  The $(a,b)$-eye is the graph $I_{a,b} = K_{a+b}-K_b$ obtained by deleting the
edges of a clique of size $b$ from a clique of size $a+b$. We show that for any
$a,b \ge 2$ and $p \in (0,1)$, if we condition the random graph $G \sim G(n,p)$
on having no induced copy of $I_{a,b}$, then with high probability $G$ is close
to an $a$-partite graph or the complement of a $(b-1)$-partite graph. Our proof
uses the recently developed theory of hypergraph containers, and a stability
result for an extremal problem with two weighted colours. We also apply the
stability method to obtain an exact Tur\'an-type result for this extremal
problem.
