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We study a dynamic-journey planning problem for multimodal transportation networks. The goal is to find a journey, possibly involving transfers between different transport modes, from a given origin to a given destination within a specified time horizon. Transport services are represented as sequences of scheduled legs between nodes in the transportation network. Due to uncertainty in transport services, we assume for each pair of adjacent legs i and j a probability of a successful transfer from i to j. If a transfer between two legs is unsuccessful, the customer needs to reconsider the remaining path to the destination. The problem is modeled as a Markov decision process, and the main contribution is a backward induction algorithm that generates an optimal policy for traversing the public transport network in terms of a given objective, e.g., reliability, ride time, waiting time, walking time, or the number of transfers. A straightforward method for maximizing reliability is also suggested, and the algorithms are tested on real-life Helsinki area public transport data. Computational examples show that, with a given input, the proposed algorithms rapidly solve the journeying problem.