What is the prosecutor's fallacy when a DNA database cold hit has a one-in-a-million match probability?

## The two conditionals Every version of this error is the same move: reading `P(E | H)` as `P(H | E)`. In the forensic setting, - `P(match | innocent)` is the **random match probability** (RMP). It is what the laboratory can estimate from allele frequencies: how often a profile drawn from the population would coincidentally match the crime-scene sample. Say 1 in 1,000,000. - `P(innocent | match)` is what a court cares about. It cannot be computed from the RMP alone, because it depends on how likely the person was to be the source *before* the DNA evidence arrived. The prosecutor's fallacy asserts that the second equals the first. It is the same structural mistake as reading a test's sensitivity as the probability of disease given a positive: the direction of conditioning has been reversed and the base rate has been dropped. ## Why the gap is enormous in a cold hit A **cold hit** means no suspect existed beforehand; a database of profiles was searched and one matched. Suppose the database holds 500,000 profiles. The expected number of coincidental matches in that search is `500,000 x (1/1,000,000) = 0.5`. In other words, if the true source were not in the database at all, you would still expect roughly a coin-flip's worth of innocent matches per search. Finding one match is therefore not astonishing on its own. Run the odds form of Bayes' rule under a deliberately simplified assumption -- that the source is certainly one of the 500,000 and all are equally likely a priori: ``` prior odds (this person is the source) = 1 : 499,999 likelihood ratio for a match = 1 / (1/1,000,000) = 1,000,000 posterior odds = 1,000,000 : 499,999 ~ 2 : 1 posterior probability ~ 2/3 ``` About two-thirds. That is meaningful evidence and it is nothing like the '99.9999% certain' that the fallacy suggests. Change the assumptions -- a larger suspect population, real doubt about whether the source is in the database at all -- and the number moves again, which is precisely the point: the RMP alone does not determine it. ## The contrast with a confirmatory test The same RMP means something very different when a suspect was already identified by independent evidence -- opportunity, motive, a witness -- that narrows the plausible source population to, say, 20 people. Now the prior odds are 1:19, the likelihood ratio is still 1,000,000, and the posterior odds are about 52,000:1. Identical laboratory statistic, radically different conclusion, because the prior differs. This is the operational lesson: a match found by searching a large database is weaker evidence than the same match on a pre-identified suspect, and the difference lives entirely in the prior. ## The mirror-image error The **defence attorney's fallacy** runs the other way: in a country of 300,000,000 people, an RMP of 1 in 1,000,000 implies roughly 300 people would match, so the evidence supposedly proves nothing. This misuses the base rate in the opposite direction, by pretending the prior suspect pool is the entire population when other case evidence has already excluded almost all of it. Both fallacies are failures to handle the prior honestly; one sets it implicitly at 1:1, the other at 1:300,000,000. ## How the statistic should be stated The defensible statement keeps the conditioning where the measurement was made: *if this person were not the source, the chance their profile would match is about one in a million*. Better still, report the likelihood ratio -- 'this evidence is about a million times more likely if the defendant is the source than if he is not' -- and leave the prior explicitly to the trier of fact, who has the rest of the case. Any sentence of the form 'there is a one-in-a-million chance he is innocent' has crossed into the fallacy. Two further honesty requirements: the RMP is a population-genetics estimate that can be badly off for close relatives or for a subpopulation with different allele frequencies, and it excludes non-coincidental sources of a match such as sample mix-up, contamination or transfer. Those laboratory error rates are frequently larger than the RMP itself, which means the realistic likelihood ratio is capped well below 1,000,000 no matter what the allele arithmetic says. ## How to answer in the room Name the error as a transposed conditional, write both conditionals side by side, run the odds calculation to show the size of the gap, and then make the structural point: a cold database search multiplies the number of comparisons, which raises the chance of a coincidental hit and lowers the prior for any individual matched entry. Finish by stating the statistic correctly.