Why Your Bus Is Always Late (And Your Friends Are More Popular)
Buses arrive every 10 minutes on average, but with random variation. You'd expect to wait 5 minutes on average... right?
Here are the intervals between bus arrivals. Notice how you're more likely to land in the longer gaps!
If one gap is 30 minutes and another is 10 minutes,
you're 3× more likely to arrive during the long gap!
You oversample long intervals simply because they occupy more time.
Same paradox, different context: Ask students their class size vs. ask the registrar.
Each student reports their class size
Average class size from records
The inspection paradox arises whenever the probability of observing something is related to its size. You don't sample uniformly — you sample proportionally to size!
For bus arrivals with mean interval T:
For exponential (Poisson) arrivals:
With regular 10-minute intervals, you'd wait 5 minutes on average.
With random Poisson arrivals (same 10-min average), you wait 10 minutes!
• Hospital stays: Surveying patients oversamples long stays
• Website visits: Active users are oversampled in traffic data
• Checkout lines: You're more likely to join a slow line (more people in it)
• Partner history: Your partners have had more partners than you (on average)
• Epidemic tracking: Friends of random people get sick earlier (used for early warning!)