Question

An experiment consists of recording boy–girl composition of families with $2$ children.
$\left(i\right)$ What is the sample space if we are interested in knowing whether it is a boy or girl in the order of their births ?
$\left(ii\right)$ What is the sample space if we are interested in the number of girls in the family ?

Answer 1

$\left(i\right)$ Given :
An experiment consists of recording boy-girl composition of families with $2$ children.
A family with $2$ children may be either boy or girl or both.
Let $B$ denote boy and $G$ denote girl.
Depending upon what is first birth and what is second birth, possible outcomes are $(BB,GG,BG,GB)$.
Hence the required sample space is $S=\{BB,BG,GB,GG\}$.

$\left(ii\right)$ Given : An experiment consists of recording boy-girl composition of families with $2$ children.
In the family with two children can have $0$ girl or $1$ girl or $2$ girls.
So, the possible outcomes are $(0,1,2)$
Hence the required sample space is $S=\{0,1,2\}$.