Why a 99% accurate test can be wrong half the time

2026-07-07

A disease affects 1 person in 100. There is a test for it, and the test is 99% accurate - it correctly flags 99% of the sick, and it correctly clears 99% of the healthy. You take the test. It comes back positive.

How worried should you be? Most people say 99%. When versions of this question have been put to practising doctors, most of them said something close to 99% too. The real answer, for the numbers above, is 50%. A coin flip. And if the disease is rarer - 1 in 1,000 - the answer falls to 9%.

No algebra is needed to see why. Just draw everyone.

The square is 10,000 people, drawn area-true: every group's patch of the square has exactly as much area as it has people. The thin slice on the left is the sick - at 1 in 100, just 100 people. The solid red part of that slice is the sick the test catches: 99 of them. And the hatched strip along the bottom of the huge healthy region is the test's mistakes - the 1% of 9,900 healthy people it wrongly flags: 99 of them.

Look at those two slivers. The strip of false alarms is just as big as the block of real catches. That is the whole story: the sick are so rare that a small error rate applied to the enormous healthy majority produces as many false positives as there are sick people found at all.

Throw away everyone else

Now press play. "You tested positive" is a piece of evidence, and using evidence means one thing: discard every person the evidence rules out. Everyone who tested negative - the pale bulk of the square - vanishes from consideration. The only people left standing are the two slivers, and they slide out into one bar: everyone who tested positive.

You are somewhere in that bar. The chance you are actually sick is simply the red share of it - and it is half, because the two slivers hold 99 people each. A 99% accurate test, a positive result, and a coin flip's worth of certainty.

Now drag how rare down to 1 in 1,000 and watch the bar. The sick slice thins, its red sliver shrinks, but the false-alarm strip barely changes - it is drawn from the healthy, and the healthy are still nearly everyone. The red share of the bar collapses to 9%. At 1 in 2,000 it is under 5%. The test never changed. Only the rarity did.

The formula was hiding in the bar

What you just read off the bar has a name. The chance of being sick given a positive test is the true positives divided by all positives:

P(sick | positive) = true positives / (true positives + false alarms)

Write each count as a fraction of the population and this is exactly Bayes' theorem: P(sick | +) = P(+ | sick) P(sick) / P(+). The forbidding formula is nothing more than "the red share of the bar". Every term is a rectangle you have already seen: the prior P(sick) is the width of the slice, the likelihood P(+ | sick) is how much of the slice is solid red, and the denominator P(+) is the whole bar.

The base rate does the work

The trap in the opening question is a confusion between two different numbers. "The test is 99% accurate" is a fact about the test - how it behaves on a sick person, on a healthy person. "Your chance of being sick after a positive" is a fact about the test and the population it was pointed at - and when the disease is rare, the population term does most of the work. Push test accuracy up to 99.9% with the disease at 1 in 1,000: even that heroic test only gets you to a 50% posterior. Rarity is not a detail. It is half the answer.

This is why screening whole populations for rare conditions produces so many frightening-then-retracted results, and why a positive result is usually followed not by a diagnosis but by a second, different test. Each test's job is to shift the base rate for the next one - to widen the slice before the next pair of slivers is compared.

One square, drawn honestly. The sick are a sliver, the false alarms are a strip, and a positive test just asks which of the two you are more likely to be standing in. Probability by counting - like the bell curve falling out of coin flips - where the counting is done with area, and the area was there all along.