  For positive integers $n$ and $k$ with $n\geq 2k+1$, the Kneser graph
$K(n,k)$ is the graph with vertex set consisting of all $k$-sets of
$\{1,\dots,n\}$, where two $k$-sets are adjacent exactly when they are
disjoint. The independent sets of $K(n,k)$ are $k$-uniform intersecting
families, and hence the maximum size independent sets are given by the
Erd\H{o}s-Ko-Rado Theorem. Let $K_p(n,k)$ be a random spanning subgraph of
$K(n,k)$ where each edge is included independently with probability $p$.
Bollob\'as, Narayanan, and Raigorodskii asked for what $p$ does $K_p(n,k)$ have
the same independence number as $K(n,k)$ with high probability. For $n=2k+1$,
we prove a hitting time result, which gives a sharp threshold for this problem
at $p=3/4$. Additionally, completing work of Das and Tran and work of Devlin
and Kahn, we determine a sharp threshold function for all $n>2k+1$.
