The concept of probability is based on the concept of measure in math => limitation in its description of things, e.g., the probability for a real number in [0, 1) being an irrational number is 1.

Hold on, what measure over the unit interval assigns probability 1 to the set of irrational numbers in the unit interval? Do irrational numbers on their own even form a proper measurable set?

Lebesgue measure [0]. I never did write part 2 and many years have gone by with that blog languishing unmaintained on free wordpress but I wrote a thing that goes through the issues around this [1].

[0] https://en.m.wikipedia.org/wiki/Lebesgue_measure

[1] https://omnicognate.wordpress.com/2013/11/04/sigma-algebras-...

Of course. [0,1] has measure 1. Rationals have measure 0. Irrationals have measure 1-0=1

This question is why measure theory exists.

The irrationals are measurable for any measure that can measure points, which is most of them.

What is funny thought, that 0% chance events don't can't happen, but must happen. Like when you pick a point on a line, or roll a dice infinitely many times.