Show that the intersection of two probabilities in a certain interval

Question

I am struggeling with the following problem:

Suppose that $P(A)= \frac{3}{4}$ and $P(B)= \frac{1}{3}$. Show that $\frac{1}{12} \leq P(A \cap B) \leq \frac{1}{3} $.

Basically I try to show this for the upper and lower bound. For the lower bound I just take $min (P(A), P(B) )$. However I am struggeling with the upper bound.

I really appreciate your answers!

Answer 1

Hint: You have probably seen the formula $\Pr(A\cup B)=\Pr(A)+\Pr(B)-\Pr(A\cap B)$. Note that $\Pr(A\cup B)\le 1$.