A couple has two children,

(i) Find the probability that both children are males, if it is known that at least one of the children is male.

(ii) Find the probability that both children are females, if it is known that the elder child is a female.

.: Sample space is $S = \{ ff,fm,mf,mm\}$

where $f =$ female, $m =$ male

(i) Let $A =$ both are male i.e., $\{ mm\} \Rightarrow P(A) = \cfrac{1}{4}$

$B =$ atleast one is male i.e., $\{ mm,fm,mf\} \Rightarrow P(B) = \cfrac{3}{4}$

$A \cap B = \{ mm\} \Rightarrow P(A \cap B) = \cfrac{1}{4}$

Required probability $= P(A|B) = \cfrac{{P(A \cap B)}}{{P(B)}} = \cfrac{{1/4}}{{3/4}} = \cfrac{1}{3}$

(ii) Let $A =$ both are female i.e., $\{ ff\} \Rightarrow P(A) = \cfrac{1}{4}$

$B =$ the elder is a female i.e., $\{ ff,fm\} \Rightarrow P(B) = \cfrac{2}{4}$

$A \cap B = \{ ff\} \Rightarrow P(A \cap B) = \cfrac{1}{4}$

Required probability $= P(A|B) = \cfrac{{P(A \cap B)}}{{P(B)}} = \cfrac{{1/4}}{{2/4}} = \cfrac{1}{2}$