Suppose I have a population of size $N$ made up of people of different U.S. political parties: $K_R$ Republicans, $K_D$ Democrats, $K_L$ Libertarians, $K_G$ Greens, $K_S$ Socialists and $K_I$ people with no political party (Independents), $K_R + K_D + K_L + K_G + K_S + K_I = N$
I draw a sample of size $n$. What is the probability that there are exactly $k_R$ Republicans, $k_D$ Democrats, $k_L$ Libertarians, $k_G$ Greens, $k_S$ Socialists, and at least $k_P$ people of any political party (non-Independents), $k_R + k_D + k_L + k_G + k_S + k_P \leq n$ (so there can also be some number of Independents also in the sample, or none)
This looks a lot like a multivariate hypergeometric distribution problem, except for the "and at least $k_P$ people of any political party". I'm trying to figure out how to add that to the formula. How would I figure out this probability?
Thanks for looking!
