  Given a large graph $H$, does the binomial random graph $G(n,p)$ contain a
copy of $H$ as an induced subgraph with high probability? This classical
question has been studied extensively for various graphs $H$, going back to the
study of the independence number of $G(n,p)$ by Erd\H{o}s and Bollob\'as, and
Matula in 1976. In this paper we prove an asymptotically best possible result
for induced matchings by showing that if $C/n\le p \le 0.99$ for some large
constant $C$, then $G(n,p)$ contains an induced matching of order approximately
$2\log_q(np)$, where $q= \frac{1}{1-p}$.
