Assume that each born child is equally likely to be a boy or a girl. If a family has two children,

Let b and g represent the boy and the girl child respectively. If a family has two children, the sample space will be

S = {(b, b), (b, g), (g, b), (g, g)}

Let A be the event that both children are girls.

$\therefore \mathrm{A}=\{(g, g)\}$

(i) Let B be the event that the youngest child is a girl.

$\therefore \mathrm{B}=[(b, g),(g, g)]$

$\Rightarrow \mathrm{A} \cap \mathrm{B}=\{(g, g)\}$

$\therefore \mathrm{P}(\mathrm{B})=\frac{2}{4}=\frac{1}{2}$

$\mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{1}{4}$

The conditional probability that both are girls, given that the youngest child is a girl, is given by P (A|B).

$P(A \mid B)=\frac{P(A \cap B)}{P(B)}=\frac{\frac{1}{4}}{\frac{1}{2}}=\frac{1}{2}$

Therefore, the required probability is $\frac{1}{2}$.

(ii) Let C be the event that at least one child is a girl.

$\therefore \mathrm{C}=\{(b, g),(g, b),(g, g)\}$

$\Rightarrow \mathrm{A} \cap \mathrm{C}=\{g, g\}$

$\Rightarrow \mathrm{P}(\mathrm{C})=\frac{3}{4}$

$\mathrm{P}(\mathrm{A} \cap \mathrm{C})=\frac{1}{4}$

The conditional probability that both are girls, given that at least one child is a girl, is given by P(A|C).

Therefore, $\mathrm{P}(\mathrm{A} \mid \mathrm{C})=\frac{\mathrm{P}(\mathrm{A} \cap \mathrm{C})}{\mathrm{P}(\mathrm{C})}=\frac{\frac{1}{4}}{\frac{3}{4}}=\frac{1}{3}$