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MENTAL MODEL · M5747

Waiting Time Paradox

Waiting Time Paradox
TechnicalmediumProbability Theory
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Updated 2026-08-08

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INTRODUCTION

English translation pending.

CORE DEFINITION

A consequence of length-biased sampling in probability, often illustrated with bus arrivals. If buses arrive at random with a fixed mean interval, a passenger arriving at an arbitrary time is more likely to land inside a long interval than a short one, so the expected wait exceeds half the mean headway. The core claim is that the observer's experience systematically exceeds the nominal average. The qualification is that the paradox disappears when arrivals are perfectly regular rather than random.

SCAFFOLDING EFFECT

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- Use Experience Check: Ask whether your own sampled experience should be expected to match the published average. - Use Distribution Check: Examine the spread of intervals, since variance drives the sampling bias. - Use Mitigation Choice: Push for regular service or real-time information instead of relying on the nominal average.

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Sampling a random moment is equivalent to sampling an interval with probability proportional to its length, so long gaps are overrepresented in what an arriving passenger encounters. Because the expected remaining wait inside a sampled interval exceeds half its length, the average experience is worse than the average interval suggests. The bias grows with the variance of the interval distribution and vanishes when all intervals are equal.

MINIMUM ACTION

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Source support: Explicit

  • link
    en.wikipedia.orghttps://en.wikipedia.org/wiki/Renewal_theoryZH · Explicit
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