We derive and analyze a novel central-upwind finite-volume for random shallow water equa- tions. In order to quantify the uncertainty in initial data and bottom topography the stochastic Galerkin method is applied. We discuss hyperbolicity of the resulting system of the moments of the truncated generalized polynomial chaos expansion and prove it’s relation to the positivity of water height. The moment system is discretized using a well-balanced and positivity preserving second-order finite-volume scheme. Extensive numerical experiments confirm the robustness of the presented approach.