Probability of a picking a given number from (0,1) = 0. Is this paradoxical?

Question

If I pick an arbitrary interval like (0,1), the probability of picking any given number (like 0.5) = $\lim_{x \to \infty}\frac{1}{x}=0$

I at first thought that this was a paradox, and questioning whether '$0$' really meant impossible. I realised that it is impossible to even pick a number between (0,1). A computer cannot do it as it would require every number in that range to chose from (infinite).

Is this really a paradox? Either interpretation of 0 being impossible or infinitesimal leads to it not being one.

Answer 1

Having probability $0$ is not the same as being impossible and having probability $1$ is not the same thing as being guaranteed. In fact, there is a term for this; if something happens with probability $1$, then we say that it happens "almost surely."

Answer 2

think of it like this - take a paper sheet and a pen, put a dot anywhere on a paper(this dot have some area, but for sake of this answer let's assume it's mathematical point with area=0), now ask a friend to put a dot in a random spot on this paper sheet, what is the probability that he will "hit" your point? well $P=0$ but it's not impossible