2.1.13 problem 15

Given the inequailities and the fact that $P(A\cap B)=P\left(A\right)+P\left(B\right)-P(A\cup B)$, to maximize $P(A\cap B)$ we maximize the smallest of the three expressions. Namely, $P\left(A\right)$. Thus, we would like to know that event $A$ occured.