Most optimization problems in applied sciences realistically involve uncertainty in the parameters defining the cost function, of which only statistical information is known beforehand. In a recent work we introduced a message passing algorithm based on the cavity method of statistical physics to solve the two-stage matching problem with independently distributed stochastic parameters. In this paper we provide an in-depth explanation of the general method and caveats, show the details of the derivation and resulting algorithm for the matching problem and apply it to a stochastic version of the independent set problem, which is a computationally hard and relevant problem in communication networks. We compare the results with some greedy algorithms and briefly discuss the extension to more complicated stochastic multi-stage problems.