This paper develops a stochastic version for the strategybased congested transit assignment problem stated by Cominetti and Correa (Trans. Sci. 35(3):250-267, 2001). As a distinctive approach, this stochastic version takes into account stochastic mean waiting times of passengers at stops and in-vehicle travel times. The model is formulated as a stochastic variational inequality derived from the formulation of the deterministic version of the problem, also stated as a variational inequality problem, for which only a single solution method is known uptodate. Closely related with the stochastic model, and as a special case of it, a consistent smoothing approximation to the deterministic model is developed and it is shown that this approximation provides an alternative way of solving the deterministic model. It is also shown that both, the stochastic model and the smoothed approximation, can be solved by means of an adaptation of a path based method for the asymmetric traffic assignment problem. Computational tests have been carried out on several medium-large scale networks showing the viability of the method and its applicability to large scale transit models.