It's not about population vs. individual. The underlying principle is the assumption that we (the people taking the bus) arrive at a random time. And we're more likely to arrive in a shit interval (because they're longer) than in a lucky interval (because they're shorter).

Here's a trivial example.

Buses are supposed to arrive at a 10 minute interval: 12:00, 12:10 and 12:20. But today the second bus arrives a bit early, at 12:07. So they arrive at 12:00, 12:07 and 12:20. We arrive at the bus stop at a random moment >12:00 and <= 12:20.

If we arrive in the interval 12:00-12:07, our average waiting time will be 3.5 minutes. What's the chance that we do arrive in this interval? 7 mins/20 mins.

If we arrive in the interval 12:07-12:20, our average waiting time will be 6.5 minutes. What's the chance that we do arrive in this interval? 13 mins/20 mins.

So our expected waiting time is not 5 minutes but 3.5 * 7/20 + 6.5 * 13/20 = 5.45.

Basically "the shit intervals are longer so we're more likely to arrive in them. the lucky intervals are shorter so we're less likely to arrive in them.". If we arrive at a random time, which is the core assumption here.

Now you might say "but 5.45 doesn't feel close to 2N". And that's where the other assumption that probably does not reflect reality comes in - the bus arrival times are simulated as uniform random numbers. I mean, it depends on where you live, haha. But it's pretty much a worst case scenario, so in reality it's not as bad. Which the writer shows using the real-world data.

Nevertheless, unless it's the Japanese subway which always arrives exactly on time, it's always going to be bigger than 2N.

And what if we don't arrive at a random time, but arrive according to some pattern guided by the bus schedule? That change everything.

Still, it's actually pretty common to arrive at a random time, and buses (and some subways, or other things in life) do tend to arrive not exactly on time, in which case it holds. To some extent.

This seems to be attacking from a different perspective, though? You are requiring the change from a bus that is not to schedule. This article pointed out that that was not necessary. And, indeed, you can presume perfectly scheduled busses and still see a distribution quirk where the average wait time of the population is at the interval level. Right?

The article says the following:

> If buses arrive exactly every ten minutes, it's true that your average wait time will be half that interval: 5 minutes.

Which means that "not arriving exactly on schedule" is indeed a requirement.

Ah, I clearly misread that spot. I don't think this changes too much of my questioning here. Will definitely be playing with this more.

If the buses are perfectly at schedule and people arrive at random (uniform), the mean (and the median) waiting time will be half the scheduled interval.

If both are completely uniformly random, the mean waiting time will be the mean interval between buses. (In fact the distribution of the passenger arrival doesn't matter anymore.)

The real world is somewhere between those two.

Isn't that at odds with the article? Quoting, "The reason is that there are (of course) more students in the larger classes, and so you oversample large classes when computing the average experience of students." This leans on the experiences of the students, which necessarily leans on the population. (Granted, this is more clearly a different question/scenario.)

I'll try and play with the simulation some. And to be clear, I don't disagree with your statements. I just feel these are still different questions/statements. How long I expect to wait at a given trip/stop is not the same as how long I expect I have waited at that stop over the days.

My comment paraphrases what the article says.

You seem to be expecting something different from the inspection paradox, but all it says is that the waiting time will be higher than half of the interval between buses. And it only applies when the time between buses is random.

One of the things I've found far more often than makes sense, is that many paraphrases also change the statement they are paraphrasing.

The article goes out of its way to call out that sampling the average experience of students. This was obviously different in class sizes. Since there are more students than there are classes, it makes sense that the average size of classes that a student attends is different than the average size of classes offered. It isn't that people are giving you two different answers, they are answering two different questions.

Also, the article specifically says the waiting time paradox makes a stronger claim that the average will tend specifically to 2N. "But the waiting time paradox makes a stronger claim than this: when the average span between arrivals is N minutes, the average span experienced by riders is 2N minutes. Could this possibly be true?" I'm trying to explore/understand how that works out.

Oh, right. I was focused on the progression, I didn't notice you were talking about a factor of 2 that I forgot there.