I’ve stayed at a fair share of different places over the last few years, and using public transportation takes the cake for being the most stressful and annoying day-to-day experience in every place I’ve been to. From riding 5-and-a-half hours every week in a packed Chevy Astro through hot Puerto Rican highways to starting your workweek at Berkeley with the fresh sight and smell of body parts, I’ve never had a positive relationship with public transportation (and don’t expect that to change anytime soon). However, for someone who can’t afford to buy a car—and who is universally described as driving “like a grandmother politely trying to get to the hospital while having a heart attack”—it is a regrettably indispensable part of my life.

As a result, I’ve had to spend a considerable amount of time thinking about how to maneuver the crowded Roman trains and smelly New York buses, and have stumbled onto some weird tricks that might be of use for both tourists and daily commuters. For this post specifically, my intent is to show you the “paradox” that, when trying to get on a packed metro train, being a dumb tourist is better than being a smart one; and I’m going to do it by using something just as annoying, stress-inducing and indispensable as public transportation. Statistics.*

*Cue Inception horns and distant screams.

If mathematics were a family, probability & statistics would be the bizarre great-uncle that won’t stop talking about how taxidermy is a spiritually fulfilling hobby at the dinner table. It is a field of study that is simultaneously too trivial for “real” mathematicians (they’re too busy writing proofs no one understands) and so strange that one of the best mathematicians of all time didn’t believe a simple statistics result until someone showed him a computer simulation proving it. But rather than go into any detailed description of this curious field of math, I’ll just give you a small primer on the basics of this strange field before we delve into any commuting weirdness.

Perhaps the two most important pieces of information in the statistical sciences are the long-term average and the single likeliest outcome. The names are pretty straightforward, but just in case, I’ll explain them with a six-sided die.

- The single likeliest outcome is just that. For one six-sided die, there isn’t any single likeliest outcome because you have an equal chance of getting any number between 1 and 6 (unless you’ve been loading your dice, you cheater). It’s easy to spot in an outcome graph, because it’s the outcome that happens the most.
- The long-time average is a little more detailed, but not very: it’s the average of your results after you obtain a very large amount of them! For a single six-sided die, that number is 3.5. You can’t spot this one in an outcome graph, but you can deduce/guess it if the shape is simple.

Now that we’ve got our statistics bases covered, allow me to illustrate the promised “dumb tourist paradox” through my experience living in the Bay Area. Trying to get on a BART train (the Bay Area’s metro system) during the busy hours was mostly a game of chance; you had to hope you picked a waiting spot close to where the train door lands or you’re looking at a 15 minute wait for the next one to roll in.

However, let’s say you knew that the train door always pops up within the same 100-foot strip of train station, but you don’t know exactly where. Assuming there’s an equal chance of it showing up anywhere in the strip, the instinctively smart thing to do would be to always wait smack-dab in the center of it; that’s the position that puts you closest to the train door in the worst-case, and it certainly *feels* like it’s your best bet.

In this scenario, you might claim you’re making the smartest choice, so let’s call this the **smart tourist scenario**. Now, instead of using some fancy math theorems to tell you what the most likely distance and long-term average distance are in this case, I’m going to be 100% thorough and actually* simulate* it! Let’s take a look at what being a smart tourist comes out to when you simulate the train arriving a million times:

There are two things to take away from this graph. First, since the graph indicates that the train stopped everywhere about the same number of times, there’s no single likeliest outcome. It’s equally likely for the train door to land right in front of you than it is for it to wind up 50 feet away! Second, if you used the train over and over, your average distance from the train door would be 25 feet (which you could calculate by finding the average of all the distance outcomes). Nothing unexpected here.

Now we’re going to go into “paradox” territory. Let’s say you take a page from your weird great-uncle’s book and, instead of carefully planning things out, you just decide to randomly pick a spot inside of the 100-foot strip to wait in.

In this case, you’re not making any decision at all about what’s best or not; you’re just randomly waiting somewhere. Let’s call this the **dumb tourist scenario**, and here’s what that looks like when you pick random spots a million times:

The simulations don’t lie: the likeliest outcome now is that the train stops right in front of you, and the average distance between you and the train will be about 33 feet.

Comparing both scenarios, there’s nothing weird going on if you commute all the time; the long-time average distance is bigger when you randomly wander around the train station (33 ft) versus when you wait in the middle (25 ft), so doing the smart thing is still your best bet in that case. But, when you’re a tourist and only plan on riding the train once or twice, this somehow seems to imply that it’s better to randomly pick a spot to wait in than to pick the best logical spot!

This “dumb tourist paradox” is profoundly counter-intuitive on many levels; how can a “dumb” action turn out to be better than a “smart” one? How can my random action cause the train to usually arrive closer to me? How can I understand this result intuitively? Well, I could try to calm you down by pointing out that being a dumb tourist has two negatives, which is that 1) your long-time average distance is larger and that 2) you have a nontrivial chance of having the train show up more than 50 feet away from you, which is impossible for the smart tourist. If you’re like me, though, you are probably still very puzzled.

The answer, however, is pretty mundane—even though it’s certainly true that the single likeliest outcome is that the train door stops directly in front of a dumb tourist, whereas there isn’t a likeliest outcome for a smart tourist, the actual chances of the train arriving directly in front of a smart tourist and a dumb tourist are effectively the same. As a result, you can’t actually game the system—it just ultimately *looks* like you can. If that wasn’t obvious to you, you can take solace from the fact that the smartest man who ever lived once said that “in mathematics you don’t understand things, you just get used to them”, and my advice is the following: get used to it. This is by no means the only “paradox” in the statistical sciences, as great many others are known to exist, and they’ve puzzled everyone just as much as this little factoid does. The best thing you can do is to learn about them and why they happen so that you don’t get surprised by them (or more importantly, make wrong assumptions because of them). And who knows! With time you may find some new ones yourself, if you decide to formally study statistics—or if you commute enough.