Interactive notebook
Bayes' Theorem
Why a 99%-accurate test can still be wrong most of the time it says yes.
The classic exam question: the answer is about 17%, and almost nobody guesses that low.
- Sick, tested positive (99)
- Healthy, tested positive (495)
- Sick, tested negative (1)
- Healthy, tested negative (9405)
You tested positive. The chance you actually have it:
16.7%
99 of the 594 people who tested positive are actually sick.
Most of the positives are blue — healthy people the test got wrong. There are simply far more healthy people to be wrong about.
The trap is that “99% accurate” describes the test, not your situation. What matters is how many healthy people get tested, because even a small error rate applied to a large healthy population can swamp the true positives entirely. Slide prevalence up and watch the answer transform without touching the test’s accuracy at all — same test, same person, completely different meaning.