  We consider the number of crossings in a random embedding of a graph, $G$,
with vertices in convex position. We give explicit formulas for the mean and
variance of the number of crossings as a function of various subgraph counts of
$G$. Using Stein's method and size-bias coupling, we prove an upper bound on
the Kolmogorov distance between the distribution of the number of crossings and
a standard normal random variable. As an application, we establish central
limit theorems, along with convergence rates, for the number of crossings in
random matchings, path graphs, cycle graphs, and the disjoint union of
triangles.
