  We prove that in all regular robust expanders $G$ every edge is
asymptotically equally likely contained in a uniformly chosen perfect matching
$M$. We also show that given any fixed matching or spanning regular graph $N$
in $G$, the random variable $|M\cap E(N)|$ is approximately Poisson
distributed. This in particular confirms a conjecture and a question due to
Spiro and Surya, and complements results due to Kahn and Kim who proved that in
a regular graph every vertex is asymptotically equally likely contained in a
uniformly chosen matching. Our proofs rely on the switching method and the fact
that simple random walks mix rapidly in robust expanders.
