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Probability problem of pairing up siblings

  1. Apr 3, 2013 #1
    1. The problem statement, all variables and given/known data

    There are 10 sister/brother pairs, (S1,B1) .... , (S10,B10). We then divide the 20 people randomly into 10 (not necessarily female/male) pairs.

    Let X be the number of female/male pairs. Find E(X)
    Let Y be the number of pairs consisting of siblings (that is, sister got matched with her brother). Find E(Y)


    2. Relevant equations



    3. The attempt at a solution

    I only know how to find out the total number of ways to select 10 pairs, that's (20,2)+(18,2)+(16,2)+.....+(2,2). (x,y) means x choose y

    Please help. Thanks!!
     
  2. jcsd
  3. Apr 4, 2013 #2
    Firstly notice whether you can have odd number of bro - sis pairs. What would be the probability of odd x then?

    Now going to even x. For example lets take x = 2. How many ways of selecting two bro - sis pairs exist? Clearly to make a pair you need to choose 1 bro and 1 sis out of 10 each. For the next pair, 1 bro and i sis out of 9 each. Then you just need to pair up 8 boys and 8 girls into boy-boy and girl-girl pairs.

    Calculate that and then divide by the total number of ways of pairing the 20 siblings. This will give you p(x=2). You can similarly find p(x) for other even x.
     
  4. Apr 4, 2013 #3
    It's often much easier to find the expectation of a random variable than all the probabilities,
    because E(X+Y) = E(X) + E(Y) always.

    suppose [itex] X_i [/itex] is 1 if sibling i is member of a pair, and 0 otherwise, what is [itex] E(X_i) [/itex] ?

    Now find the expected value of the total of pair-memberships, and find the number of pairs from that.

    The second problem can be done in the same way.
     
  5. Apr 4, 2013 #4
    When you say Xi is 1 if sibling is member of a pair, do you mean a pair of opposite sex? And does it mean both bro and sis in ith pair got assigned to another pair of opposite sex?
     
  6. Apr 4, 2013 #5

    Ray Vickson

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    For i = 1,2,..., 10, let ##I_i = 1## if sister S_i is paired with a boy, and ##I_i = 0## otherwise; and let ##J_i = 1## if sister S_i is paired with her brother, and let ##J_i = 0## otherwise. Try to express X and Y in terms of the ##I_i## and/or ##J_i##.
     
  7. Apr 4, 2013 #6
    sorry, I left something out,, I meant if the sibling is paired with someone of the opposite sex.
     
  8. Apr 5, 2013 #7
    You should think the other way to solve this question, first consider choosing 10 persons, calculate each case from 1 male to 10, then for each case, the possibilities to have siblings or couples pair. Using excel sheet can easily get the answer.
     
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