The computationally efficient search for robust feasible paths for unmanned aerial vehicles (UAVs) in the presence of uncertainty is a challenging and interesting area of research. In uncertain environments, a “conservative” planner may be required but then there may be no feasible solution. In this paper, we use a chance constraint to limit the probability of constraint violation and extend this framework to handle uncertain dynamic obstacles. The approach requires the satisfaction of probabilistic constraints at each time step in order to guarantee probabilistic feasibility. The rapidly-exploring random tree (RRT) algorithm, which enjoys the computational benefits of a sampling-based algorithm, is used to develop a real-time probabilistically robust path planner. It incorporates the chance constraint framework to account for uncertainty within the formulation and includes a number of heuristics to improve the algorithm’s performance. Simulation results demonstrate that the proposed algorithm can be used for efficient identification and execution of probabilistically safe paths in real-time.