I and my friends just had a little discussion whether the events are independent or dependent if they have no result in common. I thought that they have to be independent. When two events are independent, then $P(A)=P(A\mid B)$. Is the information given in the question enough to establish this?

If you look at it as a Venn diagram, then if there is no overlap between A and B, then they are independent. But my friend objected and said that this depends on the sample space. 

So let's say we have two events: $P(A)=6/12$, $P(A\mid B)=2/4$, and $P(B)=4/12$, then obviously $P(A\mid B)$ equals $P(A)$. But for me, something smells fishy here. 

I know this is more a stochastic question that a statistic question, but maybe someone can help.