 Solution

The classical transmission probability is $P_{\mathrm{cl}}(E)=\Theta (E-V_0)$; the convention exactly at the threshold does not affect a thermal integral. Below the barrier it is zero, whereas the quantum result permits tunnelling. Above the barrier it is one, whereas quantum reflection is generally nonzero except at transmission resonances. Both descriptions tend to unit transmission at arbitrarily high energy, and both have zero transmitted probability in the zero-energy limit for this positive barrier.