Prosecutor's Fallacy

The statistical error of confusing P(evidence|innocent) with P(innocent|evidence)—the probability of seeing this evidence IF someone is innocent versus the probability that someone IS innocent GIVEN this evidence. A DNA match occurs in 1 in a million people; a prosecutor argues this means there's a 1-in-a-million chance the defendant is innocent. But in a city of 10 million, 10 people would match—so the probability of innocence given a match is actually about 9/10, not 1/1,000,000. In business: 'only 1% of startups succeed at this, and this startup did it—they must be exceptional!' But if 10,000 startups attempted it, 100 would succeed by base rates alone.

When to use it

When rare coincidences seem to prove guilt, causation, or exceptionality; when diagnostic tests produce positive results for rare conditions; when statistical evidence is being used to make probability claims without base rate adjustment; when Bayesian reasoning is needed to correctly interpret conditional probabilities.

How it can help

When a rare match or coincidence seems to prove causation or exceptionality: apply Bayes' theorem. Ask: 'given the base rate and the population size, how many matches would I EXPECT by chance?' In hiring: a candidate passes your rare filter—but how many other candidates in the population would also pass? In diagnostics: a rare test result comes back positive—but what's the false positive rate given the base rate of the condition? The correction always requires knowing the base rate AND the population size, not just the match probability.