Question

A survey of $400$ families of a town was conducted to find out how many children are there in a family.
The result of the survey is given below

Number of families	$50$	$68$	$182$	$74$	$26$
Number of children	$0$	$1$	$2$	$3$	$4$

Find the probability that a family has
a) $3$ children
b) $2$ children

Answer 1

Hint: Probability of any event is the ratio of number of favourable outcomes to the number of possible outcomes. Here, we will, in each case, deduce these parameters and get the required probabilities.

Complete step-by-step answer:
Here, Total number of families as given in the problem $ = 400$
$\therefore $ Total number of possible outcomes or sample space , $n\left( S \right) = 400$
(a) Number of families with $3$ children $ = 74$
 $ \Rightarrow \,$Number of favourable outcomes, $n\left( E \right) = 74$
$\therefore $ Probability that a family has $3$ children,
$p(E) = \dfrac{{n\left( E \right)}}{{n\left( S \right)}}$
$ \Rightarrow \,p(E) = \dfrac{{74}}{{400}}$
$ \Rightarrow \,p(E) = \dfrac{{37}}{{200}}\, = \,0.185$
$\therefore $ Probability that a family has $3$ children is $\dfrac{{37}}{{200}}$ .
 (b) Number of families with $2$ children $ = 182$
$ \Rightarrow \,$Number of favourable outcomes, $n\left( E \right) = 182$
$\therefore $ Probability that a family has $2$ children, $p(E) = \dfrac{{n\left( E \right)}}{{n\left( S \right)}}$
$ \Rightarrow \,p(E) = \dfrac{{182}}{{400}}$
$ \Rightarrow \,p(E) = \dfrac{{91}}{{200}}\, = \,0.455$
$\therefore $ Probability that a family has $2$ children is $\dfrac{{91}}{{200}}$ .

Note: If we are asked the probability that a family has $5$ or more children, the answer will be $0$.
Also, if we are asked the probability that a family has $3$ or more children, the answer will be $\dfrac{{74 + 26}}{{400}} = \dfrac{{100}}{{400}} = \dfrac{1}{4} = 0.25$ , because the number of favourable events will be the sum of the number of families having $3$ children and the number of families having $4$ children but the sample space will remain the same.