  The asymptotic behavior of the Jaccard index in $G(n,p)$, the classical
Erd\"{o}s-R\'{e}nyi random graphs model, is studied in this paper, as $n$ goes
to infinity. We first derive the asymptotic distribution of the Jaccard index
of any pair of distinct vertices, as well as the first two moments of this
index. Then the average of the Jaccard indices over all vertex pairs in
$G(n,p)$ is shown to be asymptotically normal under an additional mild
condition that $np\to\infty$ and $n^2(1-p)\to\infty$.
