If buses arrive uniformly every 10 minutes and you arrive at a random time, your wait on average will be 5 minutes. However, if the buses arrive independently

If buses arrive uniformly every 10 minutes and you arrive at a random time, your wait on average will be 5 minutes. However, if the buses arrive independently and by a purely random process on average every 10 minutes (i.e. a poisson process), your wait will average 10 minutes. I always remember this result from probability when waiting for something. It helps me feel less unlucky: of course, I tell myself, it's more likely to land in the time interval between widely spaced busses because those intervals take up more space on the timeline.