Probability of finding a ball in a box

Question

Here is one I'm stumped on.

A ball can be in any one of $n$ boxes. It is the $i^{th}$ box with probability $p_i$. If the ball is in the $i^{th}$ box a search of that box will uncover it with probability $\alpha_i$. Given that a search of box $i$ did not uncover the ball, what is the conditional probability the ball is actually in box $j$ where $i,j=1,\ldots,n$?

I tried fooling around with the multinomial distribution but kept getting tripped up. Any thoughts?

Answer 1

• Probability ball is in box $i$ is $p_i$
• Probability ball is in box $j$ is $p_j$
• Probability ball is in box $i$ and is found in box $i$ is $a_ip_i$
• Probability ball is in box $i$ and is not found in box $i$ is $(1-a_i)p_i$
• Probability ball is not found in box $i$ is $1 - a_ip_i$
• Probability ball is in box $i$ given it is not found in box $i$ is $\dfrac{(1-a_i)p_i}{1 - a_ip_i}$
• Probability ball is in box $j \not = i$ given it is not found in box $i$ is $\dfrac{p_j}{1 - a_ip_i}$