A family has three children, and each child is equally likely to be a boy or a girl. If it is known that the eldest child is a boy, then the probability that the family has exactly two boys is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{1}{2}\)
Write the children from the eldest to the youngest. Since the eldest is a boy, the reduced sample space has 4 equally likely outcomes:
BBB, BBG, BGB, BGG
Exactly two boys means exactly one more boy among the two younger children:
BBG, BGB (2 outcomes)
So
\( \displaystyle P = \frac{2}{4} = \frac{1}{2} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{3}{8}\) — Incorrect. This is the probability of exactly two boys among all 8 outcomes, without the given condition.
Option (3) \(\displaystyle\frac{1}{4}\) — Incorrect. This is the probability of BBB only, that is, three boys given the eldest is a boy.
Option (4) \(\displaystyle\frac{3}{4}\) — Incorrect. This counts BBB, BBG and BGB, that is, at least two boys.