# Number of urns containing a ball of each color: is there a probability distribution describing this?

There are $B$ urns. There are $n$ red balls and $n$ white balls with $n\leq B$. Each ball is independently put into each urn with equal probability. An urn can get at most one ball with the same color but can get two balls of different colors. I want to know the probability distribution of the number of urns which contain the balls with different color.

Is there any existing distribution describing this? If not, do you have your own answer?

• What is the role of $k$ there? Did you mean $B$? – Clement C. Jul 9 '16 at 21:28
• It should be $B$ because we cannot distribute more than $B$ balls among $B$ urns without having an urn with two balls of equal colors. – Peter Jul 9 '16 at 21:40
• @Peter You don't seem to have understood my comment. Whether you call it $B$ or $k$, there probably should be only one extra parameter (besides $n$ -- otherwise, one of them at least serves no apparent purpose. – Clement C. Jul 9 '16 at 21:42
• What I meant is : Instead of "less than k" it should be "less than "B". Is that what you meant with your comment ? – Peter Jul 9 '16 at 21:45
• Sorry, should be B. I changed it. – user353115 Jul 9 '16 at 21:55

We get two random choices $n$ out of $B$. The number of hits (number occurs in both choices) can be calculated with the hypergeometric distribution like in the classical lotto-problem.