Probability problem of pairing up siblings

[ilvpat]

Homework Statement

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)

Homework Equations

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!

 

[Sunil Simha]

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 let's 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.

 

[willem2]

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 X_i is 1 if sibling i is member of a pair, and 0 otherwise, what is E(X_i) ?

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.

 

[ilvpat]

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 X_i is 1 if sibling i is member of a pair, and 0 otherwise, what is E(X_i) ?

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.

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?

 

[Ray Vickson]

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?

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 $J_i$.

 

[willem2]

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?

sorry, I left something out,, I meant if the sibling is paired with someone of the opposite sex.

 

[csleong]

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.