A new approximate random sampling method is described for numerical solution of the contaminant transport equation where a single uncertain parameter is present. The solution consists of the probability distribution of concentration at any point in space and time as a function of the uncertain parameter. The method is based on a limited sampling of the parameter space and subsequent interpolation of the information obtained at the sample points. This interpolation produces a complete approximation of a function that relates the random parameter and the concentration. The interpolation is performed using Hermite polynomials which require function and derivative information at each sample point. The theory is developed and an example of the computation of the distribution of concentration resulting from the distribution of a single effective conductivity is presented. Numerical results suggest that this method may be one to two orders of magnitude faster than a conventional Monte Carlo approach in solving this stochastic problem while yielding comparable accuracy.