Introduction to Probability · Problem 4.1.52

Conditional Probability — Grinstead & Snell

Mathematics — Probability · conditional-probability-search

Problem

A coin is in one of

n

boxes. The probability that it is in the

i

th box is

p_i

. If you search in the

i

th box and it is there, you find it with probability

a_i

. Show that the probability

p

that the coin is in the

j

th box, given that you have looked in the

i

th box and not found it, is

p = \begin{cases} p_j/(1 - a_i p_i), & \text{if } j \ne i, \\ (1-a_i)p_i/(1-a_i p_i), & \text{if } j = i. \end{cases}