In this study, we examine the shortest path problem under the possibility of “re-routing” when an arc that is being traversed is blocked due to reasons such as road and weather conditions, congestion, accidents etc. If an incident occurs along the arc being traversed, the vehicle either waits until all effects of the incident are cleared and then follows the same path thereafter, or returns to the starting node of that arc and follows an escape route to the destination node, the latter course of action is called as “re-routing”. Also, we consider that this arc is not visited again throughout the travel along the network when an incident occurs and the alternative of not following this arc after the event is chosen. We propose a labeling algorithm to solve this specific problem. Then, a real case problem is analyzed by the proposed algorithm and several numerical studies are conducted in order to assess the sensitivity of the probability and travel time parameters.