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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

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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]: – 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
up vote 2 down vote accepted

Probability theory, as a branch of mathematics, doesn't really purport to tell you anything about what "can or cannot occur" in the real world. It gives you probabilities (which are defined as "the kind of numbers probability theory gives you", neither more or less), and what you conclude about the real world on the basis on those numbers is up to you.

There's an argument to be made that even a countable infinity of outcomes is unrealistic as a description of something that could happen in the real world, since we can never distinguish between more than finitely many outcomes before the sun goes nova and/or the coffee gets cold anyway. From that point of view, speaking about infinite sample spaces is simply an idealization that allows us to approximate the actual finite behavior of the world very well in a mathematically tractable way.

For any real process with a finite set of possible outcomes, I would say that it would be wrong to assign a probability of zero to an outcome that can actually happen -- in the sense that if I claim that such-and-such outcome has zero probability and it then happens, this demonstrates conclusively that my claim must have been wrong. But that's not really a mathematical argument -- it is merely a boundary condition I'd like to stipulate for my real-world use of probabilities.

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Thanks for the answer. The problem araised in the context of a homework problem where I divided by the probability of some event (dealing with a deck of cards, so everything is finite here) and I wanted to state that the probability isn't zero because the event can happen..but then I remembered what I described in the question and I wan't sure how to argue that I can divide by that probability. +1 – Belgi Dec 9 '12 at 23:24

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