= Solution
Let $G$ mean that the tested person is guilty and $M$ mean a positive match. Under the specified model, $P(G)=1/N$, $P(M\mid G)=1$ and $P(M\mid G^c)=p$. <Bayes' theorem> gives
$$
\boxed{P(G\mid M)=\frac{(1/N)}{(1/N)+p(1-1/N)}=\frac1{1+(N-1)p}.}
$$
The <posterior probability for a unique source after a positive test> depends on the number of alternative possible sources as well as the false-positive probability. In particular, the false-positive probability $p$ is not the probability of innocence conditional on a positive match.
Back to article page
