P(A) = 0.7 and P(B) = 0.9. Are events A and B mutually exclusive?

Question

I was wondering if it is even possible to know if they are mutually exclusive. What I mean by this, is that there is a variable missing in order to know the answer. Would that be possible?

Answer 1

Two events $A$ and $B$ are mutually exclusive if $$\Pr[A \cap B] = 0.$$ Since we also have for any events $$\Pr[A \cup B] = \Pr[A] + \Pr[B] - \Pr[A \cap B],$$ we see that if $A$ and $B$ are mutually exclusive and $\Pr[A] = 0.7$, $\Pr[B] = 0.9$, that we would have $$\Pr[A \cup B] = \Pr[A] + \Pr[B] = 1.6,$$ which is absurd; therefore $A$ and $B$ cannot be mutually exclusive--namely, we must have $$0.6 \le \Pr[A \cap B] \le 0.7.$$