It is a common goal to improve safety and performance of mobile indoor robots by predicting the movements of people in the surroundings. In contrast to many related works which exclusively employ sensors mounted on mobile robots, this work shows a method to achieve this goal in a smart environment where external sensors are used to sense people's positions. By using probabilistic models and filters, the evolution of the environment's state is predicted and optimal paths with respect to safety and performance are planned. Experiments in reality and in a simulation environment show the applicability in real-world scenarios and the advantages over classical path planning approaches.