Go straight until you reach a Starbucks

Where I come from, Starbucks is all over the place. Their stores would make excellent recognizable landmarks if it weren’t for the fact that there are so many of them. It’s not uncommon to hear “go straight until you reach a Starbucks.” Which raises the question, will you reach one? Probably not.

When I’ve asked people this, sometimes they’ll say that “obviously you will; because eventually you go all the way around the world and go everywhere.” But, while you can pick a path that takes you through every point on the earth, it isn’t going to just happen. From here on, I’ll assume a straight line.

According to some unverified Internet fact, there are 169 Starbucks locations within five miles of one point in Manhattan. (The store locator this person used no longer shows the full number of stores, so I couldn’t check what they said.) If true, this means there are 2.15 Starbuckses per square mile there. (This is the official U.S. Customary unit of how civilized a place is, on a scale of “middle of nowhere” to “metropolis.” Measuring it in Starbuckses may be both ethnocentric and ungrammatical, but, well, this is the U.S. Customary system we’re talking about.)

Now assume the 2.15 S/mi^2 are uniformly distributed. We’re assuming this because it makes it more likely that a random line will eventually hit at least one Starbucks. Picture it as an average of one Starbucks in every square with side length of 0.68 miles. This pattern most likely doesn’t exist anywhere, not even in Manhattan, but let’s assume it not only exists but covers the whole earth. What are the odds that if you go straight and keep going, you actually do eventually reach a Starbucks?

Let’s define “at a Starbucks” as “within sixty feet of the point designated “Starbucks,”” sixty feet being the width of a two-row parking lot. And instead of having the Starbucks locations themselves be mathematical points with zero extension, let’s say they’re all 4300 square feet. (That’s the size of the largest Starbucks in Europe. Maybe it’s not the biggest, but the point is to give it the benefit of the doubt.) Since we math people are lazy, I’m going to pretend they’re all circular with a diameter of 74 feet. (For those of you keeping score, that’s 74 feet plus 60 in each direction for a total of 194 feet where you can say you’re at Starbucks. Round up to 200 for nice numbers.)

By this definition, our imaginary world is almost a quarter of a percent covered by Starbuckses. So if you pick a random location on this Earth, you have a .25% chance (rounding up again) of being at Starbucks. But we aren’t looking for the likelihood that a random point is in one, we’re looking for the likelihood that a random line intersects at least one. This isn’t going to be as simple as “Starbucks area divided by total area,” but it’s doable.

Define a line going around the world and coming back to its start point. We want to see whether it intersects any of these 200-foot disks. Any given 200-foot space has a .25% chance of containing one of those, and if the chances aren’t independent they’re at least pretty close. So each mile traveled has a (1-.0025)^(5280/200) = 93.6% chance of not arriving at a Starbucks. So for each mile you go in a random direction, you have a 6.7% chance of Starbucks. Which is surprisingly high. Maybe if we used the actual numbers instead of all these overestimates it would be closer to lottery-level unlikely. MOVING ON.

A full circle around the Earth is around (pun!) 24,900 miles. So the likelihood of arriving at a total of zero Starbuckses is (.936)^24900, which is so small that if it happened I’d assume I messed up the math and not even bother to consider that it might have been coincidence. Huh. I guess you would hit at least one Starbucks. OK, let’s try it with more realistic numbers. MOVING BACK.

Remember, we took an unusually high concentration of Starbucks stores and assumed it covered the entire earth. Let’s use the actual number of 20,366 stores worldwide instead of the four hundred and twenty-three million stores that I assumed earlier. Leave the size and the random distribution, so there’s still some benefit of the doubt.

Running through the same calculations with only this one very important number made realistic. Now there’s a total of (pi*100^2)*20,366 = 639,816,760 square feet of Starbucks area. Since the total surface area of Earth is about 5,490,232,704,000,000 square feet, this more realistic Earth is 0.0000117% Starbucks.

This time around, a one-mile distance has a 1-(1-0.000000117)^(5280/200) = 0.000031% chance of Starbucks. (That’s only five times more likely than winning the lottery! Yes!) And a circuit of the Earth has a 92.6% chance of no Starbucks. That’s surprisingly reasonable: It could happen without inhuman amounts of luck. But you can still be pretty confident that it won’t.

I don’t mind not being able to say this for certain. This is because it doesn’t undermine my point that you won’t reach a Starbucks. Why? You know why. All along, I’ve been talking about “straight lines” going around the earth. In other words, circles. If you look at lines, you won’t go around the world. In the best case, you start out going tangent to the Earth and you have as many chances as possible to intersect a Starbucks before the Earth starts curving away beneath you.

The curvature of the Earth is about eight inches per mile. So if you were to go ninety miles in a straight line, you would end up sixty feet away from the surface and out of Starbucks range. Would you hit a Starbucks within ninety miles? Maybe, maybe not, but it almost doesn’t matter. You might be thinking you would, because pretty much whenever you drive ninety miles you see some sign of a Starbucks, but remember that most of the time you’re driving, you’re driving on roads. And roads are not randomly chosen lines. Roads are in fact disproportionately likely to go near Starbuckses, though I don’t know how they know where to find them.

The math has already said that you probably wouldn’t run into a Starbucks, but for now let’s just assume that you would. In fact, let’s assume that .68 miles away from you (average distance between Starbuckses in the Manhattan estimate) was not in fact a handful of coffee shops but rather a solid ring of them. It still doesn’t really matter. Most possible lines would not be bothered by that.

Of the possible lines moving from your starting position, a large minority (actually a small majority) involve going at least slightly up. Any angle above one degree will go right above those Starbuckses and off into space. And anything going downward will take you through the Earth (ouch) and shoot out the other side. Since the Earth is “only” .05% covered by Starbuckses even from the insane estimate, you are unlikely to go through one of those points on your way out. Only angles that go parallel to the Earth’s surface, plus or minus one degree, have any serious chance of reaching a Starbucks, and even then they most likely (99.97%) won’t. Based on the likelihood of having a lucky starting position, going straight to a store, or arriving at one from underneath after being rammed through the Earth, the final odds of a random line taking you to a Starbucks are approximately:
0.000000117 + (2/360)*0.0003 + (179/360)*0.000000117 = 0.000184%

So you will probably not be able to grab a latte before beginning your long voyage through coffeeless space.


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