I don’t know that problem and don’t feel like googling it, but it’s a reasonable guess that’s because

  1/e ≈ 0.36787944

That seems unlikely. Those numbers only look related in base 10. 1/e is about a third of 1 and 37 is about a third of 100, which is why they look similar in base 10. In base 16, 1/e is about 0.5e2d58 and 37 is 25.

And prime numbers are prime numbers regardless of base.

100/e is close to 37, regardless of base.

(That 100 would be chosen as a convenient scaling factor for our notion of “percent” only in base 10 is true, but a separate point.)

No, that is precisely the point.

That's what PP was saying, in not so many words.

And thanks to 1/e being 0.367, in Pokemon Go when I catch 1000 Pokemon and each of them has a 1 in 1000 chance of being shiny, I have a 36.7% chance of having zero shinies among them, or approximately a 63% chance of the reciprocate, AKA having at least 1 shiny among them.

1000 is arbitrary of course, but the bigger the number the closer to 1/e.

See "Bernoulli trials"

It's also the most common number picked when people are asked to think of a random number between 1 and 100.

Was just about to comment this. Wonder if anyone more mathematically inclined can see a relationship.