We have an infinite $3$-ary tree, with root $R$. In coloring $C(p)$ each edge is black with probability $p$ and white with probability $1 - p$, and edges are independent.
Show that there is a $p^*$ where $0 < p^* < 1$ such that the graph with coloring $C(p)$ where $p$ is greater than $p^*$ has an infinite black binary sub-tree with probability $1$ and a graph with coloring $C(p)$ where $p$ is less than $p^*$ has an infinite black binary sub-tree with probability $0$.