  A family of sets is intersecting if no two of its members are disjoint, and
has the Erd\H{o}s-Ko-Rado property (or is EKR) if each of its largest
intersecting subfamilies has nonempty intersection.
  Denote by $\mathcal{H}_k(n,p)$ the random family in which each $k$-subset of
$\{1\dots n\}$ is present with probability $p$, independent of other choices. A
question first studied by Balogh, Bohman and Mubayi asks: \[ \mbox{for what
$p=p(n,k)$ is $\mathcal{H}_k(n,p)$ likely to be EKR?} \] Here, for fixed
$c<1/4$, and $k< \sqrt{cn\log n}$ we give a precise answer to this question,
characterizing those sequences $p=p(n,k)$ for which $$ \Pr(\mathcal{H}_k(n,p)
\textrm{ is EKR}) \rightarrow 1 \textrm{ as } n\rightarrow \infty. $$
