  An edge-colored graph $G$ is conflict-free connected if any two of its
vertices are connected by a path which contains a color used on exactly one of
its edges. The conflict-free connection number of a connected graph $G$,
denoted by $cfc(G)$, is the smallest number of colors needed in order to make
$G$ conflict-free connected. In this paper, we show that almost all graphs have
the conflict-free connection number 2. More precisely, let $G(n,p)$ denote the
Erd\H{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 $n$, $cfc(G(n,p))\le 2$ if
$p\ge\frac{\log n +\alpha(n)}{n}$, where $\alpha(n)\rightarrow \infty$. This
means that as soon as $G(n,p)$ becomes connected with high probability,
$cfc(G(n,p))\le 2$.
