  We present four models for a random graph and show that, in each case, the
probability that a graph is intrinsically knotted goes to one as the number of
vertices increases. We also argue that, for $k \geq 18$, most graphs of order
$k$ are intrinsically knotted and, for $k \geq 2n+9$, most of order $k$ are not
$n$-apex. We observe that $p(n) = 1/n$ is the threshold for intrinsic knotting
and linking in Gilbert's model.
