# A fairly technical probability question

**URL:** <https://boards.straightdope.com/t/a-fairly-technical-probability-question/555811>\
**Category:** Factual Questions\
**Created:** [October 2, 2010, 2:50am UTC](https://boards.straightdope.com/t/a-fairly-technical-probability-question/555811 "2010-10-02T02:50:11Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![OldGuy](https://avatars.discourse-cdn.com/v4/letter/o/3bc359/32.png) [@OldGuy](https://boards.straightdope.com/u/OldGuy)\
**Post date:** [October 2, 2010, 2:50am UTC](https://boards.straightdope.com/t/a-fairly-technical-probability-question/555811/1 "2010-10-02T02:50:11Z")

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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.

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**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\
**Post date:** [October 2, 2010, 7:05pm UTC](https://boards.straightdope.com/t/a-fairly-technical-probability-question/555811/2 "2010-10-02T19:05:32Z")

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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.

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<div class="post-metadata">

**Author:** ![OldGuy](https://avatars.discourse-cdn.com/v4/letter/o/3bc359/32.png) [@OldGuy](https://boards.straightdope.com/u/OldGuy)\
**Post date:** [October 2, 2010, 8:00pm UTC](https://boards.straightdope.com/t/a-fairly-technical-probability-question/555811/3 "2010-10-02T20:00:24Z")

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> [@ultrafilter](#):
>
> 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.