What is the probability that a random abelian variety over F_q is ordinary? Using (semi)linear algebra, we will answer an analogue of this question, and explain how our method can be used to answer similar statistical questions about p-rank and a-number. The answers are perhaps surprising, and deviate from what one might expect via naive reasoning. Using these computations and numerical evidence, we formulate several ``Cohen-Lenstra" heuristics for the structure of the p-torsion on the Jacobian of a random hyperelliptic curve over F_q. These heuristics are the "l=p " analogue of Cohen-Lenstra in the function field setting. This is joint work with Jordan Ellenberg and David Zureick-Brown.