Let $A$, $B$, and $C$ be events. Suppose $P(A) \ge .9$, $P(B) \ge .8$, and $P(A \cap B \cap C)=0$.
Show that $P(C) \le .3$.
Now, I tried using the inclusion-exclusion principle to solve this, but I'm getting nowhere. Perhaps that is the correct way of starting, but I'm looking at it the wrong way? It's been a little white since I've worked with this, so I'm not sure I'm on the right track.
Also, is it correct that $P(A \cap B \cap C)=0$ means that the events $A$, $B$, and $C$ are disjoint (but not necessarily $A \cap C$, $A \cap B$, and $B \cap C$)?
Any hints would be appreciated.