Probability of getting a rational number for continuum trials As far as I know, the probability of getting a rational number from the interval $(0, 1)$ is zero if we follow the Lebesgue measure. Still, it doesn't mean it's impossible.
Now, suppose we have a continuum number of independent random variables $X_{\alpha}$, where $\alpha \in \mathbb{R}$. Each of them is a sample from the interval $(0, 1)$. Would then the probability of getting at least one rational be positive?

This set of random variables can be considered as a function $f:\alpha \to X_\alpha$. So the question is what is $P(\exists \alpha | f(\alpha) is rational)$.
When the question is put this way, you can see that it relies on having a probability measure on the space of possible functions. The question, if asked for any "continuum number of random variables", is unanswerable.
If the set of possible functions is the constant functions (with the natural measure), then the answer is zero. But with many natural sets of functions the answer is one, since any nonconstant continuous function has some rationals in its image.
