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.

score 0 · Answer 1 · answered Sep 29 '19 at 08:05

0

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."

answered Sep 29 '19 at 08:05

Jonathan Dunay

1,740

Why is t h a t? – Simplex1 Sep 29 '19 at 08:06
I mean, it's like you said. In your example, you have probability $0$ of picking $0.5$ randomly in a uniform distribution of $(0,1)$, but it is not impossible to pick it either. – Jonathan Dunay Sep 29 '19 at 08:12
Are you familiar with the axioms of probability at all? – Jonathan Dunay Sep 29 '19 at 08:14
Here is what is going on rigorously: suppose $(\Omega,\mathcal{F},P)$ is a probability triple. Essentially, there is no requirement following from the axioms that states that if $P(A)=1$ then $A=\Omega$. – Jonathan Dunay Sep 29 '19 at 08:17

score 0 · Answer 2 · answered Sep 29 '19 at 08:12

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

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

2 Answers2