Bus Bunching

If Robert sees a bus leave the stop, there is a two in three chance that another bus is to come in a minute’s time.  If the bus he missed was the last of the bunch of three, he will of course have to wait thirteen minutes for the next, but the expected waiting time averages out to five minutes overall.

On the other hand, if a bus hasn’t just gone past, then Robert has arrived in the thirteen-minute interval between bunches.  He then doesn’t know how long it will take for the bus to arrive, besides that it is six and a half minutes on average.