Question

A family has 2 children,what is the probability that both children

1)are boys,given that atleast 1 of them is a boy

2)are girls,if it is known that elder child is a girl

Answer 1

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 boys.
So, $A=\left\{\left(b,b\right)\right\}$
1) Let B be the event that at least one child is a boy.
So, $B=\left\{\left(b,b\right),\left(b,g\right),\left(g,b\right)\right\}$
$\Rightarrow A\cap B=\left\{\left(b,b\right)\right\}$
$\Rightarrow P\left(B\right)=\frac{3}{4}$
$P(A\cap B)=\frac{1}{4}$
The conditional probability that both are boys, given that at least one child is a boy, is given by P(A|B).
Therefore, $P(A/B)=\frac{P(A\cap B)}{P\left(B\right)}=\frac{{\displaystyle \frac{1}{4}}}{{\displaystyle \frac{3}{4}}}=\frac{1}{3}$
2) Let C be the event that both child are girls and D be the event that an elder child is a girl.
So, $C=\left\{(g,g)\right\}$ and $D=\left\{(g,g),(g,b)\right\}$
$\Rightarrow C\cap D=\left\{(g,g)\right\}$
Now, $P\left(D\right)=\frac{2}{4}=\frac{1}{2}$
and $P(C\cap D)=\frac{1}{4}$
The conditional probability that both are girls, given that an elder child is a girl, is given by P (C|D)
Therefore, $P(C/D)=\frac{P(C\cap D)}{P\left(D\right)}=\frac{{\displaystyle \frac{1}{4}}}{{\displaystyle \frac{1}{2}}}=\frac{1}{2}$