We present a probabilistic approach to quantify parametric uncertainty in first-order hyperbolic conservation laws. The approach relies on the derivation of a deterministic equation for the cumulative distribution function (CDF) of system states, 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. We demonstrate the accuracy and robustness of our CDF approach in several settings that allow one to obtain closed-form, semi-analytical solutions for the CDF of the state variables. The results are tested and contrasted with Monte Carlo simulations and other uncertainty quantification techniques.