# Married Couple Probability Question [closed]

There are $$n$$ husband and wife couples at a party. If the $$n$$ men and $$n$$ women are randomly paired with one another, what is the expected number of pairings that are actual husband-wife couples?

Had this on a test earlier and cannot figure it out. Anyone have the answer?

FYI, the answer choices were 1, n/5, n/2, none of the above

Each couple has probability $$\frac1n$$ of being paired together. Thus, by linearity of expectation, the expected total number of paired couples is $$n\cdot\frac1n=1$$.

This is a job for Linearity of Expectation, everybody's favorite superhero.

Suppose I am at the party and I devise a random variable (e.g. my level of happiness) that is 1 when I am paired with my wife and 0 if I am paired with someone else. I guess you can compute the expectation of this variable, which I call $$X_1$$. Similarly we can think of the random variable $$X_2$$ that counts wether or not the second man on the party is paired with his wife and so on. We note that the variable $$Y = X_1 + X_2 + \ldots + X_n$$ is the random variable you want to compute the expectation of.

Again and again computing $$E(X_i)$$ is easy, but what we really want is to compute $$E(Y) = E(X_1 + \ldots + X_n)$$.

Now Linearity of Expectation swooshes in to save the day:

$$E(X_1 + \ldots + X_n) = E(X_1) + \ldots + E(X_n)$$

(This is true even when every $$X_i$$ has a completely different probability distribution than the others, which is clearly not the case here.)