I need to figure out a problem which involves multiple hypergeometric distributions.
Referring to the Urn problem, the problem can be described like the following:
We have $n$ urns $u_1,…,u_n$. Urn $u_i$ contains $n_i$ balls of which $n'_i$ are tagged.
Now, from urn $u_i$ we draw $m_i$ balls without replacement. Let $m'_i$ denote the number of tagged balls drawn from $u_i$. What is the probability of $(\sum_{i=1}^{n} m'_i) > k$, where $k$ is some integer $x\in\mathbb{N}$?
I know that drawing without replacement from a single urn can be modelled using a hypergeometrical distribution, but I couldn't figure out how to cleverly combine those observations. I'm actually interested in computing those probabilities afterwards.