The single most consequential Bayes computation there is. Physicians get this wrong. Courts get it wrong. See if your intuition survives contact with the arithmetic.
A screening test for a rare condition:
- The condition affects 1 in 1,000 people: P(condition) = 0.001. (This is the base rate — the prior, before any test.)
- If you have the condition, the test comes back positive 99% of the time (P(positive | condition) = 0.99 — the test’s sensitivity).
- If you don’t have it, the test still comes back positive 2% of the time (P(positive | no condition) = 0.02 — the false-positive rate).
A randomly screened person tests positive. What’s the probability they actually have the condition?
Before computing: write down your gut answer. Most people — including, in repeated studies, most doctors given exactly this problem — say something near 99%, reasoning “the test is 99% accurate.” Hold that gut number; you’ll want to compare.
Then do it the honest way (lesson 6): take a concrete population — 100,000 people works cleanly — and count. How many have the condition, and how many of those test positive? How many don’t have it, and how many of those test positive anyway? A positive result puts you in the combined pool; the question is what fraction of that pool is actually sick.