[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

A coin is in one of [math]n[/math] boxes. The probability that it is in the [math]i[/math]th

box is [math]p_i[/math]. If you search in the [math]i[/math]th box and it is there, you find it with probability [math]a_i[/math]. Show that the probability [math]p[/math] that the coin is in the [math]j[/math]th box, given that you have looked in the [math]i[/math]th box and not found it, is

[[math]] p = \left \{ \matrix{ p_j/(1-a_ip_i),&\,\,\, \mbox{if} \,\,\, j \ne i,\cr (1 - a_i)p_i/(1 - a_ip_i),&\,\,\,\mbox{if} \,\, j = i.\cr}\right. [[/math]]