  A set $U\subseteq \reals^2$ is $n$-universal if all $n$-vertex planar graphs
have a planar straight-line embedding into $U$. We prove that if $Q \subseteq
\reals^2$ consists of points chosen randomly and uniformly from the unit square
then $Q$ must have cardinality $\Omega(n^2)$ in order to be $n$-universal with
high probability. This shows that the probabilistic method, at least in its
basic form, cannot be used to establish an $o(n^2)$ upper bound on universal
sets.
