# Different balls in bins: What is the probability distribution of the sum of the minimum of the two types of balls over all bins?

Assume that there are $$N$$ different bins and two different kinds of balls, $$R$$ red balls and $$W$$ white balls.

The red balls and the white balls are randomly distributed across the bins (that is, for each ball it is equally likely to end up in each bin). Denote by $$R_i$$ and $$W_i$$ the numbers of red and white balls in bin $$i$$, for $$i=1,...,N$$.

What is the probability distribution of $$X=\sum_{i=1}^{N}{\min(R_i,W_i)}$$?

An exact expression for the expectation (which will be simpler) would also be helpful.

• I don't think there's a closed-form answer. The best I can see is that if you are willing to use normal approximations for R_i and W_i, perhaps because R and W are large, then there is a formula for the expectation using erf. – Matt F. Mar 7 at 16:11