Bayes' theorem and the 99%-accurate test

A medical test is 99% accurate. You test positive. What's the chance you actually have the disease? Most people say 99%. The real answer is usually far lower.

The theorem

where the denominator sums over both ways to test positive:

Plug in numbers

• Prevalence $P(D) = 0.001$ (1 in 1,000)
• Sensitivity $P(+ \mid D) = 0.99$
• False-positive rate $P(+ \mid \neg D) = 0.01$

Why so low?

Picture 100,000 people. 100 are sick and 99 of them test positive. Of the 99,900 healthy people, 1% — about 999 — also test positive. So only 99 of 1,098 positives are real.

When a condition is rare, false positives outnumber true positives — even with an accurate test.

Test twice

Use the posterior as the new prior. After a second independent positive: