Results 1 to 6 of 6

Math Help - Socks in a drawer conundrum

  1. #1
    Member
    Joined
    Mar 2009
    Posts
    182
    Thanks
    1

    Socks in a drawer conundrum

    A drawer contains 8 Red socks, 4 green socks and 4 yellow socks. Two socks
    are chosen randomly from the drawer, without replacement. Let R be the number of red socks chosen and Y be the number of yellow socks chosen.

    (a) Derive the joint probability mass function of R and Y , i.e. P(R = r, Y = y).
    (b) Calculate the marginal probability distributions of R and Y , i.e. P(R = r) and P(Y = y), respectively.
    (c) Are R and Y independent? Justify your answer.
    (d) What is the probability that a pair of the same colour is chosen?
    (e) Obtain the conditional distribution of R|Y = 1.
    (f) Evaluate E[R], Var(R) and Cov(R, Y).
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Jul 2009
    Posts
    593
    Thanks
    4
    A. Your PMF is not a function perse, in the sense you are thinking (perhaps) of it as. It is simply a chart with R and Y being as you say. To set this up you need to know the possible values of R and Y. You are told that you are choosing two socks without replacement, and so your values of R and Y can range from 0 to 2 (do you see why?)

    You then set up a grid, but how you set up the grid is the important part. In this scenario, both the R and the Y are columns and they can take on values of 0, 1, 2 since we are selecting two socks. P(R=0) would be the probability you select no red socks, which means the probability that on the first and second draw you get a green or a yellow - (12/16)(11/16). P(R=1) means you got a red on the first or second draw - (8/16)(12/15)+(12/16)(8/15). And so on and so forth.

    Once you set up your probability mass function, figuring out the rest should be easy (assuming you know how to figure out the rest).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    This is a hypergeo.

    P(R=r,Y=y)={{8\choose r}{4\choose 2-r-y}{4\choose y}\over {16\choose 2}}

    P(R=r)={{8\choose r}{8\choose 2-r}\over {16\choose 2}}

    P(Y=y)={{12\choose 2-y}{4\choose y}\over {16\choose 2}}

    I doubt that they are independent. Just show P(R=1,Y=1)\ne P(R=1)P(Y=1)
    I picked (1,1), try any spot you wish and hope they are not equal.
    Last edited by matheagle; October 26th 2009 at 10:04 AM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Jul 2009
    Posts
    593
    Thanks
    4
    Isn't a hypergeometric distribution only used for when we can classify each trial as a success or a failure? Wouldn't the above denote three possible outcomes?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    If you have a fixed chance of success and failure, that's p and q from a binomial OR multinomial.
    This is just a multi-hypergeometric.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Super Member
    Joined
    Jul 2009
    Posts
    593
    Thanks
    4
    Apologies, I was referring to success being we draw one item, and failure being we draw the other. The words I were using were not correct. I meant, isn't a hyper-geometric used when we have only two classes of objects.

    In any case I understand where you're coming from with it being a multiple hypergeometric distribution. I assumed that based on the questions being asked, perhaps the instructor didn't expect their students to use that particular method.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] socks
    Posted in the Statistics Forum
    Replies: 2
    Last Post: April 27th 2011, 09:24 AM
  2. socks in a drawer probability
    Posted in the Statistics Forum
    Replies: 4
    Last Post: September 22nd 2010, 10:15 AM
  3. Socks and probability
    Posted in the Statistics Forum
    Replies: 3
    Last Post: August 23rd 2009, 03:46 PM
  4. socks in a drawer probability problem
    Posted in the Statistics Forum
    Replies: 4
    Last Post: March 28th 2009, 10:30 AM
  5. Probability (for socks)
    Posted in the Statistics Forum
    Replies: 1
    Last Post: December 10th 2008, 08:04 AM

Search Tags


/mathhelpforum @mathhelpforum