Question

    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) 11/ 40 (B) 3 /10 (C) 13/ 40 (D) 7 /20 (E) 3 /8  

Answer 1

While transferring marble from Box 1 to box 2,
Probability that it is black = 1/2 
Probability that it is Golden = 1/2 

So,if a black marble is transferred,
Box 2 will have 1 golden marble and 3 black marbles.

And if a golden one is transferred,
Box 2 will have 2 golden and 2 black marbles 

So, probability that a golden marble is transferred to the second box = 1/2(1/2+1/4) = 3/8

Probability that a black marble is chosen from box 2 = 1/2(1/2 +3/4) = 5/8

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, 3 /8 · 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 3/8 · 2 /5 + 5/8 · 1/5 = 11/40 . 

So, the answer is (A) 11/40.

Answer 2

When a marble is chosen from Box 1, the probability is 1 2 that it will be gold and the probability is 1 2 that it will be black. Thus, after this choice is made, there is a probability of 1 2 that Box 2 contains 2 gold marbles and 2 black marbles, and there is a probability of 1 2 that Box 2 contains 1 gold marble and 3 black marbles. In the first case (which occurs with probability 1 2 ), the probability that a gold marble is chosen from Box 2 is 2 4 = 1 2 and the probability that a black marble is chosen from Box 2 is 2 4 = 1 2 . In the second case (which occurs with probability 1 2 ), the probability that a gold marble is chosen from Box 2 is 1 4 and the probability that a black marble is chosen from Box 2 is 3 4 .
Therefore, the probability that a gold marble is chosen from Box 2 is 1 2 · 1 2 + 1 2 · 1 4 = 3 8 and the probability that a black marble is chosen from Box 2 is 1 2 · 1 2 + 1 2 · 3 4 = 5 8 . Thus, after this choice is made, there is a probability of 3 8 that Box 3 contains 2 gold marbles and 3 black marbles, and a probability of 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, 3 8 · 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 3 8 · 2 5 + 5 8 · 1 5 = 11 40 .