A crime has been committed in a town of 100,000 inhabitants. The police are looking for a single perpetrator, believed to live in the town. DNA evidence is found on the crime science. Kevin's DNA matches the DNA recovered from the crime scene. According to DNA experts, the probability that a random person's DNA matches the crime scene DNA is 1 in 10,000. Before the DNA evidence, Kevin was no more likely to be the guilty person than any other person in town. What is the probability that Kevin is guilty after the DNA evidence appeared?
My answer:
Kevin is no more likely to be the perpetrator than any one else in the town. I AM assuming that they checked the DNA evidence against every town inhabitant (otherwise, they thought Kevin was more likely than any other town inhabitant to be the perpetrator and only tested it against his, and maybe against that of a few others). So, assuming 1 in 10,000 have the 100,000 came back positive, Kevin is one among ten inhabitants whose DNA matched with that found on the crime scene. Therefore, there is a $\frac{1}{10}$ probability that Kevin is the criminal.
Solution given:
Best Answer
There is some ambiguity about whether the "1 in 10,000" fraction refers to any random person or to any random innocent person. The given solution is assuming the latter (notice it says $P(A\mid G^c)=\frac1{10000}$). If you also assume the latter, then your argument will agree with the solution: the guilty person will test positive, and so will $\frac1{10,000}$ of the remaining 99,999 townsfolk, meaning 10.9999 people in total will come back positive. Kevin is one person in that total.