  A curve over a field of characteristic $p$ is called ordinary if the
$p$-torsion of its Jacobian as large as possible, that is, an $\mathbb{F}_p$
vector space of dimension equal to its genus. In this paper we consider the
following question: fix a finite field $\mathbb{F}_q$ and a family
$\mathscr{F}$ of curves over $\mathbb{F}_q$. Then, what is the probability that
a curve in this family is ordinary? We answer this question when $\mathscr{F}$
is either the Artin-Schreier family in any characteristic or a superelliptic
family in characteristic 2.
