A couple has two children

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

Here, the sample space is S= {bb,bg,gb,gg} where b stands for a boy and 'g' for a girl; first letter standing for elder child and second for the younger child.

$\therefore$ n(S)=4

Let E: both the children are females and

F: elder child is a female

$\therefore$ E={gg}, F{gb,gg} and $E\cap F$ ={gg}=E

$\Rightarrow$ n(E)=1,n(F)=2

$Requiredprobability=P\left(\frac{E}{F}\right)=\frac{P(E\cap F)}{P\left(F\right)}=\frac{P\left(E\right)}{P\left(F\right)}=\frac{1/4}{3/4}=\frac{1}{2}$