Probability and independence If (Ω,Pr) is the probability space where |Ω| is prime, and Pr is the uniform probability distribution on Ω, how would you show that any two non-trivial events A and B cannot be independent, (they must be either positively or negatively correlated). 
I know that 2 events are independent if Pr(A∩B) = Pr(A)⋅Pr(B), positively correlated if Pr(A∩B) > Pr(A)⋅Pr(B) and negatively correlated if Pr(A∩B) < Pr(A)⋅Pr(B). 

Since Pr is the uniform probability distribution on $\Omega$, $Pr(A) = |A|/|\Omega|$, and $Pr(B) = |B|/|\Omega|$, so $Pr(A) \cdot Pr(B) = |A|\cdot|B|/|\Omega|\cdot|\Omega|$. Suppose A and B are independent for a contradiction, we have $Pr(A) \cdot Pr(B) = |A|\cdot|B|/|\Omega|\cdot|\Omega| = Pr(A \cap B) = q / |\Omega|\cdot|\Omega|$ for some positive q. Then we have $q \cdot |\Omega|= |A| \cdot |B|$. Since $|\Omega|$ is prime, $|\Omega|$ divides either $|A|$ or $|B|$ by Euclid's Lemma, but A and B are two non-trivial events, so $|A| \neq p \text{ or } 0$ and $|B| \neq p \text{ or } 0$. This is a contradiction. 
