Maximum possible value for the probability of an event. Consider two events $A$ and $B$ with the following given conditions:

$$P(A)=0.4\quad\textrm{and}\quad P(B)=0.7$$

The maximum and minimum value of $P(A\cap B)$ are?

Note: $P(X)$ denotes probability of occurrence of event $X$ and the other symbols have their usual meanings as in set theory.
Venn diagram might prove useful here:
Since $A \cap B \subset A$ and  $A \cap B \subset B$, 

$$P(A \cap B) \leq \min(P(A),P(B)) = 0.4. $$

On the other hand 

$$P(A^c \cup B^c) \leq P(A^c)+(B^c) = 0.6 +0.3 = 0.9 . $$

So 

$$P(A \cap B) = 1- P((A \cap B)^c) = 1-P(A^c \cup B^c) \geq 1-0.9 = 0.1. $$

Now you can construct examples to show that both of these bounds can be attained (that is, in general, they are the best possible bounds).