Why do we always seem to just miss the bus? It’s nothing to do with luck.

Further and deeper exploration of paradoxes and challenges of intuition and logic can be found in my recently published book, Probability, Choice and Reason.

The bus arrives every twenty minutes on average, though sometimes the interval between buses is a bit longer and sometimes a bit shorter. Still, it’s 20 minutes taken as an average, or an average of three buses an hour. So you emerge onto the main road from a side lane at some random time, and come straight upon the bus stop. How long can you expect to wait on average for the next bus to arrive?
The intuitive answer is 10 minutes, since this is exactly half way along the average interval between buses, and if your usual wait is rather longer than this, then you have been unlucky.
But is this right? The Inspection Paradox suggests that in most circumstances you will actually be quite lucky only to wait ten minutes for the next bus to arrive.
Let’s examine this more closely. The bus arrives every 20 minutes on average, or three times an hour on average. But that is only an average. If they actually do arrive at exactly 20 minute intervals, then your expected wait is indeed 10 minutes (the mid-point of the interval between the bus arrivals). But if there is any variation around that average, things change, for the worse.

Say for example, that half the time the buses arrive at a ten minute interval and half the time at a 30 minute interval. The overall average is now 20 minutes, but from your point of view it is three times more likely that you’ll turn up during the 30 minute interval than during the ten minute interval. Your appearance at the stop is random, and as such is more likely to take place during a long interval between two buses arriving than during a short interval. It is like randomly throwing a dart at a timeline 30 minutes long. You could well hit the ten minute interval but it is much more likely that you will hit the 30 minute interval.
So let’s see what this means for our expected wait time. If you randomly arrive during the long (30 minute) interval, you can expect to wait 15 minutes. If you randomly arrive during the short (10 minute) interval, you can expect to wait 5 minutes. But there is three times the chance you will arrive during the long interval, and therefore three times the chance of waiting 15 minutes as five minutes. So you expected wait is 3×15 minutes plus 1x 5 minutes, divided by four. This equals 50 divided by 4 or 12.5 minutes.
In conclusion, the buses arrive on average every 20 minutes but your expected wait time is not half of that (10 minutes) but more in every case except when the buses arrive at exact 20 minute intervals. The greater the dispersion around the average, the greater the amount by which your expected wait time exceeds the average wait time. This is the ‘Inspection Paradox’, which states than whenever you ‘inspect’ a process you are likely to find that things take (or last) longer than their ‘uninspected’ average. What seems like the persistence of bad luck is actually the laws of probability and statistics playing out their natural course.
Once made aware of the paradox, it seems to appear everywhere.
For example, take the case where the average class size at an institution is 30 students. If you decide to interview random students from the institution, and ask them how big is their class size, you will usually obtain an average rather higher than 30. Let’s take a stylised example to explain why. Say that the institution has class sizes of either ten or 50, and there are equal numbers of both class sizes. So the overall average class size is 30. But in selecting a random student, it is five times more likely that he or she will come from a class of 50 students than of ten students. So for every one student who replies ‘10’ to your enquiry about their class size, there will be five who answer ’50.’ So the average class size thrown up by your survey is 5×50 + 1 x 10, divided by 6. This equals 260/6 = 43.3. So the act of inspecting the class sizes actually increases the average obtained compared to the uninspected average. The only circumstance in which the inspected and uninspected average coincides is when every class size is equal.
The range of real-life cases where this occurs is almost boundless. For example, you visit the gym at a random time of day and ask a random sample of those who are there how long they normally exercise for. The answer you obtain will likely well exceed the average of all those who attend the gym that day because it is more likely that when you turn up you will come across those who exercise for a long time than a short time.

Once you know about the Inspection Paradox, the world and our perception of our place in it, is never quite the same again.

Exercise

You arrive at someone’s home and are ushered into the garden. You know that a train passes the end of the garden every half an hour on average but the trains are actually scheduled so that half pass by with an  interval of a quarter of an hour and half with an interval of 45 minutes. Given that you have no clue when the last train passed by and the scheduled interval between that train and the next, how long can you expect to wait for the next train?

Links and References

Amir D. Aczel. Chance: A Guide to Gambling, Love, the Stock market and Just About Everything Else. 18 May, 2016. NY: Thunder’s Mouth Press.

On the Persistence of Bad Luck (and Good). Amir Aczel. Sept. 4, 2013. http://blogs.discovermagazine.com/crux/2013/09/04/on-the-persistence-of-bad-luck-and-good/#.XXJL0ihKh3g

The Waiting Time Paradox, or, Why is My Bus Always Late? https://jakevdp.github.io/blog/2018/09/13/waiting-time-paradox/

Probably Overthinking It. August 18, 2015. The Inspection Paradox is Everywhere. http://allendowney.blogspot.com/2015/08/the-inspection-paradox-is-everywhere.html