# Can an event of probability $0$ in a countable sample space occur?

I know that for a non-countable sample space there can be a situation where some event have probability $0$ yet the event may occur

For example: the event that we throw a dart at a target in it will hit a specific point on it.

Can we have a finite, or infinitely countable sample space and an event with probability zero that may actually occur ? I don't know how to phrase "can occur" in a mathematical way so I am having trouble making any progress on this thought

-
Unless you come up with a precise definition of "can occur", this cannot be answered, really! – Mariano Suárez-Alvarez Dec 3 '12 at 0:29
The definition of "can occur" which is relevant to, say, a random number in $[0, 1]$ being rational with probability $0$ is "is non-empty," and you can artificially make this happen in a sample space of size two (have one event occur with probability $0$ and one event occur with probability $1$). – Qiaochu Yuan Dec 3 '12 at 0:33
The definition of sample space, albeit informally, is the set of all possible values that the random variable may assume ([Sample Space][1]). Given this, any event with probability $0$ must not be in the sample space. [1]: mathworld.wolfram.com/SampleSpace.html – Patrick Dec 3 '12 at 0:46
I'd construe the question like this: Is there a probability space $(\Omega,\mathcal{F},P)$ for which $P(\{\omega\})=0$ for every $\omega\in\Omega$, and $\Omega$ is countable? – Michael Hardy Dec 3 '12 at 0:49
Yes. Imagine we toss up a coin, but unexpectedly a bird comes and eats it! – Berci Dec 3 '12 at 0:50