Take a bounded domain $S$ in which an explosive device is located. A team is deployed to find and disable the device before time $t^{*}$, when it will explode.
There are certain constraints in place. The domain $S$ is partitioned into subdomains $i$, with $i \in {1, . . . . , n}$. Each subdomain can be searched and re-searched in any order, but each search must last for the same length of time $t^{*} /t$, for some positive integer $t$. A priori before the search is began, there is a probability distribution assigning a probability $p_{i}$ to the event of explosive device being in subdomain $i$. Given that the device is in $i$, the probability of finding the device there is $q_{i}\in [0, 1]$, and these conditional probabilities may vary from one subdomain to another and are independent of each other across subdomains.
How to design the sequence of subdomain searches to get the maximum probability of finding the device by time $t^{*}$?
My attempt at a solution. The order of searches does not matter. Suppose that subdomain $1$ is searched for all time. Fix some $t$. Then the number of searches in total is $k = t^{*}/t$ (where we take $k$ to be the floor function if $t^{*}/t$ is not an integer). Let $A_{1}$ be the event that device is found in $1$. Let $B_{1}$ be event that the device is in $1$. Then
$$P(A_{1}) = P(A_{1}|B_{1})P(B_{1}) = \sum_{l=1}^{k} (1-q_{1})^{l-1} q_{1} p_{1}.$$
And at this point, I don't know how to proceed at all. This looks like a "knapsack problem" to me, but I have no idea what to do. Further, I am not sure if my expression for probability so far is correct.
To clarify, I am not sure if the probabilities $p_{i}$ are always fixed, or must they change depending on the outcome of the previous subdomain search. For example, suppose $q_{i}=1$ for some $i$, but the search there does not reveal the device - then surely $p_{i}$ must change?
