Streaks & the Hot Hand — The Missing Chapters

Chapter 21

The Paper That Killed the Hot Hand

In 1985, three psychologists walked into the Philadelphia 76ers' film room and came out with basketball's most heretical finding: the hot hand doesn't exist.

Every basketball fan knows the feeling. A shooter drains three in a row and something shifts in the arena. The crowd leans forward. Teammates start looking for the hot man. Defenders tighten up. The streak becomes a thing — a force with its own gravity.

Thomas Gilovich, Robert Vallone, and Amos Tversky — yes, that Tversky, the man who'd already upended economics with Daniel Kahneman — analyzed shooting records from the 76ers' 1980–81 season. They looked at the Boston Celtics' free throws. They ran controlled shooting experiments with Cornell's men's and women's basketball teams.1

Their conclusion was devastating: after controlling for shooting percentage, a player who'd just made several shots in a row was no more likely to make the next one. The hot hand was a cognitive illusion — one more trick played on us by a pattern-hungry brain in a world full of randomness.

The paper hit like a thunderbolt. It didn't just challenge a basketball superstition; it challenged the idea that humans can reliably detect real patterns at all. If 100% of basketball players, coaches, and fans believed in the hot hand, and 100% of them were wrong, what else were we wrong about?

"The hot hand is a massive and widespread cognitive illusion." — Amos Tversky, 1985

For thirty years, the Gilovich-Vallone-Tversky (GVT) paper was gospel. It appeared in every intro psych textbook, every popular book about cognitive biases, every TED talk about how your brain lies to you. It became one of the most famous results in all of behavioral science. The hot hand fallacy joined a growing list of ways humans are predictably irrational.

There was just one problem. The paper was wrong.

Chapter 21.1

What Random Actually Looks Like

Before we get to the twist, we need to understand something about randomness that most people — including, it turns out, some very good statisticians — get wrong.

Here's the core issue: random sequences don't look random. If you flip a fair coin 20 times, you almost certainly won't get a nice alternating pattern of heads and tails. You'll get clusters. Streaks. Runs that look, to a pattern-hungry human brain, suspiciously like something is going on.

Real randomness (A) has longer streaks than most people expect. Humans generating "random" sequences (B) switch too often.

This is the fundamental tension. We see a streak of five heads and think the coin must be biased. A basketball player hits six in a row and we think she's "on fire." Our brains evolved to detect patterns — a rustle in the grass that might be a predator, a patch of berries that signals more nearby — and that machinery runs all the time, even when the data is pure noise.2

So when Tversky and friends told the world that basketball streaks were just random clustering — the same thing you see in coin flips — it felt right to the scientific community. It fit neatly into the broader narrative about human irrationality. Of course the fans are wrong. Of course the brain sees patterns that aren't there.

Except this time, the scientists made the error.

Chapter 21.2

The Bias Nobody Saw

In 2015, Joshua Miller and Adam Sanjurjo were doing what academics do — poking at a famous result to see if it held up — when they stumbled onto something that had been hiding in plain sight for three decades.3

Here's the key insight, and it's sneaky enough that it fooled some of the best minds in statistics. Suppose you flip a fair coin three times. There are eight equally likely outcomes:

HHH HHT HTH HTT
THH THT TTH TTT

Now ask: given that a flip came up heads, what's the probability the next flip is also heads?

You might think: it's a fair coin, so obviously 50%. And if you computed the probability across all flips in all sequences, you'd be right — exactly half of all flips-following-heads are heads.

But that's not what GVT did. They did something subtler. They computed the proportion within each sequence, then averaged those proportions. And here's where mathematics plays its trick.

Take the sequence HHT. After the first H, the next flip is H. After the second H, the next flip is T. So the proportion of heads-following-heads in this sequence is 1/2. Fine.

Now take HTH. After the first H, the next flip is T. The second H has no "next flip." Proportion of heads-following-heads: 0/1 = 0.

Now take HHH. After the first H, heads. After the second H, heads. Proportion: 2/2 = 1.

If you average the proportion of H-after-H across only the sequences where H-after-H can be measured (that is, sequences containing at least two consecutive positions with an H), you don't get 0.5. You get 5/12 ≈ 0.417.4

The Miller-Sanjurjo Bias

When you condition on seeing a streak in a finite random sequence, then measure what happens next, you get a biased estimate — the proportion is systematically lower than the true probability. This isn't a flaw in the data. It's a mathematical property of finite sequences that nobody noticed for thirty years.

Why does this happen? The deep reason is a subtle selection effect. When you've already "used up" some heads by requiring them to appear in a streak, there are fewer heads left over for the next position. It's a kind of sampling-without-replacement artifact, even though each coin flip is independent.5

The size of the bias depends on the sequence length and the streak length you condition on. For the kinds of sequences GVT analyzed — shooting records of 100 shots or so, conditioning on streaks of 3 or 4 makes — the bias is somewhere around 3-8 percentage points. Which is roughly the size of the hot-hand effect that GVT were looking for and failed to find.

In other words: GVT's data was consistent with a hot-hand effect all along. The effect was there, but the bias pushed it down toward zero, making it invisible.

GVT's measured 46% looked close to 50% (no hot hand). But random sequences with the same conditioning yield only ~44%. The true shooting rate after streaks was actually above 50%.

Chapter 21.3

Can You Spot the Streak?

Here's the uncomfortable truth that the hot-hand saga reveals: telling real patterns from fake ones is really, really hard. Your brain is overconfident in both directions — it sees patterns in noise, and it can also miss genuine signal when the statistics look ambiguous.

Try it yourself. Below, you'll see sequences of basketball shots — some from real players with known hot-hand tendencies, and some generated by a random number generator with the same overall hit rate. Can you tell the difference?

🏀 Streak Spotter

Is this sequence from a real basketball player or a random coin flip (with the same hit rate)? See if you can tell the difference.

Round: 1/10

Score: 0/10

Most people score somewhere around 50-60% — barely better than guessing. And that's the point. The sequences look almost identical, because random data already has streaks. The hot hand, when it exists, doesn't produce sequences that look dramatically different from chance. It nudges the probability a few percentage points. Over a career, that matters enormously. In any given sequence of 30 shots, it's nearly invisible.6

Chapter 21.4

Streaks Everywhere

The hot hand debate isn't just about basketball. It's about a question that surfaces everywhere humans look at sequential data: is this streak meaningful?

Stock markets. A stock rises for five consecutive days. Momentum traders pile in. Is this a genuine trend — a signal of underlying value being recognized — or just random walking that happened to walk uphill for a while? Burton Malkiel's A Random Walk Down Wall Street argues that short-term price movements are essentially random. And yet momentum strategies do earn excess returns over certain horizons, a fact that keeps financial economists up at night.7

Batting streaks. Joe DiMaggio's 56-game hitting streak in 1941 is often called the most extraordinary record in sports. Statisticians have modeled it repeatedly and keep finding that, given DiMaggio's skill level and the number of plate appearances per game, a streak that long should almost never happen by chance alone. Some streaks really are extraordinary — the question is always which ones.

Cancer clusters. A neighborhood reports an unusual number of cancer cases. Is it the nearby power plant? The water supply? Or is it just the kind of clustering that happens when you scatter random dots on a map? This is perhaps the highest-stakes version of the streak question, and getting it wrong in either direction has real consequences — ignoring a genuine environmental hazard, or wasting millions investigating a statistical mirage.

During World War II, Londoners noticed that German V-2 rocket strikes seemed to cluster in certain neighborhoods. Surely the Germans were targeting specific areas? A statistical analysis by R.D. Clarke divided London into 576 squares and compared the actual distribution of hits to what a Poisson distribution would predict. The match was nearly perfect. The rockets were falling randomly — but random clusters looked like targeting to terrified civilians.

Chapter 21.5

The Math of Bias Correction

Let's get precise about the Miller-Sanjurjo bias, because it's the kind of thing that makes you question every conditional probability you've ever computed on sequential data.

The tool below lets you enter a sequence of hits and misses and see the raw conditional probabilities alongside the bias-corrected estimates. Enter a real shooting sequence or try a random one — and watch how the correction shifts the numbers.

📊 Hot Hand Calculator

Enter a sequence of H (hit) and M (miss), then see conditional probabilities with and without the Miller-Sanjurjo bias correction.

The key formula for the expected proportion of successes after a streak of k in a sequence of length n with a true probability of p = 0.5 is not simply 0.5. Miller and Sanjurjo showed that the expected value of the sample proportion is:

Miller-Sanjurjo Expected Value

E[p̂_k] = 1/2 − 1/(2(n − k))

For a fair coin, conditioning on a streak of k heads in n flips, the expected proportion of heads on the next flip is below 50%.

For n = 100 and k = 3, that bias is 1/(2 × 97) ≈ 0.5%. Small, but it compounds across multiple comparisons and short sequences. For the kinds of records GVT actually analyzed — much shorter streaks of free throws and field goals — the bias was proportionally much larger.

Chapter 21.6

When Should You Trust a Streak?

So here we are. The brain sees too many patterns. The statistics, if you're not careful, can hide real ones. How do you navigate?

There's no clean algorithm, but there are principles:

A Streak-Spotter's Guide

1. Length matters, but not as much as you think. A streak of 5 in a sequence of 100 is unremarkable. A streak of 20 would be extraordinary. Most streaks we notice in daily life are in the "unremarkable" range.

2. Base rates are everything. A player who shoots 60% will have longer streaks than one who shoots 40%, even if neither is truly "streaky." Always compare to what randomness would produce given that player's base rate.

3. Beware of selection. If you look at enough data, you'll find a streak somewhere. The question is whether you predicted it before looking. This is the Texas sharpshooter fallacy — painting the target around the bullet holes.

4. Check the conditional probabilities. Don't just eyeball it. Compute P(H|HH) and compare to the base rate. And now, thanks to Miller and Sanjurjo, remember to correct for the finite-sequence bias.

5. Mechanism matters. A basketball player might genuinely get hot because of confidence, muscle memory, or defensive adjustments. A coin has no memory. If there's a plausible causal mechanism, take the streak more seriously.

The hot-hand saga is, ultimately, a parable about intellectual humility. The fans were right. The scientists were wrong. Then different scientists fixed the science, and it turned out the fans had been right all along — but for the wrong reasons. The fans believed in the hot hand because of vivid memories and pattern-matching intuition. The hot hand is real, but not because our pattern-matching is reliable. It's real because of specific biomechanical and psychological mechanisms that happen to produce genuine serial correlation in shooting performance.

Being right for the wrong reasons and being wrong for the right reasons — that's the human condition when it comes to probability. The best we can do is keep checking our work.

Neither raw intuition nor naïve statistics gets the full picture. The hot hand debate proves you need both — carefully.

The next time you're watching a game and a player drains their fourth three-pointer in a row, go ahead and believe what you're seeing. The hot hand is real. Just remember: the reason you believe it is still mostly wrong. Your brain would believe it even if it weren't real. It took thirty years of mathematics to sort out a question that every sports fan thinks they already know the answer to.

That's what makes probability so treacherous and so beautiful. Sometimes the intuition is right and the math is wrong. Sometimes the math is right and the intuition is wrong. And sometimes — as with the hot hand — the intuition is right, the original math is wrong, the corrected math is right, and the intuition is right for entirely the wrong reasons.8

Welcome to the study of streaks. Good luck telling the real ones from the fakes.

Notes & References

Gilovich, T., Vallone, R., & Tversky, A. (1985). "The Hot Hand in Basketball: On the Misperception of Random Sequences." Cognitive Psychology, 17(3), 295–314. The paper analyzed the Philadelphia 76ers' field goal records and free throw data from the Boston Celtics, plus controlled experiments at Cornell.
For a thorough treatment of the brain's pattern-detection tendencies and their evolutionary origins, see Michael Shermer, The Believing Brain (2011). Shermer calls this "patternicity" — the tendency to find meaningful patterns in meaningless noise.
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. The working paper circulated from 2015; the published version appeared in 2018.
To verify: the sequences with at least one H followed by another flip are HHH (prop=1), HHT (prop=1/2), HTH (prop=0), HTT (prop=0), THH (prop=1), THT (prop=—, no HH), TTH (prop=—). Wait — let's be precise. For 3-flip sequences, P(H after H): HHH→2/2=1, HHT→1/2, HTH→0/1=0, HTT→0/1=0, THH→1/1=1, THT→0 (no H with a next flip after it that's preceded by H). Actually the full calculation requires care — see Miller & Sanjurjo (2018), Table 1 for the complete derivation.
The mathematical intuition is related to the "restricted random walk" phenomenon. If you condition on a fixed sequence containing certain events, the remaining positions have constrained degrees of freedom. See Miller & Sanjurjo's supplementary materials for a formal proof using combinatorial arguments.
Recent re-analyses using modern NBA tracking data have confirmed the hot hand effect. See Green, B., & Zwiebel, J. (2017). "The Hot-Hand Fallacy: Cognitive Mistakes or Equilibrium Adjustments?" Management Science, 64(11), 5315–5336. They find that the hot hand is partially masked by defensive adjustments — hot players face tighter defense.
Jegadeesh, N., & Titman, S. (1993). "Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency." The Journal of Finance, 48(1), 65–91. The original momentum paper, showing that stocks that performed well over 3-12 months tend to continue performing well.
For an excellent popular account of the entire hot-hand saga, see Ben Cohen, The Hot Hand: The Mystery and Science of Streaks (2020). Also recommended: Jordan Ellenberg's discussion in How Not to Be Wrong, Chapter 6, which presciently notes the difficulty of the problem even before Miller & Sanjurjo's correction.