Question

A couple has two children.

Find the probability that both children are males, if it is known that atleast one of the children is male.

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

Answer 1

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 males

F: atleast one of the children is a male

$\therefore$ E={bb},F={bg,gb,bb}

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

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

$\Rightarrow$ n(F)=3

and $E\cap F=bb=E\Rightarrow n(E\cap F)=n\left(E\right)=1Requiredprobability=P\left(\frac{E}{F}\right)=\frac{P(E\cap F)}{P\left(F\right)}=\frac{P\left(E\right)}{P\left(F\right)}=\frac{1/4}{3/4}=\frac{1}{3}$