We develop a probabilistic approach to quantify parametric uncertainty in first-order hyperbolic conservation laws (kinematic wave equations). The approach relies on the derivation of a deterministic equation for the cumulative density function (CDF) of a system state, in which probabilistic descriptions (probability density functions or PDFs) of system parameters and/or initial and boundary conditions serve as inputs. In contrast to PDF equations, which are often used in other contexts, CDF equations allow for straightforward and unambiguous determination of boundary conditions with respect to sample variables. The accuracy and robustness of solutions of the CDF equation for one such system, the Saint-Venant equations of river flows, are investigated via comparison with Monte Carlo simulations.