Suppose you have the set of the rationals between 0 and 1, let's call it $\mathbb{Q}_1$. Now I want the probability of choosing a number in $\mathbb{Q}_1$ and getting 1/2. If I add this to the probability of getting every other rational in $\mathbb{Q}_1$ e.g. 1/3, 0, 2/7 ... (which would each be the same as getting 1/2 in $\mathbb{Q}_1$) then I should get back 1 because I'm basically asking what is the probability of choosing a rational in $\mathbb{Q}_1$ and getting back a rational in $\mathbb{Q}_1$.
The probability of getting 1/2 from $\mathbb{Q}_1$ must then be non-zero and must be equal to 1 / the countably infinite. This is an infinitesimal probability, but does not exist within $\mathbb{R}$.
Is this maths correct?