hand at all, who makes half of about 100 shots, would be expected to hit around 8 percentage points less often after three hits than after three misses. The Cornell players had hit slightly more often. Correcting for the bias, Miller and Sanjurjo estimated that the players shot about 13 percentage points better after three hits than after three misses, and a close replication of the experiment with elite players reversed the same way. They also report that, pooled across bettors, the Cornell players shot about 7 percentage points better when a bet predicted a hit than when it predicted a miss. They concluded that belief in the hot hand “is not a cognitive illusion”, while noting that people might still believe in a stronger hot hand than exists. Benjamin’s review describes their work as reopening, but not answering, the question of whether a hot hand bias exists in basketball.

Challenges to the correction’s conclusions. Ritzwoller and Romano (2022) developed exact tests for streakiness and applied them to four controlled shooting experiments. In the Cornell data, after correcting for testing 26 players at once, they could reject pure chance for only one shooter, and the overall result depended on including him. They concluded that the existing experiments are too small to say whether the hot hand in basketball is small or substantial, and that the available measures of people’s beliefs can’t be compared directly with measures of actual streakiness.

Game data gives mixed answers, and games add complications (defenders and shot choices respond to streaks).

• Miller and Sanjurjo (2021) analyzed 29 years of the NBA Three-Point Contest, where defense is absent, and reported considerable evidence of hot hand shooting.
• Lantis and Nesson (2021), using 12 NBA seasons, found a small hot hand for free throws, but none for field goals, where longer streaks of makes were followed by lower success.
• Pelechrinis and Winston (2022) found that some individual players show a hot hand in games, but that the average player shot below expectations after consecutive makes.

Settings without a real hot hand. Where outcomes are known to be independent, the error is clearer. As summarized in Benjamin’s review, Danish lottery players bet less on a number just after it won once, but more on numbers the longer their streak of wins, and similar patterns have been reported in roulette and sports betting. The fans’ judgments of random sequences in the 1985 study are also unaffected by the statistical flaw.

What remains uncertain. How large a real hot hand is in basketball and other skills, whether fans, players and coaches believe in a stronger one than exists, and how much the belief costs in decisions. The popular summary that “the hot hand is a myth” is out of date; the claim that people see too many streaks in genuinely random sequences is not.