Greetings,

I have a problem similar to the following:

Suppose at some call center, there are a lot of people available to take calls. They are grouped according to their average success rate. There are $\displaystyle m$ groups of call takers $\displaystyle A_1,...,A_m$ with $\displaystyle a_1,...,a_m$ people in each respectively. Each group has a different average success rate. If each person takes a call at exactly the same time, what is the probability that they will succeed with exactly $\displaystyle k$ callers (across the entire call center)?

My approach is to sum over all possible solutions to a diophantine equation in $\displaystyle m$ variables where the sum must equal $\displaystyle k$. Hence I have:

$\displaystyle \displaymode \sum_{b_1+...+b_m=k} \left( \prod_{i=1}^m {\left( \binom{a_i}{b_i} \text{Pr}(A_i)^{b_i} (1-\text{Pr}(A_i))^{a_i-b_i} \right)} \right)$

Any ideas for another approach that might be solvable in polynomial time? Or possibly a book I might take a look at that might offer some clue?

Edit: I am not a statistician. I have some experience with combinatorics, and I would prefer a combinatorial approach, but in this instance, it seems like statistics may be a more appropriate field to estimate the results. I really don't know enough about statistics to know where to begin looking, though.