Ben participates in a prize draw. He receives one prize that is equally likely to be worth $\$ 5, \$ 10$ or $\$ 20$. Jamie participates in a different prize draw. She receives one prize that is equally likely to be worth $\$ 30$ or $\$ 40$. What is the probability that the total value of their prizes is exactly $\$ 50$ ?
(A) $\frac{1}{6}$
(B) $\frac{1}{3}$
(C) $\frac{1}{2}$
(D) $\frac{2}{5}$
(E) $\frac{2}{3}$
Since there are two possible prizes that Jamie can win and each is equally likely, then the probability that Jamie wins $\$ 30$ is $\frac{1}{2}$ and the probability that Jamie wins $\$ 40$ is $\frac{1}{2}$. If Jamie wins $\$ 30$, then for the total value of the prizes to $\$ 50$, Ben must win $\$ 20$. The probability that Ben wins $\$ 20$ is $\frac{1}{3}$, since there are three equally likely outcomes for Ben.

If Jamie wins $\$ 40$, then for the total value of the prizes to be $\$ 50$, Ben must win $\$ 10$. The probability that Ben wins $\$ 10$ is $\frac{1}{3}$.

Since Ben's and Jamie's prizes come from different draws, we can assume that the results are independent, and so the probability that Jamie wins $\$ 30$ and Ben wins $\$ 20$ is $\frac{1}{2} \times \frac{1}{3}=\frac{1}{6}$.

Similarly, the probability that Jamie wins $\$ 40$ and Ben wins $\$ 10$ is $\frac{1}{2} \times \frac{1}{3}=\frac{1}{6}$.

Therefore, the probability that the total value of their prizes is $\$ 50$ is $\frac{1}{6}+\frac{1}{6}=\frac{1}{3}$.

ANSWER: (B)