  The main purpose of this paper is to provide threshold functions for the
events that a random subset of the points of a finite vector space has certain
properties related to point-flat incidences. Specifically, we consider the
events that there is an $\ell$-rich $m$-flat with regard to a random set of
points in $\mathbb{F}_q^n$, the event that a random set of points is an
$m$-blocking set, and the event that there is an incidence between a random set
of points and a random set of $m$-flats. One of our key ingredients is a
stronger version of a recent result obtained by Chen and Greenhill (2021).
