  Let $\Sigma^p_g$ denote a surface of genus $g$ and with $p$ punctures. Our
main result is that the fraction of degree $n$ covers of $\Sigma^p_g$ which
have the Putman-Wieland property tends to $1$ as $n\to \infty$. In addition, we
show that the monodromy of a random cover of $\Sigma^p_g$ is asymptotically
almost surely tangle free.
