Distribution of irrational or transcendental numbers

Question

Lets say we select a random number $0.abc.......$ between $[0,1]$ where each digit after decimal is independently drawn from a fixed discrete probability distribution ($\sum P(X=n)=1$ for $n\in\{0,1,...9\}$). Obviously, $0=0.\bar{0}$ and $1=0.\bar{9}$ are included in this range. What is the probability that a random number is irrational/transcendental? Are there infinitely more transcendental numbers than irrational numbers? Can we get a closed-form solution for uniform distribution? Is it possible to select a distribution such that it always generates an irrational or rational number? Clearly, it is possible to generate only rational numbers by fixing the distribution to a fixed point mass.

Answer 1

The rational numbers are merely countably infinite, while the real numbers are uncountably infinite, so the probability of a number in the interval [ 0 , 1 ] being irrational is "certainty".

The algebraic numbers are also countably infinite, so the probability of a number in that interval being transcendental is also "certainty".