The Prosecutor's Fallacy

Say a murderer leaves a strand of hair at the scene of the crime, and it shows that the murderer had a specific combination of genes that only 1 in 1000 people have. The person accused of the murder has that combination of genes. Does it follow that there's a 99.9% chance that he's guilty?

No! You can see this quite clearly if you use a trait that's more common. Say there's incontrovertible evidence that the murderer was left handed, and the accused is also left handed. Only 10% of people are left handed. Does it follow that there's a 90% chance that this left-handed person committed the murder. Of course not -- that would imply that there's a 90% chance that any given left handed person committed the murder, which obviously isn't true. Even though it's rare to be left handed, it's still the case that the vast majority of left-handed people didn't commit the murder. The odds that this specific left-handed person committed the murder depend on the prior odds that he was guilty -- it's not fully determined by the incidence of left-handedness in the population.

This is a very common error that people make when reasoning about statistics. I think it's a fun one to think about, and maybe you do too!