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
Best Answer
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.