TITLE: This is question 3.3 from Alan Karr's Probability
QUESTION [5 upvotes]: What is the minimum number of points a sample space must contain in order that there exist $n$ independent events none of which has probability zero or one?
I'm thinking the answer is $2^n$, but this is just from checking by hand for the values $n=2$ and $n=3$.  I thought maybe a proof by induction would be appropriate, but didn't make it terribly far with it.  
I'm merely studying probability because I want to be better at it, and this seems to be a pretty classic problem. So it seemed like a good problem to understand.

REPLY [2 votes]: Consider non-trivial independent events $\{A_i\}_{i=1}^n$. Now, use the fact that for any $\alpha\in\{0,1\}^n$, the family $\{B_i^{\alpha_i}\}$ is independent (see, for example, Theorem 2.5. in Klenke, Prob. Theory), where $B_i^{\alpha_i}=A_i$, if $\alpha_i=0$ and $A_i^c$, if $\alpha_i=1$.
Namely, for any $\alpha\in\{0,1\}^n$, $\bigcap_{i=1}^nB_i^{\alpha_i}$ must be non-empty (otherwise independence fails). By construction, these sets are disjoint and must contain at least one element. Hence, $|\Omega|\ge|\{0,1\}^n|=2^n$.