# Probability of obtaining x balls when drawing y bags from a set

I need help determining what statistical test is appropriate for my analysis.

• We have a set of $N$ objects, say a bag.
• From that set we draw $n$ bags at random.
• Each bag may contain $0 \text{ to } m$ smaller objects within itself, for example a ball.
• There are $M$ total small objects distributed unevenly in the $N$ bags.

What is the probability of drawing $x$ balls at random when we draw $n$ bags?

My initial guess was that I need to somehow use a multivariate hypergeometric distribution, where there are several categories of "bags" (i.e. they can either contain $0,1,2,3$ balls, etc.).

One additional clarification: the number of balls a bag can contain is very small in comparison to the number of bags in the set.

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Some questions: (1) "we draw $n$ bags at random" v. "when we draw $y$ bags"? (2) How are the $M$ small objects distributed between the $N$ bags? – Henry Jan 21 '13 at 15:46
@Henry (1) Thanks for pointing out the mistake. $y$ is actually $n$. I'll correct the question. (2) Most bags contain 0 balls. A small number of bags contain 1-3 balls. I can get a histogram of the number of balls per bag (since I'm working with an actual dataset), but I don't know how to evaluate what distribution $m$ follows. – Drosophila Jan 21 '13 at 15:52