We propose efficient numerical algorithms for approximating statistical
solutions of scalar conservation laws. The proposed algorithms combine finite
volume spatio-temporal approximations with Monte Carlo and multi-level Monte
Carlo discretizations of the probability space. Both sets of methods are proved
to converge to the entropy statistical solution. We also prove that there is a
considerable gain in efficiency resulting from the multi-level Monte Carlo
method over the standard Monte Carlo method. Numerical experiments illustrating
the ability of both methods to accurately compute multi-point statistical
quantities of interest are also presented
