Chapter 3, Problem 3.4TE

A ball is in any one of n boxes and is in the ith box with probability $p_i$. If the ball is in box i, a search of that box will uncover it with probability ${\alpha }_i$. Show that the conditional probability that the ball is in box j, given that a search of box i did not uncover it, is $\begin{array}{l}\frac{p_j}{1-{\alpha }_i{p}_i}\text{if }j\ne i\\ \frac{\left(1-{\alpha }_i\right)p_j}{1-{\alpha }_i{p}_i}\text{if }j=i\end{array}$