Box 1 contains one gold marble and one black marble. Box 2 contains one gold marble and two black marbles. Box 3 contains one gold marble and three black marbles. Whenever a marble is chosen randomly from one of the boxes, each marble in that box is equally likely to be chosen. A marble is randomly chosen from Box 1 and placed in Box 2. Then a marble is randomly chosen from Box 2 and placed in Box 3. Finally, a marble is randomly chosen from Box 3 . What is the probability that the marble chosen from Box 3 is gold?
(A) $\frac{11}{40}$
(B) $\frac{3}{10}$
(C) $\frac{13}{40}$
(D) $\frac{7}{20}$
(E) $\frac{3}{8}$
When a marble is chosen from Box 1 , the probability is $\frac{1}{2}$ that it will be gold and the probability is $\frac{1}{2}$ that it will be black.

Thus, after this choice is made, there is a probability of $\frac{1}{2}$ that Box 2 contains 2 gold marbles and 2 black marbles, and there is a probability of $\frac{1}{2}$ that Box 2 contains 1 gold marble and 3 black marbles.

In the first case (which occurs with probability $\frac{1}{2}$ ), the probability that a gold marble is chosen from Box 2 is $\frac{2}{4}=\frac{1}{2}$ and the probability that a black marble is chosen from Box 2 is $\frac{2}{4}=\frac{1}{2}$. In the second case (which occurs with probability $\frac{1}{2}$ ), the probability that a gold marble is chosen from Box 2 is $\frac{1}{4}$ and the probability that a black marble is chosen from Box 2 is $\frac{3}{4}$.

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-091.jpg?height=250&width=293&top_left_y=669&top_left_x=816)

Box 2: 2G, 2B

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-091.jpg?height=265&width=792&top_left_y=949&top_left_x=726)

Box 3: 2G, 3B

Box 3: 1G, 4B

Therefore, the probability that a gold marble is chosen from Box 2 is $\frac{1}{2} \cdot \frac{1}{2}+\frac{1}{2} \cdot \frac{1}{4}=\frac{3}{8}$ and the probability that a black marble is chosen from Box 2 is $\frac{1}{2} \cdot \frac{1}{2}+\frac{1}{2} \cdot \frac{3}{4}=\frac{5}{8}$.

Thus, after this choice is made, there is a probability of $\frac{3}{8}$ that Box 3 contains 2 gold marbles and 3 black marbles, and a probability of $\frac{5}{8}$ that Box 3 contains 1 gold marble and 4 black marbles.

Finally, the probability that a gold marble is chosen from Box 3 equals the probability that Box 3 contains 2 gold marbles and 3 black marbles times the probability of choosing a gold marble in this situation (that is, $\frac{3}{8} \cdot \frac{2}{5}$ ) plus the probability that Box 3 contains 1 gold marble and 4 black marbles times the probability of choosing a gold marble in this situation.

In other words, this probability is $\frac{3}{8} \cdot \frac{2}{5}+\frac{5}{8} \cdot \frac{1}{5}=\frac{11}{40}$.

ANSwer: (A)

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