One class contains 5 girls and 10 boys and a second class co

One class contains 5 girls and 10 boys and a second class contains 13 boys and 12 girls. A student is randomly picked from the second class and transferred to the first one. After that, a student is randomly chosen from the first class. What is the probability that this student is a boy?

Solution

There are two mutually exclusive possibilities:

(a) A boy is transferred, and then a boy is chosen.

(b) A girl is transferred, and then a boy is chosen.

(a) A boy is transferred with probability 13/25, after which the first class now has 5 girls and 11 boys. Then a boy is chosen from this class with probability 11/16, so the total probability of the chain is

13/25 * 11/16 = 143/400

(b) A girl is transferred with probability 12/25, after which the first class now has 6 girls and 10 boys. Then a boy is chosen from this class with probability 10/16, so the total probability of the chain is

12/25 * 10/16 = 120/400

Thus the total probability that a boy is eventually chosen is 143/400 + 120/400 = 263/400