Consider an island with n + 2 inhabitants. One of the inhabitants is murdered. The police know for sure that one of the remaining n + 1 inhabitants must be the killer and forensic experts find a particular DNA profile at the crime scene. It is known that this particular DNA profile occurs in a fraction p of the population, i.e., every inhabitant has probability p of having this particular DNA profile, independently of the other inhabitants. The police decides to screen all inhabitants for the DNA profile. The first person they screen is Rick Random and he turns out to have the DNA profile.
Let E be the event of finding Rick Random’s particular DNA profile at the crime scene.
Let R be the event that Rick Random is the killer
Given $\mathbb P (E|R) = 1$. Find $\mathbb P (E|R^C)$
My first attempt is $\mathbb P (E|R^c) = \frac{(n+2)p-2}{n}$ however I'm not sure.
Notice: " every inhabitant has probability $p$ of having this particular DNA profile, independently of the other inhabitants. "
So what is the probability that the killer had the profile under the condition that Rick Random is not the killer?
(PS: $\mathsf P(E\mid R)=1$ indicates that the police are assuming that the evidence was certainly left by the killer, not possibly by an uninvolved islander.)