I have found answers to similar, much simpler questions on this site before (such as here How do you calculate probability of rolling all faces of a die after n number of rolls?), but can't find anything that deals with this exact generalization. After thinking about it some, this seems headache-inducingly more complicated to solve than these simpler variants, because the combinatorics are more nested and interfere with the simpler approaches used to solve those simpler problems (e.g. the fact that you want to roll each number b times instead of just once as in the linked question means that you can't use such a simple expression for not rolling that number in c rolls).
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- $\begingroup$ Consider the random vector $(N_1,\ldots,N_a)$ having the multinomial distribution with $c$ trials and probability vector $(\frac 1a , \ldots, \frac 1a)$. Then you're looking for $\mathbb P(N_1\geq b ,\ldots ,N_a\geq b)$. I don't know how this can be computed, even numerically. $\endgroup$Gabriel Romon– Gabriel Romon2023-11-22 13:59:28 +00:00Commented Nov 22, 2023 at 13:59
- 1$\begingroup$ This is called the "double dixie cup problem". You should be able to find some references. $\endgroup$awkward– awkward2023-11-22 14:28:55 +00:00Commented Nov 22, 2023 at 14:28
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