[prmph]

Wouldn't it be simpler to explain it this way: there are infinitely many reals between any two different reals, and thus the theoretical probability of picking any number between them at random is 1/infinity, which we think if as 0.

But in that case, why is is more probable to pick an irrational number?

[hn_throwaway_99]

I don't think it's simpler that way at all, mainly because the ways in which some "infinities" are larger than others, which explains why it's more probable to pick an irrational number. The rational numbers are countable, while the real numbers are not.