A fairly technical probability question

If I have a finite number of independent uniformly distributed (on 0,1) random variables, then the probability that none of them falls in a sub interval of length x is ((1-x)/x)[sup]n[/sup]. This goes to zero as n becomes large. If I have a countably infinite number of such variables, then with probability one any non-infinitesimal interval has at least one occurrence. Nevertheless, the set of points with an occurrence has measure zero. So the probability that a specific point has an occurrence is zero

Now suppose there are a continuum of such independent random variables with the same cardinality as the interval 0,1.

a) is the probability that a given point has an occurrence 1?

b) Is the probability that every point has an occurrence 0, 1, or something else?

I’m not even quite sure I used the right specific terms to ask this question.

The axioms of probability theory only allow us to reason about countable collections of events, so the short answer is that you can pick whatever answer you like.

Thanks, I guess that’s why I couldn’t figure out the answer nor find one on line. Also why my intuition would tell me different things depending on how I looked at it.