A mail can be sent with equal probability to three mailboxes. If you check mailbox $i$ you find the mail with probability $0<p_i<1$ if the mail was indeed sent to this mailbox. (Imagine you did not read all mails but only did a quick carelss scan ...)
Let's assume you have checked mailbox $1$ and did not find the mail, what is the probability that the mail has been sent to mailbox $1$ although you have checked mailbox $1$?
Let be $M_i$ the event that the mail was sent to mailbox $i$,
$S_i$ the event that you have checked mailbox $i$ and found the mail and
$S_i^c$ is the event that you have checked mailbox $i$ and did not find the mail.
We are looking for the conditional probability $P(M_1\mid S_1^c)$. We know that $P(S_1\mid M_1)=p_1\iff P(S_1^c\mid M_1)=1-p_1$. Then
$$\begin{align*} &P(M_1\mid S_1^c)=\frac{P(M_1\cap S_1^c)}{P(S_1^c)}\frac{P(M_1)}{P(M_1)}=(1-p_1)\frac{\frac{1}{3}}{P(S_1^c)},\\ &P(S_1^c)=1-P(S_1)=1-(P(S_1\mid M_1)P(M_1)+P(S_1\mid M_1^c)P(M_1^c))=1-\frac{p_1}{3}-0,\\ &\implies P(M_1\mid S_1^c)=\dots=\frac{(1-p_1)}{3-p_1}. \end{align*}$$
I am not sure if I correctly translated the conditional probabilities. Is this correct?