n white and k black balls are randomly and independently distributed amongst m boxes. There is no limit to the number of balls a box can contain.

As a result, there are four possible states for each box:

1. Empty
2. black only
3. white only
4. black and white

What is the expected fraction of boxes in each state given n, m and k?

Obviously the expected distribution of states depend on the expected frequency of boxes containing at least one black or one white ball, but I do not even know how to calculate that given that there are no restrictions on the number of balls in a box. 

I had calculated this whithout taking into account the possibility that multiple balls with the same colour would end up in the same box. In that case it would simply be:

1. 1 - α - β + α∗β
2. α − α∗β
3. β − α∗β
4. α∗β

with α = m/k and β = m/n. But this is obviously wrong if there is no limitation on the number of balls per box.