A fairly technical probability question

Factual Questions
1

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.

2

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.

3

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.