AHSME 1983 Problem 15

Three balls marked $1,2$ and $3$ are placed in an urn. One ball is drawn, its number is recorded, and then the ball is returned to the urn. This process is repeated and then repeated once more, and each ball is equally likely to be drawn on each occasion. If the sum of the numbers recorded is $6$, what is the probability that the ball numbered $2$ was drawn all three times? $\textbf{(A)} \ \frac{1}{27} \textbf{(B)} \ \frac{1}{8} \textbf{(C)} \ \frac{1}{7} \textbf{(D)} \ \frac{1}{6} \textbf{(E)}\ \frac{1}{3}$