Bayes update on a performer's home city

Question

Before a festival, organisers believe a rapper is in Philadelphia with probability $0.8$ and in Baltimore with probability $0.2$ . If he is in Baltimore he performs with probability $0.8$ ; if he is in Philadelphia he performs with probability $0.1$ . Given that he performs, what is the probability he was in Philadelphia?

Short answer

$1/3$ . By Bayes' rule, the joint probability of Philadelphia and a performance is $0.08$ , while Baltimore and a performance has $0.16$ ; so the posterior for Philadelphia is $\tfrac{0.08}{0.08+0.16}=\tfrac13$ .

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