the inspection (waiting-time) paradox

You arrive at a bus stop at a random moment. Buses come every 10 minutes on average. You might expect to wait about 5 minutes — half the average gap. Yet again and again you wait longer, and the gap you land in feels longer than typical. This is the inspection paradox, and it is not bad luck: it is a real and inevitable bias built into how you sample.

The mechanism is length-biased sampling. By showing up at a random instant, you are more likely to land inside a LONG gap than a short one, simply because long gaps cover more of the timeline — a 20-minute gap gives you twice the chance of arriving during it as a 10-minute gap does. So the gap you actually experience is not a typical gap; it is size-biased toward larger values, with mean E[T^2]/E[T], which is always at least E[T] (and strictly larger whenever gaps vary at all). Your wait is on average half of THAT inflated gap, not half of the ordinary mean. For the Poisson process the numbers are vivid: the mean gap is 1/lambda, yet the gap you inspect averages 2/lambda — twice as long — and your residual wait still averages 1/lambda by memorylessness, not 1/(2*lambda).

This is genuinely counterintuitive and shows up everywhere you sample 'whatever interval is in progress': the family you meet at random is larger than the average family (you are more likely to meet someone from a big family), the call you interrupt is longer than the average call, the file you grab at random is bigger than the average file. The fix is to be honest about the sampling: averaging over gaps and averaging over time are different operations, and length-biased sampling systematically favors the long ones. It is a bias, not a fallacy or an error in the underlying rate.