Question

In a housing society, half of the families have a single child per family, while the remaining half have two children per family. The probability that a child picked at random, has a sibling is

• A. 0.667

Answer 1

The correct option is A 0.667
$\text{Let}{E}_1=\text{One children family}$
$E_2=\text{Two children family}$
A = Picking a child
By Baye's theorem

$P\left(E_2/A\right)=\frac{P\left(E_2\right)P\left(A/E_2\right)}{P\left(E_1\right)P\left(A/E_1\right)+P\left(E_2\right).P\left(A/E_2\right)}$

$=\frac{\frac{1}{2}.x}{\frac{1}{2}.\frac{x}{2}+\frac{1}{2}.x}=\frac{2}{3}=0.667$

Method II:
Let there are 50 families of each type. So 50 children have no siblings and 100 children have siblings.

So, required probability $=\frac{{}^{100}{C}_1}{{}^{150}{C}_1}=\frac{100}{150}=\frac{2}{3}$