Suppose we sample (uniformly, with replacement) $t$ times a set of $N$ items. What is the probability $x$ that the sample contains $y$ different items?
This is representative of a real-life scenario I am facing, trying to determine if a certain issue impacts the whole population, or is specific to a certain subset. For example, I have a population of 73 specimens ($N$). One issue has manifested 250 times ($t_1$) on 47 different specimens ($y_1$), while another has manifested 64 times ($t_2$) on 17 different specimens ($y_2$). Intuitively, issue 2 is much more likely to be specific to a subset of the specimens, but I would like to quantify this before trying to analyse further.
One way forward I struggled with for a while was to use the inclusion probability. There are formulas for including one item $x_k$ or two items $x_{kl}$ (below), but I couldn't generalize for $y$ items.
$$x_k = (1-\frac{1}{N})^t $$ $$x_{kl} = 2(1-\frac{1}{N})^t - (1-\frac{2}{N})^t $$
I also have a little simulation to get some taste of what the distribution would look like.
Any help or direction would be appreciated!