For $s>1$ the Riemann Zeta-function is $$\zeta(s)=\sum_{n=1}^{\infty}n^{-s}$$
For a random variable $X$ with values in $\Bbb{N}$, its distribution is given by
$$\Bbb{P}[X=n]=\frac{n^{-s}}{\zeta(s)}$$
First I have shown that the probability that an $m \in \Bbb{N}$ divides $X$ (this event is called $E_m$) is $\Bbb{P}(E_m)=m^{-s}$.
Then I have shown that the events in the family $\{E_p\}_{p \text{ prime}}$ are independent.
Next, I was asked to determine $\Bbb{P}(\bigcap_{p \text{ prime}}E^c_p)$, that is the probability that $X$ is not divisible by any prime. I have determined it to be the Euler product $$\prod_{p \text{ prime}}(1-p^{-s})=\frac{1}{\zeta(s)}$$
Here is my question: How does the last question even make sense? How can we ask the probability that a natural number $X$ is not divisible by ANY prime? Clearly, it must be divisible by SOME prime and the probability is $0$. Or where have we made an assumption that would make it possible?