Hot-Hand Fallacy

The hot-hand fallacy is the mistaken belief that success in a sequence of independent attempts is self-sustaining, so that a person who has just succeeded several times in a row is more likely to succeed again. The name comes from basketball, where a player on a scoring run is described as having a hot hand. Thomas Gilovich, Robert Vallone and Amos Tversky gave the term its meaning in "The Hot Hand in Basketball: On the Misperception of Random Sequences," published in Cognitive Psychology in 1985, which reported that shooting records showed no such dependence while almost everyone involved in the game believed they did. The claim was never that streaks do not happen. Streaks happen in any random sequence. The claim was that they happen about as often as chance predicts, and that people read meaning into them anyway.

The original work looked in three places and found the same answer in each. Amos Tversky and Thomas Gilovich summarized it for a general audience in "The Cold Facts About the 'Hot Hand' in Basketball," in Chance in 1989. They examined the field-goal records of nine Philadelphia 76ers players across 48 home games of the 1980-81 season, the free-throw records of nine Boston Celtics players across the 1980-81 and 1981-82 seasons, and a controlled experiment in which 14 men and 12 women from the Cornell University varsity teams each took 100 shots from a distance at which they made roughly half. The reported result in each setting was that the chance of hitting was about the same after a hit as after a miss. They also asked people to classify sequences of hits and misses, and found that sequences with the alternation rate of a fair coin were routinely read as streak shooting, which is the perceptual half of the finding: a genuinely random sequence looks streaky to us.

The misreading underneath both fallacies is the same one. Tversky and Kahneman had already named it in 1971 as a belief in the law of small numbers: people expect a short stretch of a random process to look like the process as a whole. A run of six heads in twelve flips looks wrong, so something must explain it. From there the mind can go two ways. If you assume the process must balance out, you get the gambler's fallacy and expect a reversal. If you assume the run reveals a hidden state, a player who is on, a manager with a touch, a fund that has found something, you get the hot-hand fallacy and expect continuation. The two predictions are opposite and the error is identical, which is why they are treated as a pair.

Then the statistics turned out to be harder than anyone realized. In 2018 Joshua Miller and Adam Sanjurjo published "Surprised by the Hot Hand Fallacy? A Truth in the Law of Small Numbers" in Econometrica. Their result is a mathematical point about measurement rather than about basketball. If you take a finite sequence, find the outcomes that immediately follow a streak of successes, and compute the proportion of those that are also successes, that proportion is a biased estimate: its expected value is below the true probability, and the bias grows with streak length and shrinks as the sequence gets longer. Their abstract states the consequence plainly, that the canonical study and its replications "are vulnerable to the bias," and that "upon correcting for the bias, we find that the longstanding conclusions of the canonical study are reversed." That is a genuine reversal of the original reading of the original data, and it is not a claim that the hot hand is large or reliable.

The current state of the evidence is genuinely unsettled, and the honest summary has three parts. The original finding stands as a description of what the uncorrected measure showed. The correction stands, and means the canonical data cannot be used to rule a hot hand out. And larger modern datasets give a mixed picture: Robert Lantis and Erik Nesson, in National Bureau of Economic Research working paper 26510, examine free throws and field goals for the 2004-05 through 2015-16 NBA seasons and report a small hot-hand effect for free throws that more than doubles for longer streaks, while for field goals a made shot leaves the next attempt no more likely and longer made streaks slightly reduce it. A plausible reading is that a controlled repeated motion behaves differently from a contested one, where a player who feels hot takes harder shots and the defense adjusts.

What survives the argument is the part that matters for money. Whether or not a small hot hand exists in basketball, the reason people believe in it is not evidence. The belief precedes the data and survives its absence, and it attaches to any sequence a person watches closely: a fund's run of good years, a stock picker's calls, a rising local property market, a personal run of good decisions. The statistical lesson generalizes even where the basketball answer does not. A short run is a small sample, small samples produce streaks by chance, and distinguishing skill from chance in a handful of periods requires far more data than intuition suggests. Treating a three-year record as informative is the error, whether the inference drawn from it is that the run will continue or that it is due to end.

1. A sequence of outcomes is observed, each one largely independent of the last.

2. Chance produces runs, because runs are a normal feature of random sequences rather than a departure from them.

3. The run is noticed and explained, by attributing a state to whoever produced it: confidence, form, a system that is working.

4. The explanation generates a prediction, that the next outcome is more likely to be a success.

5. The prediction is acted on, by giving the hot shooter the ball or by moving money toward the recent winner.

Take an example that shows why the measurement is so slippery. Flip a fair coin three times and write down the proportion of the flips that immediately follow a heads which are themselves heads. There are eight equally likely sequences. Two of them, TTH and TTT, have no flip after a heads at all, so they produce no number. Of the remaining six: HHH gives two heads out of two following flips, so 1; HHT gives one of two, so 0.5; HTH gives zero of one, so 0; HTT gives zero of one, so 0; THH gives one of one, so 1; and THT gives zero of one, so 0. Averaging the six equally likely values gives 2.5 divided by 6, which is about 0.4167. The coin is fair and the flips are independent, yet this measure expects about 42 percent rather than 50. That gap is the bias Miller and Sanjurjo identified, and it is why a study using this measure could find "no hot hand" in data that in fact contained one.

This is a reasoning error rather than a product, so what follows is where the intuition has some merit and where it misleads.

Where the intuition is not simply wrong

• Streaks are real events. Nothing in the research says a run of successes did not happen, only that it may not predict the next one.
• In tasks that genuinely are repeated under constant conditions, such as free throws, careful modern work finds a small effect.
• Skill is real and persistent in many domains, so treating past performance as informative is not always an error; the question is how much data it takes.
• Noticing a pattern is the first step of a hypothesis, which is useful as long as it stays a hypothesis.

Where it misleads

• A short sequence produces streaks by chance often enough that a streak is almost no evidence of anything.
• The belief attaches most strongly where outcomes are least controlled, which is where independence is most likely.
• It converts a small sample into confidence, and confidence into a decision made at the point where prices or expectations already reflect the run.
• It is the mirror of the gambler's fallacy, so the same person can hold both errors about different sequences without noticing.
• Even measuring the effect correctly is difficult enough that a published finding stood for over thirty years before the measure itself was shown to be biased.