DNA evidence is often introduced with a striking number — the profile would occur in perhaps one person in a billion. It sounds like it settles the case. But interpreting that figure is where trials are genuinely won and lost, because a very natural reading of it is simply wrong. That error has a name: the prosecutor's fallacy.
What the number actually means
The statistic is a random match probability: the chance that a random, unrelated person would happen to share the same profile. It quantifies coincidence — nothing more. It is not the probability that the defendant is innocent.
The fallacy
The prosecutor's fallacy swaps those two things. It reasons: "the match is one in a billion, so there's only a one-in-a-billion chance he's innocent." But those are different quantities:
- P(match, given innocent) — the chance an innocent person matches: this is the small statistic.
- P(innocent, given a match) — the chance the defendant is innocent despite matching: this is what the jury actually cares about, and it is not the same number.
In a large population, even a tiny random-match probability can mean that several people share the profile. The match alone cannot say which of them left the sample; it must be weighed with all the other evidence.
It has changed verdicts
This isn't academic. English appeal courts have grappled with exactly this confusion — the case of R v Adams turned on how jurors should reason about DNA statistics — and convictions elsewhere have been overturned where a compelling-sounding number was presented as if it were the probability of guilt.
The expert's proper role
Good practice keeps the roles separate: the scientist reports the match and the random-occurrence ratio, and may say roughly how many people would be expected to share the profile — but must not offer an opinion that the defendant is guilty. That judgement belongs to the court. Understanding the difference is the single most valuable statistical skill in forensic evidence.