Let M players, P = {p1, p2, ..., pM}, earn raffle tickets as a reward for completing various tasks.
For each p in P, let n(p) equal the number of tickets earned by p.
Assume tickets will be drawn randomly without replacement until K unique winners, W = {w1, w2, ..., wK}, have been drawn. Assume K < M, thus making W a proper subset of P.
Question:
Given M, K, and n(p) for each p in P, what is the probability that each player, p, gets drawn, i.e. p is in W.
I'm seeking a general solution to the problem as stated above, but I would appreciate an instructive solution to the example provided below. The example also best demonstrates the problem's appropriate interpretation.
Example:
Let P = {A, B, C}, so that M = 3. Assume K = 2, and
n(A) = 3, n(B) = 2, n(C) = 1.
Imagine the situation as 6 tickets in a 'bucket' with names attached.
'bucket' : [ A, A, A, B, B, C ]
Tickets will be drawn from this bucket w.o. replacement until K = 2 unique winners have been drawn. These two players will comprise W, the set of winners.
What is the probability that A is in W? that B is in W? that C is in W?
Plato, I agree with what you've posted. It shows you understand the problem statement, as intended. This is but a small piece though, as I'm sure you know. You aren't far from a solution to the simple example, but I think you know that isn't the hard part. My work on this problem is not yet ready to put into a post. (I'm still getting used to posting mathematical work on a bulletin board. My work is all in pen on paper.) I'll be checking back often to see your progress and that of anyone who joins in the fun.
Any help on this would be greatly appreciated. And if anyone knows the relevant branch of mathematics this type of problem would fall under, that may point toward a solution to these that I have yet to find.
I've found the coupon collector problem (see below), and I think it may be relevant to the problem in the OP.
Coupon collector's problem - Wikipedia, the free encyclopedia