  We show that there is a constant $K > 0$ such that for all $N, s \in \N$, $s
\le N$, the point set consisting of $N$ points chosen uniformly at random in
the $s$-dimensional unit cube $[0,1]^s$ with probability at least
$1-\exp(-\Theta(s))$ admits an axis parallel rectangle $[0,x] \subseteq
[0,1]^s$ containing $K \sqrt{sN}$ points more than expected. Consequently, the
expected star discrepancy of a random point set is of order $\sqrt{s/N}$.
