Journey planning in public transport networks is inherently stochastic, not only in case of delays. Classical pareto-optimal algorithms do not take into account reliability of transfers and alternative continuations. In addition to the static timetable, we incorporate discrete departure and arrival time distributions and cancellation probabilities for all connections, which are estimated based on given non-stochastic realtime delay predictions. For a query, given a transfer strategy minimizing the mean destination arrival time, entire destination arrival distributions are propagated through the network. This induces a set of alternative continuations at the origin and all possible intermediate stops. Our stochastic algorithm takes well below one second on the complete timetable of Switzerland when using heuristic approaches, depending on the exact variant. Compared to a user following a fixed journey returned by a classical non-stochastic algorithm, a nimble user may arrive on average 9.5 minutes earlier with our algorithm. However, depending on the assumed transfer times and with continuous updates from the non-stochastic algorithm, the gains may also become negligible. Compared to another stochastic approach, CSA MEAT from Dibbelt et al., about three minutes are gained on average. Our algorithm will be particularly helpful for flexible users in more delay-prone environments.