I'm trying to think of situations in which two events are independent, but one event can be predicted perfectly once one has the information about whether or not the other event has happened.
I think these situations arise when two events are disjoint, but one event has probability $0$. For instance, consider the sample space $(0,1)$ with Lebesgue measure. Then consider $A=(0,1) \cap \mathbb{Q}$ and $B=(0,1)\backslash A$. Now if we consider the event of choosing a random number from $(0,1)$, it falls on either $A$ or $B$, so the event that it falls on $A$ and the event that it falls on $B$ are independent but disjoint, hence if we know that it fell on $A$ then we know it didn't fall on $B$ and vice versa.
I'm wondering what other situations we can think of, since this was the best I could come up with so far.