Lesson 7 showed the error in a hospital. This lesson shows it in a courtroom, where it has a name: the prosecutor’s fallacy.
A crime is committed in a city of 10 million people. The only evidence is a DNA sample from the scene. The forensic lab reports:
“The probability that a random innocent person would match this sample is one in a million.” — that is, P(match | innocent) = 0.000001.
A database sweep finds a match: the defendant. At trial, the prosecutor argues:
“The probability of a match if he were innocent is one in a million. Therefore the probability he is innocent is one in a million.”
That sentence performs the classic swap — it reads P(match | innocent) as P(innocent | match). Lesson 2 showed those live in different slices; lesson 7 showed the gap between them is set by the base rate. Here the “base rate” is brutal: before the DNA evidence, the defendant was just one resident among 10 million, exactly one of whom is the true source.
Do the count, exactly like the medical test: in a city of 10 million, how many innocent people match a one-in-a-million profile, in expectation? How many guilty people match? A matching person sits in that combined pool. (Assume the true source is in the city and would certainly match, and — generously to the prosecutor — that the lab never makes clerical errors.)