probability that a number defined by a random process is irrational

Question

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?

Answer 1

Your probability measure is actually equivalent to the standard Lebesgue measure on the real line.

The probability of getting a rational number is zero, and the probability it is irrational is one.

Answer 2

What do you mean "depending on the result"? Would you stop the process or just write down the result?

Generalizing the answer of Stephen Montgomery-Smith (who is a much better mathematician than I), the probability that the result is transcendental or normal or confused or traumatic is also one.

The probability that I am normal is much closer to zero, although I am not sure about many aspects of the process that created me (certain aspects are well-known and common to many others).