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 atleast one boy is:

$\frac{3}{7}$

The correct answer is Option (2) → $\frac{3}{7}$

Sample space for 3 children (B = boy, G = girl):

$S = \{BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG\}$

Total outcomes = 8

Favorable outcomes with at least one boy = all except GGG

$A = \{BBB, BBG, BGB, BGG, GBB, GBG, GGB\}$

$n(A) = 7$

Among these, outcomes where eldest child is a girl: $GBB, GBG, GGB$

$n(A \cap E) = 3$

Required probability:

$P(E \mid A) = \frac{P(E \cap A)}{P(A)} = \frac{3}{7}$