What if we write $0$. and then throw a coin and depending on the result continue the number with 1 or $0$ and continue this process indefinitely.
It is clear that the result of this procedure is a real number. There is an infinity of infinite sequences of $0s$ and $1s$ that are rational (for example $0.101010...,0.101101101...$) I have also come to the conclusion that every infinite sequence (rational or irrational) has probability $0$.
My question is: What is the probability that the number produced is irrational? Or is it the case that like with the question: what is the probability that a number is prime, we cannot meaningfully assign a probability?