I have the following scenario that I have to model a probability mass function to:
Suppose that there are $2m$ objects in a bag that form $m$ pairs. There is a hole in the bag, leading to $k$ objects being lost. Note that here $m, k \in \mathbb{N}$ and $0 \leq k \leq 2m$. Assuming that these decreased $k$ objects were chosen at random and with equal probabilities, what is the probability mass function for the number of pairs of objects that were not lost?
Initially, I thought that this would be binomially distributed with probability of success:
$$\frac{2m - 2 \choose k}{2m \choose k}$$
But this does not seem to work. For example, if you try to find the probability that $m$ pairs were not lost while setting $k = 1$, you would expect the probability to be zero, seeing how one pair would be guaranteed to be lost. But you would get something along the lines of
$$P(X = m) = {m \choose m}\left(\frac{2m - 2 \choose 1}{2m \choose 1} \right)^m \left(1 - \frac{2m - 2 \choose 1}{2m \choose 1}\right)^{m - m} = \left(1 - \frac{1}{m}\right)^m$$
which is not zero.
So I don't think that the problem could be modelled with the binomial distribution, but now I'm at a loss. Could anyone help me out with this problem please? Thanks!