Suppose you choose a rational number at random and reduce it; what is the probability that the denominator is odd?
I looked around and heard you can't define a uniform probability measure on the rationals so the question is useless, but I ran some monte carlo simulations and no matter what bounds I chose, I always saw $\frac{2}{3}$ as the answer.
This makes me think that the question is well defined, but I'm not sure, because the question of the probability of choosing two random integers such that they are coprime does have a well defined answer, $\frac{6}{\pi^2}$.
An formulation that I think is equivalent is:
Let $A,B \sim U(1,n)$ be discrete random variables, then consider the new random variable $X = \frac{A}{\gcd(A,B)} \pmod{2}$, then what is $\mathbb{P}(X = 1)$ as $n \to \infty$?