Posterior probability for a unique source after a positive test ($P(G\mid +)=[1+(N-1)p{]}^{-1}$)

Suppose exactly one of $N$ equally likely candidates is the source of an observation. A test is always positive for that source and is positive with probability $p$ for any nonsource candidate. For one randomly selected candidate, Bayes' theorem gives $P(G\mid +)=(1/N)/[(1/N)+p(1-1/N)]$, proving the displayed result. Even a small false positive rate can leave a substantial chance of a different source when there are many alternatives. The model concerns one selected candidate with the stated prior; searching many candidates and selecting a match requires accounting for that selection procedure.