Nonlinear conservation laws subject to random initial conditions pose fundamental problems in the evolution and interactions of shocks and rarefactions. Using a discrete set of values for the solution, we derive a hierarchy of equations in terms of the states in two different methods. This hierarchy involves the n-point function, the probability that the solution takes on various values at different positions, in terms of the (n+1)-point function. In the first approach, these equations can be closed but the resulting solutions do not persist through shock interactions. In our second approach, the n-point function is expressed in terms of the (n+1)-point functions, and remains valid through collisions of shocks.