Why Random Arrivals at a Bus Stop More Often Land in Long Gaps

When gaps between buses vary, a passenger arriving at random does not have an equal chance of “selecting” each gap. They have an equal chance of landing in each minute. A longer gap contains more possible arrival minutes, so it is more likely to be encountered. This is the inspection paradox, or length-biased sampling.

Suppose bus gaps alternate between 5 and 15 minutes. The operator's mean headway is 10 minutes. Over a long period, however, only one quarter of all moments lie in a 5-minute gap, while three quarters lie in a 15-minute gap. If passenger arrivals are independent of the buses, their position within a gap is approximately uniform. The mean remaining waits are therefore 2.5 and 7.5 minutes. Weighted together, the overall mean wait is 6.25 minutes, not the intuitive 5.

No particular bus has become more likely to be late. The difference comes from how observations are sampled: the operator counts intervals, while passengers sample time. Mean headway alone is insufficient; variability between gaps also affects waiting. In standard renewal-process models, greater variability usually makes the random passenger's mean wait longer.

The claim has clear limits. With perfectly regular headways, mean waiting is half the interval. If passengers consult a timetable and deliberately time their arrival, their arrivals are no longer random. The inspection paradox is not a law of every bus trip. It is a reminder that observed samples can be naturally weighted by how long each case lasts.