# Probability - Independent events with minimal points in sample space

1. Oct 16, 2012

### nrush

1. The problem statement, all variables and given/known data
What is the minimum number of points a sample space must contain in order that there exists n independent events A_1, ..., A_n , none of which has probability zero or one?

2. Relevant equations
None at this time

3. The attempt at a solution
I was thinking that if each A_i consisted of one point in the sample space, that then they would all be independent. But it seems that this is definitely not the minimum number of points. Any hints at this would be greatly appreciated.

Thanks!

2. Oct 16, 2012

### LCKurtz

Hmmm. A set of size n has $2^n$ subsets, counting the empty subset and the whole set. So if you want $n+2$ subsets you need $\ln_2(n+2)$, rounded up to an integer, elements in the set. Whether or not you can make them independent is another question.