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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

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

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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.)

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