Turn up at a random moment and you're more likely to land in a long gap than a short one — because long gaps cover more of the timeline. So the interval you experience is bigger than the average interval, and your wait beats half the mean. The more irregular the arrivals, the worse it gets.
Random timing drops you into the long gaps more often, so the gap you land in averages 20 minutes — versus a true mean of 10 minutes — and your wait is half of that landed gap. At CV = 1 (Poisson — the classic random bus) that works out to a whole average gap: you wait 10 minutes even though one is due every 10 minutes.
It's size-biased samplingSize-biased sampling: when you draw a random point from the timeline, the probability it falls inside a given interval is proportional to that interval's length — so long intervals are over-represented in what you observe.. A random probe is likelier to fall inside a long interval because long intervals fill more of the line. The same effect explains why the bus feels late even on schedule, why your gym is crowded whenever you visit (you visit on busy days, when there are more people to observe you), and why your university class felt bigger than the school's average class size — you're one of the many students in a large class, not one of the few in a small one. At CV = 1 (Poisson arrivals — the default random bus) your wait equals the full mean gap, not half of it.