And it’s not “overwhelmingly likely” as in there’s a 99% chance or whatever. If you choose a random point on the line, the probability of choosing a rational is zero.
Wat?
If a thing is in my pocket, there's an above zero probability of me picking it when I randomly take a thing out of my pocket.
What are math people doing that's different?
It's a pretty basic thing covered in undergrad prob/stats classes. We don't re-explain it every time we use it for the same reason computer scientists don't re-explain the halting problem every time it comes up.
This is only true if there are only a finite number of things in your pocket, though… I think an analogy is how we always have 1/n>0 for any finite (positive) number n—and yet, 1/infinity=0.
For something more precise, I think the corresponding Wikipedia page (FWIW) is https://en.wikipedia.org/wiki/Almost_never.