Let's assume that taxi journeys are all of an equal distance (10km, say), and that they go straight in a random direction. At the end of one journey, the driver picks up another passenger at that very drop-off spot.
After her first journey, the taxi driver is ten kilometres from her start point. But what is her expected distance from her original start point after a second journey? And is there a generic formula that tells us how far she is from his origin after n journeys?
With regard to the second journey, there is a tiny chance (infinitely small) that she will end up exactly at the start point; and there is an equally miniscule possibility that she'll be 20km from home. But I want the average distance (i.e. the expected distance, given that the angle of journey 2 is random). I have worked out that after the second journey, she has a 35% 33% chance of being closer to home than she was after the first journey.
