3. A ball is in any one of *n* boxes. It is in the *i*th box with probability *P_i*. If the ball is in box *i*, a search of that box will uncover it with probability *α_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

$$P_j$$

$${1 - α_iP_i}$$ if *j ≠ i*

$$(1 - α_i)P_i$$ if *j* = *i*

$${1 - α_iP_i}$$