StackExchange has been amazing in the past, and I want to thank the collective hive-mind in advance. I have a pretty basic probability problem that I have no idea how to solve. I'm not looking for the answer, but rather, how to tackle it.
"Suppose that an experiment leads to events A and B with the following probabilities: $P(A) = 0.6$ and $P(B) = 0.7$. Show that $P(A \cap B) \geq 0.3$."
I suppose we know the events cannot be mutually exclusive, since $P(A) + P(B) > 1$. If the events are independent, then $P(A \cap B) = P(A)*P(B) = 0.42$, which I suppose is a (lower or upper?) bound for the joint probability.
Am I on the right track? Assuming that $A$ and $B$ are not completely independent, but also not mutually exclusive, what would my next step be?