About 91% likely innocent. The count:
Innocent matchers: ~10 million innocent residents × 1-in-a-million match probability ≈ 10 people, in expectation.
Guilty matchers: the one true source, who matches for sure: 1 person.
A person known only to match is one of ≈ 11, of whom 10 are innocent:
P(innocent | match) ≈ 10/11 ≈ 0.91
The prosecutor’s “one in a million” and the correct “91%” differ by six orders of magnitude, using the same evidence. Nothing was miscalculated in the lab; the number was simply transposed — P(match | innocent) sworn in as P(innocent | match). This isn’t hypothetical: the fallacy has produced real convictions, real appeals, and a small canon of infamous cases (People v. Collins, Sally Clark — lesson 4’s independence abuse and this transposition often travel together).
Three habits worth extracting:
- Count the reference class. “How rare is this evidence?” means nothing until you ask how many opportunities the evidence had to occur. One-in-a-million events are everyday events in a city of ten million. (Same reflex, different clothes: “this A/B test hit p < 0.001!” — out of how many metrics × segments × daily peeks? A dashboard is a database sweep.)
- The match did enormous work — just not all of it. The defendant went from 1-in-10-million (a random resident) to 1-in-11 (a matcher). That’s evidence multiplying plausibility by nearly a million. The fallacy isn’t overrating the evidence’s strength; it’s forgetting how far away the starting line was. Strong evidence + tiny prior can still equal “probably not.”
- What would legitimately convict? More evidence: opportunity, motive, a second independent forensic line. Each shrinks the innocent-matcher pool further. And “found by sweeping a database” vs “matched after being arrested on other grounds” are different procedures with different priors — lesson 5’s information-in-the-telling, now with liberty at stake.
Where this goes: you’ve now reversed conditionals three times by rebuilding a population and counting. The odds form of Bayes’ theorem does the same update as pure multiplication — prior odds × likelihood ratio = posterior odds — and turns all three of these lessons into one-line arithmetic. That’s next.