When it comes to probabilities, our intuitions are not reliable, as I have written about before (see here and here). On so many occasions, I have thought that the result to a problem was so obvious as to not be worth thinking about more deeply, only to find myself proven wrong. And the new solution seems also so obvious that you wonder why you ever believed the earlier wrong answer.
Another example of this comes from the ‘inspection paradox’. It works like this. Suppose you know that the bus or train on a route that you take regularly does not always conform to the schedule. But while its arrival times are unreliable, it comes on average every ten minutes. If one goes to the bus stop at random times, one would think that the average wait time would be five minutes. But anyone who has actually experienced this has the feeling that on average the wait time is longer than that. We tend to put that down to being misled by psychology, by the fact that time drags when we are waiting for something, making it seem longer than it actually is
But it turns out that psychology is not to blame, that the wait time is in fact longer than half the average time between bus arrivals and this is what the inspection paradox is about.
To take a concrete example, suppose that the time interval between buses on a route is two minutes followed by 18 minutes, then two minutes, then 18 minutes, and so on, so that the average time interval is ten minutes. If one marks those on a time line, then one can easily see that if one arrives at random (equivalent to picking a random spot on the line or ‘inspecting’ it), then one is nine times more likely to arrive during the 18 minute interval than the two minute one. Hence on average, the wait time will be longer than five minutes.
One can make it more concrete. The probability of arriving during the two-minute interval is 2/20 and will have an average wait time of one minute, while the probability of arriving during the 18 minute interval is 18/20 and the average wait time will be nine minutes. So the expected wait time overall is (2/20)x1+(18/ 20)x9, which works out to 8.2 minutes.
I learned about this from Amir Aczel who goes into it in more detail and provides a general proof of the result, and also uses it to explain why it is that countries with a lot of immigrants have higher life expectancies than those with less, other things being equal of course.