KöMaL 2012 B4496

An absent minded professor is travelling in a train along a subway line having infinite length in both directions. He would like to get from Alpha station to Tau station. At every station, the probability that he looks up from his notes is the same $p$>0 (independently of each other). If he looks up and observes that he has reached his destination, then he gets off. If he can see that he has passed his destination, then he also gets off and gets on the train travelling in the opposite direction. If he looks up and sees that he has not reached his destination yet, he stays in his train. If he does not look up, he will stay on the train wherever he is. What is the probability that he will get off at Tau station after a while?