Given a set containing n unique members, what are the odds that all members of a subset m are sampled at least once when non-replacement samplings of size k are repeated i times? Members are replaced between samplings.
Using the binomial coefficient, we can figure out the probability that a single member of the subset is never sampled in the i samplings. With that probability and the complement rule, we can find the probability that a single member of m is sampled at-least once in the i samplings. However, I am struggling to find a way to represent the probability of sampling each of the members of m at least once.
I know that it's not the complement of the probability of sampling m raised to i because the members of m could be spread out across multiple sampling events and still be represneted.
Examples:
n = {1,2,3,4,5}, m = {1,2}, k = 3, i = 2,
"Good": ({1,3,5}, {2,4,5}) - 1 and 2 sampled (different samplings)
"Good": ({1,2,3}, {3,4,5}) - 1 and 2 sampled (same sampling)
"Bad": ({3,4,5}, {3,4,5}) - neither 1 nor 2 sampled
"Bad": ({1,3,4}, {1,4,5}) - 2 never sampled