Probability of a number in the real line

Question

I have read that the probability to pick a rational number in the real line is null. My problem is:

If $S$ is a dense set in the real line, what is the probability to pick an element of $S$?

There are references for this type of problems(research papers, books etc.)?

Answer 1

$\mathbb{Q}$ is dense in $\mathbb{R}$, but the probability of randomly choosing a rational number from the reals is zero. $\mathbb{R}\setminus\mathbb{Q}$ is also dense in $\mathbb{R}$ and the probability of choosing such an element is one.

If you are interested in these things, you should study basic measure theory.

(To make statements about chance you actually need a probability measure, so it's easier to restrict yourself to some interval of finite length.)

Answer 2

Indeed, another probability measure in on the real linea is, for fixed $s>1$, $P(X=r)=\left\{\begin{array}{ccc}\frac{r^{-s}}{\zeta(s)}&\mbox{if}&r\in\mathbb{N}\\0&\mbox{if}&r\notin\mathbb{N}\end{array}\right.$

Thus, $P(x\in\mathbb{N})=1$, and $\mathbb{N}$ is indeed a very small topological subset of $\mathbb{R}$.