  Given a graph $G$ and a real number $0\le p\le 1$, we define the random set
$B_p(G)\subset V(G)$ by including each vertex independently and with
probability $p$. We investigate the probability that the random set $B_p(G)$ is
a zero forcing set of $G$. In particular, we prove that for large $n$, this
probability for trees is upper bounded by the corresponding probability for a
path graph. Given a minimum degree condition, we also prove a conjecture of
Boyer et.\ al.\ regarding the number of zero forcing sets of a given size that
a graph can have.
