I've come across the statement which says that if two events $A$ and $B$ are *disjoint*, then they are *de*pendent because if, for example, $A$ occurs, it gives us the information that $B$ didn't occur. So far so good.

Hence I draw the conclusion (possibly erroneously) that if two events are *not* disjoint, they are *in*dependent.

I'm trying to test it with the artificial example a friend of mine came up:

Consider the sample space in the picture which is the unit square.

Event $A$ - the blue rectangle

Event $B$ - the brown triangle

$P(A)=0.5$

$P(B)=0.5$

$P(A\cap B)=0.125$

Obviously, $P(A)*P(B)\neq P(A\cap B)$ and I conclude that $A$ and $B$ are also dependent.

So, I am wrong saying that if $A$ and $B$ are *not* disjoint, they are always *in*dependent?