Overview
Doctors, judges and engineers all fall for this one. When a rare event meets an imperfect test, the false positives outnumber the true ones — often by a lot.
How to solve The 95% Accurate Test
- Never reason in percentages; reason in people. Take 100 of them.
- 1 person is ill and tests positive. Of the 99 healthy people, 5% — about 5 — also test positive.
- That is 6 positive results, only 1 of which is real: P(ill | positive) = 1/6 ≈ 17%.
The key insight
The test's '95% accuracy' is P(positive | healthy) = 5%, which is not what you asked. You want P(ill | positive), and Bayes' theorem says that answer is weighted by how rare the disease is — the base rate you were never given a reason to think about.
Variations & echoes
- The same arithmetic explains why mass screening for rare diseases produces so much needless anxiety, and why a second independent test changes the picture completely.
- In court it appears as the prosecutor's fallacy: P(match | innocent) is not P(innocent | match).
Frequently asked questions
So should I ignore a positive result?
No — it raised your odds seventeen-fold, from 1% to 17%. It just is not the near-certainty it feels like, which is exactly why a confirmatory test exists.