Why The Bus is Always Late and You Live in a Strangely Atypical World

Ever heard of the Inspection Paradox? In some contexts it just explains why you have to wait a long time for the bus. But if we started working with it, we could make the world a more extraordinary place - at least, on paper.

When you first hear about the Inspection Paradox, you might confuse it for something confirmed by MISPWOSO, The Maximegalon Institute for Slowly and Painfully Working Out the Surprisingly Obvious - from Douglas Adams' books. The math behind it shows that, for any periodically occurring event, the average wait time for the next event will always be longer than the average wait time for the next event.

Let's make it simpler. Buses, trains, or shuttles will come a certain number of times per hour, say once every fifteen minutes. That means that your wait time for the next bus should be, on average, seven and a half minutes. Anyone who has been on public transport will not be surprised to note that their average wait time is longer than the average wait time. They'll also probably guess at the answer as to why. Whenever there's a long wait for a bus, it arrives packed to the roof with disgruntled people. And it is about to be packed further, as anyone who has had a long wait time at a stop will be alone at first, then joined by more and more people until there is a crowd. The longer the wait between buses, the more likely people are going to show up and be annoyed by the longer wait.

Why The Bus is Always Late and You Live in a Strangely Atypical World

In short, you are more likely to experience a longer-than-average wait instead of a shorter-than-average wait, because you are more likely to show up during a long stretch of wait time than during a short stretch of wait time. The bus riders are considered "inspectors" because they show up at random intervals and check out the wait time for the next bus. They don't sit all day long and survey the time between buses. Inspectors are always more likely than average to experience longer-than-average waits, because delays are likely to take up more than the average amount of time.

But the paradox doesn't always work against you. Your light bulb is probably going to last longer than the average light bulb. Your watch will probably last longer than the average watch. So will your laptop. If you were to measure various things in your life, you would probably think that your life was startlingly atypical, with the various appliances, delays, waits, and stop-gap solutions lasting longer than they should. Instead, your life is typically paradoxical. You are always more likely to stumble into a longer situation than you should, on average, stumble into.

[Via Renewal Theory and Its Applications, Illustration of the Inspection Paradox,On the Persistence of Bad Luck.]