Another "paradox": even though it's possible to randomly pick a rational number from the reals, the probability of this happening is 0.
A much more surprising result is that most irrational are normal numbers, but we know almost no normal number (morally speaking, a normal number is an irrational number where each digit is equiprobable in any base).
One of the axioms of probability is that if you have an event (i.e. a set), then the probability of a countable union of disjoint sets is the sum of the probability of each set (event) occurring.
Assume a uniform distribution between 0 and 1. Now consider point sets of the rationals (i.e. the number 0.5 is represented by a set with just 0.5 in it). Since the distribution is uniform, each set has the same probability (i.e. the likelihood of picking a random rational).
Now consider this question: What is the probability of picking any rational between 0 and 1? Well, that's just the sum of the probabilities over all rationals (because it is a countable sum of disjoint sets). If the probability of picking any particular rational was non-zero, this sum would be infinite, which violates the laws of probability.
Thus, by convention, it's just simpler to define it to be 0.
There's no magic here. These properties were picked merely to make analysis with measure theory clean. Don't try to ascribe any real world meaning to picking a point.
[edit] strictly speaking, you would restrict yourself to a bounded interval, e.g. if you pick a random number from a uniform distribution on [0, 1], the probability that this number is rational is 0.
You are talking about the probability of a single point event, which is also always 0 on that same sigma algebra.
The OP point is not completely trivial because the event contains an infinite (but countably) number of elements. It is fairly easy to understand though since by its very definition, the P[{X in Q}] = sum P[{x}] taken over every rational number (since Q is countable), and each P[{x}] is 0.
A deeper statement is that there exists uncountable sets of probability 0.
I don't think anybody even has a procedure for gathering that kind of evidence.
Infinities are problematic too.
The here is the non-constructive nature of the real numbers. That is not to say the reals are useless, but they are not much more than a formalism. It's rather useful though because it's hard to get numbers like pi or e. Its really nice that any real interval is compact, but that too is hard to replicate.
In terms of implementation, I'm not aware of an algorithm that can randomly pick a real number on an actual computer. Perhaps a mathematician could show how to pick one on some abstract machine with infinite resources, and not constrained by finite bit representations of numbers.
An actual (finite in time and space) computer can't even represent arbitrary real numbers, much less randomly choose them.
Once you accept that you have an entropy source in the physical world, you can easily be injecting random real numbers (from some range) and in fact, usually are, which are then being binned into integers by ADCs.
https://en.wikipedia.org/wiki/Almost_surely
This strange property comes from strange properties of the real numbers (and uncountably infinite sets) that give rise to things like:
https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
Measure theory deals with resolving this:
Banach Tarski Banach Tarski.
Seriously though, you can do math without invoking the axiom of choice. The formulation of probability doesn't strictly depend on it.
Infinity Paradoxes - Numberphile - https://www.youtube.com/watch?v=dDl7g_2x74Q
Perhaps you meant the interval from 0 to 1?