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(C) Let $B$ denote a boy and $G$ denote a girl. The sample space $S$ for a family with $3$ children is: $S = \{BBB, BBG, BGB, GBB, BGG, GBG, GGB, GGG\

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$\frac{1}{7}$

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(C) Let $B$ denote a boy and $G$ denote a girl. The sample space $S$ for a family with $3$ children is: $S = \{BBB, BBG, BGB, GBB, BGG, GBG, GGB, GGG\}$. Total number of outcomes $n(S) = 8$. Let $A$ be the event that all three children are girls,so $A = \{GGG\}$. Let $E$ be the event that at least one child is a girl,so $E = \{BBG, BGB, GBB, BGG, GBG, GGB, GGG\}$. The number of outcomes in $E$ is $n(E) = 7$. The intersection $A \cap E = \{GGG\}$,so $n(A \cap E) = 1$. The conditional probability $P(A|E)$ is given by: $P(A|E) = \frac{n(A \cap E)}{n(E)} = \frac{1}{7}$.

$A$ family has $3$ children. The probability that all the three children are girls,given that at least one of them is a girl is:

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