If you have never missed a flight, you are probably wasting your time.

On the other hand, I am an early bird. I naturally rise early in the morning and do my most creative work before the sun rises. My parents taught me to always be early and it has stuck. It makes me physically queasy anything I might be late, let alone the rare times when I am. One of my favourite quotes (I am have favourite quotes) is

If you're early: you're on time.
If you're on time: you're late.
If you're late: you're forgotten.

So imagine my shock when I read that I've probably been wasting my time arriving at airports so early. I read this in How Not To Be Wrong, a book by Jordan Ellenberg. How Not To Be Wrong teaches the reader how to utilize the power of mathematical thinking. As an absolute nerd and a math major myself, I really enjoyed the book. But the section on being early made me stop and think.

If you never miss a flight, you are arriving too early at airports.

When I first read this sentence, it sounded absurd. Given my disposition to arriving ahead of schedule, I took a long hard look at his argument.

This argument is based on utilitarian concepts, where every event can be given a positive or negative score, the utility that the event creates. These utility units are offered referred to as "utils". Since they are an arbitrary unit, we can define them in terms of something concrete. For example, we can say that one hour of my time is worth one util. Of course I could also say it's worth 100 utils, but then the rest of my calculations would only differ by a factor of 100, instead of any meaningful difference.

The earlier you arrive at the airport, the less likely it is that you will miss your flight. We can represent this by associating a percentage with the amount of time you arrive at the airport prior to your flight. For example, if you arrive 1 hour before your flight leaves, there might be a 15% chance you miss your flight. If you arrive 1.5 hours before that percentage drops to 10%, and arriving 2 hours before it could be a 2% chance you miss your flight. We are assuming that time spent at the airport is wasted time, and thus a negative utility.

We can also associate a utility with the cost of missing a flight. Perhaps you don't mind missing your flight, there's another in an hour and it doesn't cost you that much money. That's a very low utility compared to a very expensive rebooking the next day when you are flying internationally. One more thing to factor into the utility calculation for missing a flight is your personal comfort missing a flight. Personally, I have never missed a flight yet and I have no idea what I would do. It would make me quite uncomfortable.

Now we just have to weigh these two utility calculations to determine when you should arrive at the airport. The final result will also include a probability, the probability that you miss your flight. If you have a low tolerance for missing flights and you don't fly often, me and me, then in a year you probably won't miss a flight. However, if you time is very valuable to you and you fly often: you are probably going to miss a flight. Not only that, if you are not missing flights then you are wasting time at the airport.

The essence of this argument can be applied to other situations to illuminate exactly what it is getting at. The point is that in order to be efficient, you must be willing to accept some level of risk. For example, the safest way to drive a car is to never leave your driveway. This argument is akin to saying if it is impossible to get into a crash you are wasting time. It does not advocate to miss flights or crash cars, but to be aware of balancing risk and efficiency.

I happen to be writing this in an airport (that I arrived at 2 hours early), but at least my utils haven't gone to waste.