# Very bothersome probability problem. Multi-winner raffles.

• Nov 15th 2012, 01:28 PM
RBowman
Very bothersome probability problem. Multi-winner raffles.
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?
• Nov 15th 2012, 02:31 PM
Plato
Re: Very bothersome probability problem. Multi-winner raffles.
Quote:

Originally Posted by RBowman
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.
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?

Your correct, this is a beast.
The hard part is constructing a model.
Just on your simple example there are twelve elementary events. Each with a different probability. I have absolutely no idea how to generalize that model.
• Nov 15th 2012, 03:26 PM
RBowman
Re: Very bothersome probability problem. Multi-winner raffles.
Quote:

Originally Posted by Plato
Your correct, this is a beast.
The hard part is constructing a model.
Just on your simple example there are eleven elementary events. Each with a different probability. I have absolutely no idea how to generalize that model.

Yes, I think beast describes this well. I've been working on it in spurts, but I suspect that it may be very difficult, especially in the general case. I'll post some of my partial analyses soon. Thanks for any help you can provide.
• Nov 15th 2012, 03:59 PM
Plato
Re: Very bothersome probability problem. Multi-winner raffles.
Quote:

Originally Posted by RBowman
Yes, I think beast describes this well. I've been working on it in spurts, but I suspect that it may be very difficult, especially in the general case. I'll post some of my partial analyses soon. Thanks for any help you can provide.

Now that I know that you are serious, I will give you what I have found.
In your simple example the events are:
$\displaystyle (A,B),~(A,C),~(A,A,B),~(A,A,C),~(A,A,A,B),~(A,A,A, C),$
$\displaystyle ~(B,A),~(B,C),~(B,B,A),~(B,B,C),~(C,A),~(C,B)$

Note that I have used order in this model.

Also, $\displaystyle \mathcal{P}(A,A,B)=\frac{3}{6}\cdot\frac{2}{5} \cdot\frac{2}{4}$
• Nov 15th 2012, 04:18 PM
RBowman
Re: Very bothersome probability problem. Multi-winner raffles.
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.
• Apr 16th 2013, 05:33 AM
RBowman
Re: Very bothersome probability problem. Multi-winner raffles.
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.
• Apr 23rd 2013, 05:24 PM
RBowman
Re: Very bothersome probability problem. Multi-winner raffles.
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