Letter location after an unproductive mailbox check

Question

A letter is delivered to one of three mailboxes, numbered $1$ , $2$ , $3$ , each with probability $\tfrac13$ . If it is in mailbox $i$ and you check that box, you notice it with probability $\tfrac{i}{3}$ . You check mailbox $1$ and find nothing. What is the probability the letter is in fact in mailbox $1$ ?

Short answer

$1/4$ . By Bayes: the letter is in box $1$ and missed with probability $\tfrac13\cdot\tfrac23=\tfrac29$ , and it is elsewhere with probability $\tfrac23$ . Hence the posterior is $\tfrac29\div\left(\tfrac29+\tfrac69\right)=\tfrac14$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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