Consider the following problem: There are $N$ urns each containing different numbers of balls, colored in $k$ different colors. Refer to this picture for an example with $N=2$ and $k=3$: https://i.stack.imgur.com/8nRlY.png
First question: We draw $n$ balls without replacement from the urns with an equal probability for each ball to be drawn. For a given set of drawn balls what is the probability distribution over which urns these balls came from?
That is we want to be able to answer questions like: We draw two green and two red balls. What is the probability that one of these balls was drawn from urn 1 and three from urn 2?
Second question: Fix a total number of draws $n$. We draw $c_j$ balls with color $j$ from the urns without replacement, multiple times for each color $j \in \{1,...,k\}$ so that $\sum_{j=1}^kc_j=n$. For each draw all balls of that color have an equal probability of being drawn. Our sample space in this experiment is the number of balls drawn from every urn i.e. let's denote by a tuple $(n_1, ..., n_N)$ the case that $n_1$ balls were drawn from urn 1 etc. The parameter for the distribution is the family of tuples $(c_1, ..., c_k)$ with $\sum_{j=1}^kc_j=n$ denoting how many balls of every color were drawn. For a given tuple $(n_1, ..., n_N)$, what is the maximum likelihood estimator for the parameter $(c_1, ..., c_k)$?
That is we want to be able to answer questions like: If 4 balls were drawn from urn one and none from urn two, what are the most likely colors of the balls that were drawn?
I would appreciate any ideas on how to solve this (I'm also interested in approximate heuristic solutions).