A couple has 3 children each child is equally likely to be a boy or a girl. The probability that the eldest child is a girl given that they have at least one boy is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{3}{7}\)
Let A: the eldest child is a girl, and B: there is at least one boy. By conditional probability, \(P(A|B) = \frac{P(A \cap B)}{P(B)}\).
Writing the children from eldest to youngest, the 8 equally likely outcomes are
BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG
B is every outcome except GGG, so \(n(B) = 7\).
\(A \cap B\): the eldest is a girl and there is at least one boy: GBB, GBG, GGB, so \(n(A \cap B) = 3\).
\(\displaystyle P(A|B) = \frac{n(A \cap B)}{n(B)} = \frac{3}{7}\)