  We consider the energy-critical defocusing nonlinear wave equation on
$\mathbb{R}^4$ and establish almost sure global existence and scattering for
randomized radially symmetric initial data in $H^s_x(\mathbb{R}^4) \times
H^{s-1}_x(\mathbb{R}^4)$ for $\frac{1}{2} < s < 1$. This is the first almost
sure scattering result for an energy-critical dispersive or hyperbolic equation
with scaling super-critical initial data. The proof is based on the
introduction of an approximate Morawetz estimate to the random data setting and
new large deviation estimates for the free wave evolution of randomized
radially symmetric data.
