AHSME 1983 Problem 26

The probability that event $A$ occurs is $\frac{3}{4}$; the probability that event B occurs is $\frac{2}{3}$. Let $p$ be the probability that both $A$ and $B$ occur. The smallest interval necessarily containing $p$ is the interval $\textbf{(A)}\ \Big[\frac{1}{12},\frac{1}{2}\Big] \textbf{(B)}\ \Big[\frac{5}{12},\frac{1}{2}\Big] \textbf{(C)}\ \Big[\frac{1}{2},\frac{2}{3}\Big] \textbf{(D)}\ \Big[\frac{5}{12},\frac{2}{3}\Big] \textbf{(E)}\ \Big[\frac{1}{12},\frac{2}{3}\Big]$