I'm taking a class in Probability Theory, and I was asked this question in class today:
Given disjoint events $A$ and $B$ for which $$ P(A)>0\\ P(B)>0 $$ Can $A$ and $B$ be independent?
My answer was:
$A$ and $B$ are disjoint, so $P(A\cap B)=0$.
$P(A)>0$ and $P(B)>0$, so $P(A)P(B)>0$.
$P(A\cap B)\not =P(A)P(B)$, so $A$ and $B$ are not independent.
However, I was told that I am wrong and we cannot know whether or not $A$ and $B$ are independent from the given information, but I did not receive a satisfactory explanation. Is my argument valid? If not, where do I go wrong?