The Prosecutor's Fallacy

A city has 1,000,000 residents. A crime is committed by one person. The police use a forensic test that has a false positive rate of 1 in 10,000 (i.e., $P(\text{positive} \mid \text{innocent}) = 10^{-4}$) and a true positive rate of 100% ($P(\text{positive} \mid \text{guilty}) = 1$). A suspect tests positive. (a) The prosecutor argues: 'The probability of a false positive is only $10^{-4}$, so there is only a 0.01% chance this person is innocent.' What is wrong with this argument? (b) Compute the actual probability the suspect is guilty given the positive test, assuming the suspect was selected uniformly at random from the city before the test. (c) Now suppose eyewitness evidence independently narrows the suspect pool to 5 people before the forensic test is applied. Recompute the posterior probability of guilt.