  Probabilistic zero forcing is a coloring game played on a graph where the
goal is to color every vertex blue starting with an initial blue vertex set. As
long as the graph is connected, if at least one vertex is blue then eventually
all of the vertices will be colored blue. The most studied parameter in
probabilistic zero forcing is the expected propagation time starting from a
given vertex of $G.$ In this paper we improve on upper bounds for the expected
propagation time by Geneson and Hogben and Chan et al. in terms of a graph's
order and radius. In particular, for a connected graph $G$ of order $n$ and
radius $r,$ we prove the bound $\text{ept}(G) = O(r\log(n/r)).$ We also show
using Doob's Optional Stopping Theorem and a combinatorial object known as a
cornerstone that $\text{ept}(G) \le n/2 + O(\log n).$ Finally, we derive an
explicit lower bound $\text{ept}(G)\ge \log_2 \log_2 n.$
