  A path in an edge-colored graph is called a proper path if no two adjacent
edges of the path are colored the same. For a connected graph $G$, the proper
connection number $pc(G)$ of $G$ is defined as the minimum number of colors
needed to color its edges, so that every pair of distinct vertices of $G$ is
connected by at least one proper path in $G$. In this paper, we show that
almost all graphs have the proper connection number 2. More precisely, let
$G(n,p)$ denote the Erd\"{o}s-R\'{e}nyi random graph model, in which each of
the $\binom{n}{2}$ pairs of vertices appears as an edge with probability $p$
independent from other pairs. We prove that for sufficiently large $n$,
$pc(G(n,p))\le2$ if $p\ge\frac{\log n +\alpha(n)}{n}$, where
$\alpha(n)\rightarrow \infty$.
