Thought Toys · Data & inference · Exhibit 104
A bus route promises one bus every 10 minutes, on average — and it delivers. Yet the riders' average wait is 10 minutes, not 5. Nobody is lying. Long gaps are simply bigger targets, and a random arrival lands in them more often.
A stretch of the timetable — every dot is a rider who landed here bus rider
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The line across the top is time at a bus stop, and the amber ticks are buses. Their gaps average 10 minutes — the first readout keeps checking, and the timetable never breaks its promise. Press rain 150 riders: each cyan dot walks up at a purely random moment and lands wherever it lands. Every rider you see is counted, and every rider counted is on screen.
Look at where the dots pile up. A 30-minute gap is a target three times as wide as a 10-minute one, so it catches three times the riders. Averaging over riders therefore over-weights exactly the gaps that are longest — and the riders' average gap floats above the timetable's, by precisely a factor of 1 + CV², where CV measures the schedule's spread. Same buses, same line, two honest averages that disagree. The only question is who you average over.
The amber tick marks what the theory predicts, and the bar rarely lands exactly on it — because this stretch is one stretch, fourteen gaps drawn once, not an eternity of them. Press new timetable a few times: the bar hops around the tick and settles nowhere. The prediction is about the process, not about any one afternoon at the stop. The stretch is scaled so its own average gap is exactly ten minutes, so the left bar never wanders — the route is not secretly running late, and everything you see is the rider-weighting alone.
At spread 1 the schedule is fully random — bus arrivals with no memory at all. Here the paradox hits its famous landmark: riders' gaps average 20 minutes, and the average wait is the full 10, double the naive guess. It is worse than that, in a way the wait bar can't show: with a memoryless schedule, after 10 minutes of waiting your expected remaining wait is still 10 minutes. Your patience buys nothing, because the process doesn't know you're there.
Now drag the spread to 0 and watch the negative control run live: clockwork buses, every gap exactly 10, and the two gap bars snap together while the wait bar drops onto the naive 5. The paradox was never about buses being slow. It is about sampling — any inspection that lands on a line weighted by length meets the long stretches first. The same arithmetic explains why the average class a student sits in is bigger than the college's average class, and why your call reaches the hold queue at its worst. One exhibit over, it is why your friends have more friends than you: sampling by membership is sampling by size.
improve/verify/104-inspection-paradox.js, 31 checks)
confirms the closed forms by Monte Carlo on six schedules (Gamma, uniform, a two-point mixture), checks
the memoryless case in distribution (P(wait > μ) = 1/e), and isolates the mechanism with negative
controls: the paradox vanishes exactly at CV = 0, and vanishes when gaps are sampled by index
instead of by landing time — length-biased sampling is the whole story.
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