the prosecutor's fallacy
The prosecutor's fallacy is mistaking P(evidence given innocent) for P(innocent given evidence) — treating the probability of the evidence assuming innocence as if it were the probability of innocence given the evidence. In a courtroom: 'the chance an innocent person's DNA would match by accident is one in a million, therefore there is only a one-in-a-million chance the defendant is innocent.' That conclusion does not follow.
The two conditional probabilities can be wildly different because they ignore the base rate — the prior chance of guilt before the match. Suppose a DNA database has 10 million innocent people and the random-match probability is one in a million. Then among the innocent alone you expect about 10 accidental matches. If the only evidence against a defendant is that they matched, they are just one of those ~10 (plus possibly the true culprit) — so the probability of innocence given the match is nowhere near one in a million; it is closer to 10 in 11. The small number P(match given innocent) said nothing direct about P(innocent given match); Bayes' theorem is the only honest bridge between them, and it forces in the prior.
This fallacy has overturned real convictions and is exactly the base-rate fallacy wearing a courtroom suit. The same transposition appears whenever a striking coincidence is offered as proof: 'the odds of this happening by chance are tiny, so it cannot be chance.' To answer such a claim you must compare it against how many opportunities there were and the prior plausibility of the alternative — not just stare at one small likelihood.
Random DNA match probability one in a million sounds damning. But search a city of 2 million innocent people and you expect about 2 accidental matches. A match alone then leaves the suspect roughly a 1-in-3 chance of innocence (sharing the spotlight with the true culprit), not one in a million.
A tiny match probability times a large innocent population yields many accidental matches.
P(evidence given innocent) and P(innocent given evidence) are different numbers; swapping them is the 'transposed conditional'. Only Bayes' theorem with an honest prior connects the two.