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 .