The Metro Passenger's Curious Adventures
A subway rider who travels at random times notices that trains heading in his preferred direction seem to arrive less often than trains going the other way, even though the timetable runs both directions with exactly equal frequency — how can this be?
History and references
Settled by probability theory of renewal/interval processes applied to transit scheduling.
This is a specific transit-flavored instance of what is generally called the inspection (or 'waiting-time') paradox in probability: random sampling of an interval process is biased toward longer intervals. It is a recurring theme in popular mathematics and has been rediscovered many times (e.g., as the 'uptown/downtown dating paradox' in New York City folklore). Lange's version, framed as a Moscow-metro commuter's puzzle, is confirmed as the opening problem excerpt shown on a hosting site for the book, illustrating that the book blends genuinely classic paradoxes with original probability-flavored puzzles.
It shows that 'equally frequent on average' and 'equally likely to be seen next by a random observer' are not the same statement, a distinction with real consequences for scheduling and perception.
Explanations that circulated
- Assuming that because both directions run trains at the same long-run average frequency, a passenger who arrives at a random moment must be equally likely to see either direction's next train first.
- Ignoring that if the two directions' departures are not perfectly interleaved (e.g., a downtown train leaves 1 minute after an uptown train, then there is a long gap before the next uptown train), a randomly-arriving passenger is more likely to land inside the long gap before one direction than the other — biasing which direction 'shows up next' far more often than the raw frequencies would suggest.
References
- Inspection Paradox, Sheldon M. Ross, Encyclopedia of Statistical Sciences (Wiley), 2006
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