I'm interested in understanding the probability that given a prime $p$, $p$ divides the order of the torsional part of $H^k(X,Z)$, where $X$ is a finite simplicial complex.

Lets say you have a uniform distribution over all finite simplicial complexes on $n$ vertices. Given $p > 0$ what is the probability the complex you get has $H_K(X,Z)$ or $H^K(X,Z)$ having $p$ torsion?