The resolution to the apparent paradox is that Old Faithful does not erupt exactly every 90 minutes. If she did, your average waiting time would be 45 minutes. However, she only erupts on average every 90 minutes which means that some arrivals between eruptions are shorter than 90 minutes and some are longer, and when you arrive at random, you are more likely to hit one of the longer intervals. For an unrealistic but informative example, suppose that times between eruptions are either half an hour or two and a half hours, with equal probabilities. When you arrive, you are five times as likely to arrive in a two-and-a-half hour interval, which gives an average wait of 75 minutes, as you are to arrive in a half-hour interval, with an average wait of 15 minutes. Thus, with probability 5/6 you wait on average 75 minutes and with probability 1/6, you wait on average 15 minutes and your average wait at a random arrival is (5/6)*75 + (1/6)*15 = 65 minutes, quite a bit more than 45.

This phenomenon is known as the waiting time paradox which you can read more about in .