Week 2: Probability feels wrong

Why intuitions about chance are reliably, predictably bad — and what that means for anything you're tempted to call a coincidence.

Last week was about why the detector fires. This week is about a second, compounding problem: even when someone tries to reason carefully about chance, the reasoning itself tends to be wrong, in specific and repeatable ways.

The birthday paradox in the activity below is one example. Another is the gambler’s fallacy — the belief that a coin “due” for tails after five heads is more likely to land tails, when in fact each flip is independent and the coin has no memory. A third is the sheer size of the “reference class” a coincidence is drawn from. If you notice one striking coincidence a month, across a 20,000-day life that is over 650 opportunities for something mundane to line up in a way that feels impossible — not because your life is unusually fated, but because the population of “days in which something could coincide” is enormous, and only the hits get remembered.

Two more named errors belong next to the gambler’s fallacy, because all three share one root cause. The clustering illusion is the tendency to see meaningful clumps or streaks in data that is actually independent and random — five heads in a row looks like a pattern, even though a truly random process produces streaks like that constantly. The law of small numbers is the related mistake of expecting a small sample to look as “balanced” as a huge one — assuming that ten coin flips should come out close to five-five, when in fact ten flips routinely produce seven-three or eight-two runs purely by chance. All three errors come from the same place: human intuition expects randomness to look orderly-random — evenly spread, streak-free — when real randomness is actually clumpier and streakier than that intuition allows.

The hot hand: a case that resists a clean verdict#

The clustering illusion has a famous applied case. In 1985, psychologists Thomas Gilovich, Robert Vallone and Amos Tversky analysed shooting records from the Philadelphia 76ers and a controlled shooting experiment with Cornell’s basketball teams, looking for the “hot hand” — the belief that a player who has just made several shots in a row is more likely to make the next one. They found no such effect: a player’s hit streaks were statistically indistinguishable from what a fixed-percentage random shooter would produce by chance. The finding became one of the most cited demonstrations of the clustering illusion in the entire behavioural-science literature — thirty years of “the hot hand is a myth.”

Then, in 2015, economists Joshua Miller and Adam Sanjurjo found a subtle statistical bias hiding inside the original method: when you look only at shots that followed a streak, you are sampling in a way that slightly understates the true hit rate, even in genuinely random data. Correcting for it, some real hot-hand effects reappeared in the same kinds of data the 1985 study used. The honest state of the field is not “the hot hand is real” or “the hot hand is a myth” — it’s that a famous debunking turned out to have its own hidden assumption, and the correct answer sits somewhere between the two confident-sounding headlines. That unresolved middle is exactly the kind of outcome this course keeps returning to, and exactly what a flat true/false verdict would have hidden.

Try the arithmetic yourself#

Before reading on: if a fair coin is flipped five times, what’s the chance of getting heads all five times? It’s 1⁄2 multiplied by itself five times — 1⁄32, or about 3%. That means in any run of about 32 five-flip stretches, you should expect to see one all-heads run purely by chance. A five-streak isn’t a sign the coin (or the shooter, or the pattern) is doing something special. It’s what randomness looks like often enough that everyone should expect to personally witness one sooner or later.

What this changes#

This is not an argument that probability explains away every coincidence, or that every striking event is secretly mundane once you do the arithmetic — the hot hand case above shows real effects can hide inside data that looks like noise, in both directions. It is an argument that “how surprising does this feel” is a bad instrument for measuring “how surprising is this,” and the gap between the two is wide, consistent, and well studied. Kahneman and Tversky’s research on heuristics and biases — much of which the course returns to in week 10 — is the intellectual backbone of this claim, and it applies before any interpretation gets added: an event can be statistically unremarkable and still feel enormous, long before anyone starts calling it a sign.

Further reading#

• Thomas Gilovich, Robert Vallone & Amos Tversky, “The Hot Hand in Basketball: On the Misperception of Random Sequences” (1985).
• Amos Tversky & Daniel Kahneman, “Judgment Under Uncertainty: Heuristics and Biases” (1974).

Estimate first#

Estimate before you look. The gap between your guess and the answer below is the whole lecture in miniature.