The hot hand: thirty years a fallacy, then the fallacy was the mistake

The 1985 result

Ask anyone who has watched basketball and they will tell you that shooters get hot: make three in a row and the fourth feels more likely. In 1985 Thomas Gilovich, Robert Vallone and Amos Tversky set out to test that belief against records. They took shooting sequences from the Philadelphia 76ers, free throws from the Boston Celtics and a controlled shooting experiment with Cornell players, and asked a precise question: is a player’s hit rate after a run of hits higher than his hit rate after a run of misses?

It was not. In their data the probability of a hit after three hits was, if anything, slightly lower than after three misses. Fans, players and coaches all believed in the hot hand; the numbers did not show one. The paper concluded that the hot hand was a cognitive illusion, a cousin of the gambler’s fallacy: people expect random sequences to look more alternating than they are, so an ordinary cluster of hits reads as a streak.

The result was influential far beyond basketball. For three decades “hot hand fallacy” was shorthand, in economics and psychology, for the mind manufacturing momentum out of noise. It sat beside the gambler’s fallacy as the two ways people misread independent events: expecting a reversal, or expecting a continuation, when the process has no memory either way.

The counting error

In 2018 Joshua Miller and Adam Sanjurjo published a paper with a modest title and an immodest consequence. The 1985 test, they showed, has a bias built into it. Take a sequence of independent fair coin flips, find every flip that follows a run of heads, and compute the share of those flips that are heads. Intuition says that share should average 50%, because the coin has no memory. It does not. Averaged over all possible sequences, it is below 50%, and for short sequences it is well below.

The 4-flip figure is an exact enumeration of the 16 sequences; the others are simulation averages over 200,000 sequences each, computed for this article. In every case the true probability of heads on every flip is 50%.

The reason is a selection effect, and the four-flip case shows it in miniature. Write out all sixteen sequences of four flips, and for each one look at the flips that come after a head. In a sequence like HHHH every counted flip is a head, but that sequence contributes one observation, the number 100%, to the average. In a sequence like HTHT the two counted flips are both tails, contributing 0%. The sequences that have many heads to count also tend to have used up their heads in the run itself, so heads-after-heads are systematically under-represented when you average the proportions sequence by sequence. The 1985 study computed exactly that kind of average, and compared it to the player’s overall rate as if the fair baseline were 50%.

Correct the baseline and the conclusion flips. Against the biased expectation, the 1985 shooters’ hit rate after runs of hits was not equal to their normal rate; it was several percentage points above it. The data that founded the hot hand fallacy contained, on a fair reading, a hot hand. Thirty years of citations had been describing a mistake in the arithmetic as a mistake in the mind.

The intuition was right, the test was wrong, and the test had been trusted precisely because it contradicted the intuition.the shape of the 2018 correction, in one sentence

What it means for a gambler

It is tempting to read the correction as vindication for every player who has ever felt hot at a table, and that reading is wrong in an instructive way. The hot hand in basketball is a claim about a human being: a shooter’s form, confidence or rhythm can carry from one attempt to the next because the same body takes both shots. The correction says the evidence against that claim was flawed. It says nothing about dice.

• A streak in a skill process can mean something. Shooting, poker decisions, sports betting handicapping: where a person’s state affects the outcome, runs can carry information, and the 2018 result says the classic evidence that they do not was mismeasured.
• A streak in a chance process means nothing. A roulette wheel, a die or a shuffled shoe has no state that a run can express. Twenty-six blacks at Monte Carlo told the room nothing about the twenty-seventh spin, and the correction does not change that by a hair.
• The bias runs the other way from the fallacy. The gambler’s fallacy expects a reversal after a run; the hot hand expects a continuation. On independent events both are wrong, and the 2018 paper adds a third error: a statistician measuring runs in short sequences will see a spurious reversal even in perfect randomness, and should not mistake it for the gambler’s fallacy being right.

The practical test is whether anything carries between rounds. In a game where it does, watch the streaks. In a game where it does not, the streak is a fact about the past with no vote on the future, and the feeling that it has one is the near-miss effect and its relatives doing their work.

No state to carry

On this site the question has an exact answer, because the process that produces each round is written down. Every round of every original is derived from the same three inputs: the server seed the house committed to, your client seed and a round number that increases by one each bet. The result of round 41 is a hash of those inputs with 41 in them; round 42 is the same hash with 42. There is no variable that round 41 sets and round 42 reads.

That makes a streak here something stranger than random: it is fully determined and fully uninformative at once. Every result under a seed pair already exists the moment the pair does, and none of them says anything about its neighbours, because a cryptographic hash is built so that changing one input bit scrambles the whole output. A run of five wins is not momentum and not a debt coming due. It is five hashes that happened to land on one side.

The one thing a streak on a seeded game can legitimately make you do is check it. After the pair is rotated and the server seed revealed, every round in the run can be recomputed in the browser, which is a better use of a hot feeling than raising the stake. The verification walkthrough does it step by step, and it works the same on a run of losses.