  For a polynomial $P_n$ of degree $n$, Bernstein's inequality states that
$\|P_n'\| \le n \|P_n\|$ for all $L^p$ norms on the unit circle,
$0<p\le\infty,$ with equality for $P_n(z)= c z^n.$ We study this inequality for
random polynomials, and show that the expected (average) and almost sure value
of $\Vert P_n' \Vert/\Vert P_n\Vert$ is often different from the classical
deterministic upper bound $n$. In particular, for circles of radii less than
one, the ratio $\Vert P_n' \Vert/\Vert P_n\Vert$ is almost surely bounded as
$n$ tends to infinity, and its expected value is uniformly bounded for all
degrees under mild assumptions on the random coefficients. For norms on the
unit circle, Borwein and Lockhart mentioned that the asymptotic value of $\Vert
P_n' \Vert/\Vert P_n\Vert$ in probability is $n/\sqrt{3},$ and we strengthen
this to almost sure limit for $p=2.$ If the radius $R$ of the circle is larger
than one, then the asymptotic value of $\Vert P_n' \Vert/\Vert P_n\Vert$ in
probability is $n/R$, matching the sharp upper bound for the deterministic
case. We also obtain bounds for the case $p=\infty$ on the unit circle.
