Binomial distribution where probability of success is dependent on another binomial distribution

Question

How does one model the Binomial distribution where the probability of success is the result of another Binomial distribution.

For example, say I make 10 coin tosses many times and record the number of heads (H). Then for each set (i) of 10 coin tosses I put Hi black marbles, and 10-Hi white marbles in a jar and make 50 draws with replacement. How would I model the distribution of the black marble draws taking into account their dependence on the previous Binomial distribution that generated their probability of success.

Answer 1

This is known as a compound distribution. Some compound distributions simplify, and can be recognized as another well-known distribution, but I don't think this binomial-binomial compound is one of those. (There is another type of binomial-binomial compound which does simplify to just a binomial, where you toss all coins, and then reflip the heads.)

I think the simplest way to handle the distribution is as a mixture of $11$ different binomial distributions parametrized by the number of heads in the initial batch, from $0$ to $10$. There are other possibilities based on recognizing this as a compound of a multinomial distribution.

Answer 2

Thanks D. Zare for the response. I'm the OP. I spent a little more time looking into this and I believe another solution is to use a beta-binomial distribution (http://en.wikipedia.org/wiki/Beta-binomial_distribution). The beta-binomial requires alpha and beta shape parameters like a beta distribution. In the context of the situation described above, alpha = beta = 5, which corresponds to the size of the first "coin toss" binomial. For small values of alpha and beta the beta-binomial distribution is more dispersed than a binomial. As alpha and beta get larger the distribution converges on the binomial.