Suppose that a house is robbed and police find a strand of the burglar’s hair at the scene of the crime. A suspect is in custody, and tests show that the strand matches his hair. A forensic scientist testifies that the chance of a random person producing such a match is 1/2000. Does this mean that there’s a 1999/2000 chance that the suspect is guilty?

No, it doesn’t. In a city of 5 million there will be 1/2000 × 5,000,000 = 2,500 people who produce a match, so on the basis of this evidence alone the probability that the suspect is guilty is only 1/2500.

In a 1987 article, William Thomson and Edward Schumann dubbed this “prosecutor’s fallacy.” Unfortunately, it’s matched by the “defense attorney’s fallacy,” which holds that the hair-match evidence is worthless because it increases the likelihood of the suspect’s guilt by a negligible amount, 1/2500. In fact it drastically narrows the range of possible suspects, from 5 million to 2,500, while failing to exclude the defendant, hardly cause for confidence.

Worryingly, Thompson and Schumann found an experienced prosecutor who insisted that if a defendant and a perpetrator match on a blood type found in 10 percent of the population, then there’s a 10 percent chance that the defendant would have this blood type if he were innocent and hence a 90 percent chance that he’s guilty. “If a prosecutor falls victim to this error,” they write, “it is possible that jurors do as well.”

(William C. Thompson and Edward L. Schumann, “Interpretation of Statistical Evidence in Criminal Trials,” *Law and Human Behavior*, 11:3 [September 1987], 167-187)