In a two-child family, given that at least one child is a boy born on Tuesday, what is the probability that both children are boys?

• A. 1/2
• B. 1/3
• C. 7/27
• D. 13/27

Answer: (D)

The situation is about conditional probability: given that at least one child is a boy born on Tuesday, what is the chance both children are boys?

Treat each child as having 14 equally likely outcomes (boy or girl, each with 7 possible days). The event we’re conditioning on is that at least one child is a boy on Tuesday. It’s easier to find the complement: no child is a boy born on Tuesday. For one child there are 13 outcomes that are not “boy on Tuesday” out of 14, so for two independent children the probability both miss boy-on-Tuesday is (13/14)². Therefore P(at least one boy on Tuesday) = 1 − (13/14)² = 27/196.

Now look at the event that both children are boys and at least one is born on Tuesday. Among two boys there are 7×7 = 49 day combinations; those with no Tuesday are 6×6 = 36, so at least one Tuesday gives 49 − 36 = 13 favorable combinations. Thus P(both boys and at least one Tuesday) = 13/196.

So the conditional probability is (13/196) ÷ (27/196) = 13/27.