Bayes’ theorem · reading a lab result

WHEN THE PROBLEM IS RARE, MOST POSITIVES ARE FALSE

P(contaminated | +) = p s_e p s_e + (1 − p)(1 − s_p)

Ten thousand lots, one in a thousand truly contaminated. The test is good: it catches 99 of every 100 it should and clears 99 of every 100 it should. Watch what comes back positive — about ten real, and about a hundred false. A single positive is right roughly nine percent of the time, and nothing is wrong with the test. Rarity does this to any test, however good. It is also why a positive is confirmed rather than acted on: retest, and the false ones fall away.