The waiting time paradox: why is my bus always late? (2018)
Random arrival times and irregular bus schedules create a “waiting time paradox,” where passengers tend to experience longer-than-average gaps between vehicles even when the official headway is short. Commenters connect this to broader statistical effects like lottery jackpots and class sizes, where sampling from individuals’ experiences skews perceived averages, and to real-world transit issues such as unreliable prediction apps and bus bunching. Several posts also highlight operational fixes — from better dispatching and ticketing to GPS tracking and self-coordinating routes — that can mitigate both the mathematical bias and the practical pain of long waits.
Everyday experiences & humor
- Many describe always seeing the bus or train just leaving, or the bus arriving only once they give up and start walking.
- “Lighting a cigarette makes the bus arrive” is a widely recognized running joke; some treat it as an almost deterministic law.
- Several doctor/patient and TV show jokes spin off from the “if it hurts, don’t do it” attitude toward watching the station.
Intuition for the waiting time paradox
- Key assumption: passengers arrive at random times, not synchronized to the timetable.
- Because long gaps contain more time, you’re more likely to arrive during a long interval than a short one.
- With perfectly punctual buses, average wait is half the headway; with random (Poisson-like) headways, average experienced gap doubles, so the average wait approaches the nominal interval.
- Some confusion appears around notation (N vs 2N vs wait time); others clarify: average span riders experience is 2N, so average wait is N.
Related paradoxes and probability debates
- Analogies:
- Class-size paradox: the “average student” sees larger classes than the average class size.
- “Your friends have more friends than you” phenomenon.
- Bitcoin block times: memoryless exponential waiting.
- Large subthread debates the claim: “if you win the lottery jackpot, you win less than the average lottery winner.”
- One side emphasizes conditional expectation: given that you win, you tend to be in drawings with more co‑winners.
- Others argue about what “average winner” and “average prize” mean (per draw vs per winner, conditioning on at least one winner, etc.).
- Consensus: the paradox is real at the level of conditional expectations, but precise statements depend heavily on definitions.
Transit reliability, operations, and bus bunching
- Many note that in practice, apps and GPS-based predictions can be wildly inaccurate, sometimes no better than random.
- Others report fairly accurate real-time systems where knowing “4 minutes away” matters more than schedule adherence.
- Bus bunching is discussed: late buses get more passengers and fall further behind; following buses catch up and form clumps.
- Proposed mitigations: multi-door boarding, off-bus or app ticketing, random inspections, mandatory vs optional stops, dispatch control, and self-coordinating headway control (including approaches that hold buses at control points).