Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the condition probability that both are girls given that (i) the youngest is a girl, (ii) at least one is the girl?

.: Let $b$ denote the boy and $g$ denote the girl.

$S = \left\{ {{b_1}{b_2},bg,gb,{g_1}{g_2}} \right\}$

which contains four equally like sample points.

$S = \left\{ {{b_1}{b_2},bg,gb,{g_1}{g_2}} \right\}$

which contains four equally like sample points.
Let $E$ : 'both children are girls' $\Rightarrow E = \left\{ {{g_1}{g_2}} \right\} \Rightarrow P(E) = 1/4$

(i) Let $F$ : the youngest is a girl', then $F = \left\{ {bg,{g_1}{g_2}} \right\}$

$\Rightarrow P(F) = 2/4 \Rightarrow E \cap F = \left\{ {{g_1}{g_2}} \right\} \Rightarrow q(E \cap F) = 1/4$

$\therefore$ Required probability $= P(E|F) = \cfrac{{P(E \cap F)}}{{P(F)}} = \cfrac{{1/4}}{{2/4}} = \cfrac{1}{2}$

(ii) Let $F:$ atleast one is a girl', then $F = \left\{ {bg,gb,{g_1}{g_2}} \right\}$

$\Rightarrow$ $P(F) = 3/4$

$\therefore$ Required probability $= P(E|F) = \cfrac{{P(E \cap F)}}{{P(F)}} = \cfrac{{1/4}}{{3/4}} = \cfrac{1}{3}$