Module II of VI

Bayes by counting

A test for a rare disease is advertised as “90 percent accurate.” You take it. It comes back positive. How worried should you be? Most people — including, in study after study, a sobering share of physicians — answer somewhere near 90 percent. The real answer, for the numbers we’re about to use, is closer to 9 percent. You can reach it without a single formula. You only have to count.

Take 1,000 people. The disease has a prevalence of 1 percent, so 10 of them are sick and 990 are healthy. The test catches 9 of the 10 sick people — that’s its sensitivity. It also correctly clears 91 percent of the healthy — its specificity — which means it misfires on the other 9 percent, flagging 89 healthy people. Now count everyone holding a positive result: 9 sick, 89 healthy, 98 in all. A positive test puts you in a room of 98 people where 9 have the disease.

Panel II·1 — The population grid

1 dot = 1 person · n = 1,000

Prevalence 1 %Sensitivity 90 %Specificity 91 %

sick, tests positive — 9

sick, missed — 1

healthy, false alarm — 89

healthy, negative — 901

9 sick + 89 false alarms = 98 positives → 9 of 98 are real = 9.2 %

Try this Drag prevalence from 1 % to 10 % and watch how many of the positives are real. Then push specificity toward 99 % instead — which slider rescued the test more? Most people guess wrong before trying it.

Why does intuition crash here? Because “90 percent accurate” answers a different question than the one you care about. The test’s accuracy says: if you’re sick, how often does it say so? You want to know: it said so — am I sick? Swapping those two questions feels harmless and quietly reverses the arithmetic. The rarity of the disease does the damage: with only 10 sick people in the room, even a small error rate applied to the 990 healthy ones produces a crowd of false alarms that swamps the true ones.

Notice what made the puzzle easy: we counted people. Stated as percentages — “P(positive | sick) = 0.9” — the problem defeats most educated adults. Stated as natural frequencies — “9 of the 98 positives are sick” — schoolchildren can solve it. The psychologist Gerd Gigerenzer built a career on this observation — the physician studies above are his and Ulrich Hoffrage’s — and it’s the reason this tutorial keeps reaching for grids of dots: our species is old friends with counting and a recent, awkward acquaintance of percentages.

The number you computed — the share of positives that are real — is your updated belief, and from here on it gets its proper name: the posterior. What you believed before the test (1 percent, the prevalence) is the prior. The test result is the evidence that carried you from one to the other. That’s the entire skeleton of Bayesian reasoning, and you just did it by sorting dots into piles.

Retell itA positive result from a good test for a rare condition is usually a false alarm — check how rare, then count.

When a formula finally appears in Module IV it will only wrap words around what you did here. First, though, Module III pushes on the number everyone ignores: the one you start from.