  We describe a probability distribution on isomorphism classes of principally
quasi-polarized p-divisible groups over a finite field k of characteristic p
which can reasonably be thought of as "uniform distribution," and we compute
the distribution of various statistics (p-corank, a-number, etc.) of
p-divisible groups drawn from this distribution. It is then natural to ask to
what extent the p-divisible groups attached to a randomly chosen hyperelliptic
curve (resp. curve, resp. abelian variety) over k are uniformly distributed in
this sense. For instance, one can ask whether the proportion of genus-g curves
over F_p whose Jacobian is ordinary approaches the limit that such a heuristic
would predict. This heuristic is analogous to conjectures of Cohen-Lenstra type
for fields k of characteristic other than p, in which case the random
p-divisible group is defined by a random matrix recording the action of
Frobenius. Extensive numerical investigation reveals some cases of agreement
with the heuristic and some interesting discrepancies. For example, plane
curves over F_3 appear substantially less likely to be ordinary than
hyperelliptic curves over F_3.
