Can someone help me with a basic Bayesian Search Theory question? I'm not sure how to approach these questions and after reading up about the Bayesian search theory I'm still wondering how I should approach these questions. The question is:
A ball is in one box out of $n$. For $i = 1,..,n$ the ball is in the $i$-th box with probability $p_i$. For each value of $i$, there is a probability $\alpha _i$ that a search of the $i$-th box will reveal the ball if it is actually in that box.
What is the probability that the ball is in box 1 given that a search of box 3 fails to reveal the ball? Also, what is the probability the ball is in box 1 given that a search of box 1 fails to reveal the ball?
Any advice on how to tackle these sort of questions will be appreciated.