I'm working on a variant of the coat check problem (each person checks their coat, and at the end of the night each person is handed a random coat) in which each person has a friend who has an identical coat, so in total they are $2n$ people and $2$ coats that they consider 'their coat'.
It seems fairly straight forward, except for one thing - when I'm trying to calculate the variance, (given by $\sum\sum \mathrm{Var}(X_i,X_j)-\mathrm{Var}(X_i)\mathrm{Var}(X_j)$, where $i\neq j$ and $X$ is an indicator variable), I'm calculating $\mathrm{Var}(X_i,X_j)$ as $\frac{2}{2n} \cdot \frac{2}{2n-1}$... What i'm not sure of is if the $E(X_j)=\frac{2}{2n-1}$ is correct; since it does not account for the chance that $X_i$ is the other person with $X_j$'s coat.
Any thoughts?
Thanks!