The Hot Hand That Was Real All Along

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In 1985, three psychologists set out to test something every basketball fan already knew. Thomas Gilovich, Robert Vallone and Amos Tversky asked fans whether a player has a better chance of making a shot after making their last two or three than after missing them, and 91 percent said yes. Players and coaches agreed. Then the researchers went to the data. They examined a season of Philadelphia 76ers field-goal records and Boston Celtics free throws, and they ran a controlled experiment: 26 Cornell varsity players, 14 men and 12 women, each took 100 shots while the researchers logged every make and miss.

The streaks weren't there. After a run of hits, players were no more likely to hit the next shot than after a run of misses. The sequences looked statistically like coin flips. Published in Cognitive Psychology, the paper concluded that the hot hand was a cognitive illusion: people see patterns in randomness because they expect short random sequences to look more orderly than they do. For three decades, the "hot hand fallacy" sat in textbooks as a showcase of human irrationality. Then, in 2018, the economists Joshua Miller and Adam Sanjurjo published a paper in Econometrica showing that the original test had a hidden bias, and that once it was corrected, the long-standing conclusion of the canonical study reversed. The fans had a point. The statisticians had made the error.

The mechanism: looking after a streak tilts the count

The test sounds airtight. Take each player's sequence, find every shot that came right after a streak of hits, and compute the hit rate on those shots. Do the same after streaks of misses. If there's no hot hand, the two rates should be about equal. Right?

Not in a finite sequence. Try it with a fair coin. Flip it four times, find every flip that immediately follows a head, and record the share of those flips that came up heads. Average that share across all 16 possible sequences (dropping the two with no head in the first three flips) and you get 17/42, about 40.5 percent, not 50. The coin is perfectly fair. The procedure isn't.

Here's the intuition. A short sequence holds only so many heads. Every time you pick out a flip because the one before it was a head, you've already spent one of those heads, so the flips left to choose from lean slightly toward tails. It's the same reason that after you draw a red card from a small deck, the next card is a little more likely to be black. Averaging the proportion sequence by sequence adds to the tilt, because a streaky sequence with many qualifying flips counts no more than one with a single head followed by a tail. The result is a streak selection bias: conditioning on a past streak in a finite sample pulls the measured hit rate down, even when every shot is independent.

The bias shrinks as sequences get longer and grows as the streak you condition on gets longer. At 100 shots and streaks of three, it is not small. Run the arithmetic for a 50 percent shooter whose shots are truly independent, and the expected hit rate after three straight hits comes out around 46 percent, while the rate after three straight misses comes out around 54. A truly streak-free shooter should look roughly 8 points colder after a hot run. So when Gilovich and colleagues found "no difference," that null result was actually evidence of a positive effect hiding under a negative bias. The lesson isn't that the 1985 team was careless. It's that a test almost anyone would accept had a flaw nobody noticed for 30 years.

None of this means streaks are as large as fans feel they are; researchers still debate how big the hot hand is and when it appears. What the case settles is narrower: the evidence that was supposed to prove people irrational was itself built on a biased estimator.

Where else this shows up

• Academic and grant-funded research. Any analysis that conditions on what just happened in a short sequence is exposed: trial-by-trial learning in experiments, momentum in diary or experience-sampling studies, a patient's next symptom score after a run of bad days. Per-person proportions averaged across many short sequences are exactly where this bias lives. Simulate your analysis on data with no effect built in, and see whether it returns zero.
• Non-profit program evaluation. "Clients who attended three sessions in a row were more likely to attend the fourth" sounds like evidence of engagement momentum. Depending on how it's computed, the raw comparison can be biased in either direction relative to what pure chance would produce. Before reporting streak or persistence effects from attendance logs or case notes, check what your method produces when attendance is random.
• AI and LLM evaluation. Teams ask whether a model that got the last few items right is more likely to get the next one right, whether an agent "spirals" after an error, or whether a human rater drifts after a run of passes. These are streak questions on finite sequences, often short ones like a single conversation or agent run. The fix is the same: establish the null distribution by simulation or permutation rather than assuming the naive comparison should come out to zero.

What to ask for instead

Whenever someone reports a pattern that depends on what came just before, ask one question: what does this exact analysis show on data where we know there's no effect? Shuffle the sequence, or simulate independent outcomes with the same length and base rate, run the identical calculation, and compare. If the "no effect" version doesn't come out at zero, the benchmark you've been comparing against is wrong. That one check would have caught the hot hand bias in 1985, and it costs an afternoon.

If your study, evaluation, or model analysis looks for momentum, streaks, or sequence effects and you want to be sure the method isn't quietly answering for you, talk to us.

Related reading

• The Asylum Judges Who Couldn't Say Yes Twice — the hot hand's mirror image: decision-makers who expected chance to even out, and leaned against their own last call.
• The Best Schools Were Small. So Were the Worst. — another case where short samples behave in ways intuition (and sometimes the analysis) doesn't expect.

Sources and further reading

• Gilovich, T., Vallone, R., & Tversky, A. (1985). The hot hand in basketball: On the misperception of random sequences. Cognitive Psychology, 17(3), 295–314.
• Miller, J. B., & Sanjurjo, A. (2018). Surprised by the hot hand fallacy? A truth in the law of small numbers. Econometrica, 86(6), 2019–2047.

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