The Inspection Paradox Revisited

The Inspection Paradox Revisited 

In a Nutshell

Imagine a bus that arrives, on average, every 20 minutes. Sometimes it comes a little sooner, sometimes a little later. But it’s 20 minutes on average. 

You arrive at the bus stop at a random time. How long should you expect to wait?

Ten minutes seems the obvious answer – half way between the average 20 minute interval between buses. If the buses arrived precisely every 20 minutes, that would be the right answer. But in reality they don’t – sometimes they arrive a little early and sometimes a little late. 

The Bus Stop

Let’s make the example very simple. Suppose half the intervals between buses are 10 minutes and the other half are 30 minutes.

The average interval is still 20 minutes.

But now when you turn up at the bus stop, you are three times as likely to arrive during one of the 30-minute gaps as during one of the 10-minute gaps. It’s simply a longer time period in which you could pitch up.

If you do arrive during a 30-minute interval, your average wait will be 15 minutes. If you arrive during a 10-minute interval, it will on average be just 5 minutes.

So your expected waiting time isn’t 10 minutes at all. It is:

(3 × 15 + 1 × 5) / 4 = 12.5 minutes.

Something rather strange has happened. The average gap between buses is 20 minutes, yet if you arrive at a random time you should expect to wait 12.5 minutes rather than 10.

You haven’t been unlucky. You’ve encountered the Inspection Paradox.

What’s Going On?

The explanation is actually quite simple.

When we inspect something at random, we are more likely to encounter the things that take up more space or more time. And this doesn’t just apply to buses.

How Big Is Your Class?

Suppose a college tells you that its average class size is  30 students. You check this by stopping students at random and asking how many are in their class. Now, let’s say half the university’s classes contain 10 students and the other half contain 50. The average class size really is 30.

But you aren’t equally likely to come across someone from the classes of 10 and the classes of 50. There are five times as many students available to be interviewed from a class of 50 as from a class of 10.

So for every student answering “10”, we should expect five to answer “50”.

The average class size reported by the students will therefore be:

(5 × 50 + 1 × 10) / 6 = 43.3.

The university says the average class size is 30. The students say it is more than 43.

And both are right!

They are simply answering different questions. The university is averaging across classes. Our survey is effectively averaging across students.

How Long Do Students Spend in the Library?

Here’s another example. Take your clipboard into a university library and ask the students you come across how long they usually stay in the library. Obviously, your sample will contain a disproportionate number of long-stay students, simply because they are more likely to come into contact with you. The short stay students have in large part left the library by the time you arrive and start and finish asking questions – or else they haven’t yet arrived. 

It’s the bus stop story all over again, in another guise. Similarly, you might experience longer restaurant waits or supermarket queues more frequently simply because longer waits and queues are around for longer and therefore provide more opportunities for you to encounter them. 

Why Do I Always Hit the Biggest Potato?

There’s an even simpler version of the same idea. You’re digging potatoes and accidentally slice through one with your spade. It’s an unusually big potato. Coincidence? Not at all. There’s more chance of the spade striking a bigger potato for the very simple reason it takes up more space. Same idea when your internet connection fails while you’re downloading that monster size file. There’s simply more time in which something can go wrong.

The Lesson

The Inspection Paradox is one of those statistical ideas that does seem rather odd until you see what’s going on. When there’s more time to enter the scene, more space to bump into, we’re more likely to be there. And that has implications which are sometimes counterintuitive. So next time you’ve been standing at a bus stop for 15 minutes waiting for a bus that supposedly comes “every 20 minutes”, don’t immediately conclude that you’ve been unlucky. It may just be the Inspection Paradox.

Then again, the bus might simply be late!