Results 1 to 7 of 7
Like Tree1Thanks
  • 1 Post By Plato

Math Help - Very bothersome probability problem. Multi-winner raffles.

  1. #1
    Junior Member
    Joined
    Nov 2012
    From
    Jacksonville, FL
    Posts
    30
    Thanks
    4

    Question 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?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1573
    Awards
    1

    Re: Very bothersome probability problem. Multi-winner raffles.

    Quote Originally Posted by RBowman View Post
    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.
    Last edited by Plato; November 15th 2012 at 03:39 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Nov 2012
    From
    Jacksonville, FL
    Posts
    30
    Thanks
    4

    Re: Very bothersome probability problem. Multi-winner raffles.

    Quote Originally Posted by Plato View Post
    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.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1573
    Awards
    1

    Re: Very bothersome probability problem. Multi-winner raffles.

    Quote Originally Posted by RBowman View Post
    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:
    (A,B),~(A,C),~(A,A,B),~(A,A,C),~(A,A,A,B),~(A,A,A,  C),
    ~(B,A),~(B,C),~(B,B,A),~(B,B,C),~(C,A),~(C,B)

    Note that I have used order in this model.

    Also, \mathcal{P}(A,A,B)=\frac{3}{6}\cdot\frac{2}{5} \cdot\frac{2}{4}
    Thanks from topsquark
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Nov 2012
    From
    Jacksonville, FL
    Posts
    30
    Thanks
    4

    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.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member
    Joined
    Nov 2012
    From
    Jacksonville, FL
    Posts
    30
    Thanks
    4

    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.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Nov 2012
    From
    Jacksonville, FL
    Posts
    30
    Thanks
    4

    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
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. 20.000 Bingo Cards... Just 1 winner (cheating??)
    Posted in the Statistics Forum
    Replies: 2
    Last Post: March 25th 2011, 01:55 PM
  2. Replies: 2
    Last Post: March 3rd 2010, 01:25 PM
  3. Multi variable problem.
    Posted in the Calculus Forum
    Replies: 4
    Last Post: January 28th 2009, 03:02 AM
  4. Help with finding an Average Rating Winner
    Posted in the Math Software Forum
    Replies: 1
    Last Post: October 7th 2008, 01:58 PM
  5. Limit Problem (Multi)
    Posted in the Calculus Forum
    Replies: 2
    Last Post: February 25th 2007, 07:41 PM

Search Tags


/mathhelpforum @mathhelpforum