Suppose that police pick up a suspect and match his or her DNA to evidence collected at a crime scene. Suppose that the likelihood of a match, purely by chance, is only 1 in 10,000. Is this also the chance that they are innocent? It’s easy to make this leap, but you shouldn’t.
Here’s why. Suppose the city in which the person lives has 500,000 adult inhabitants. Given the 1 in 10,000 likelihood of a random DNA match, you’d expect that about 50 people in the city would have DNA that also matches the sample. So the suspect is only 1 of 50 people who could have been at the crime scene. Based on the DNA evidence only, the person is almost certainly innocent, not certainly guilty.
--Mark Buchanan, NYT, on the prosecutor's fallacy