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:

No. of Families	50	68	182	74	26
No. of Children	0	1	2	3	4

Find the probability that a family (a) 3 children, (b) 2 children.

Answer 1

Hint: Here, we need to find the probability that a family has 3 children or 2 children. First, we will find the total number of outcomes. Using the table, we will obtain the number of families that have 3 children, and the number of families that have 2 children. Finally, we will use the formula for probability to get the required probabilities.

Formula Used:
The probability of an event is given by \[P\left( E \right) = \dfrac{{{\text{Number of favourable outcomes}}}}{{{\text{Number of total outcomes}}}}\].

Complete step-by-step answer:
(i)
First, we will find the number of favourable outcomes and total outcomes.
The total number of families is 400.
Therefore, the total number of outcomes is 400.
Next, we will use the table to find the number of families that have 3 children.
We can observe that 74 families have 3 children.
Therefore, the number of favourable outcomes is 74.
Finally, we will use the formula for probability of an event to calculate the probability that a family has 3 children.
We know that the probability of an event \[E\] is given by \[P\left( E \right) = \dfrac{{{\text{Number of favourable outcomes}}}}{{{\text{Number of total outcomes}}}}\].
Let \[E\] be the event that that a family has 3 children.
Substituting 74 for the number of favourable outcomes, and 400 for the number of total outcomes in the formula, we get
\[ \Rightarrow P\left( E \right) = \dfrac{{74}}{{400}}\]
Simplifying the expression, we get
\[ \Rightarrow P\left( E \right) = \dfrac{{37}}{{200}}\]
Therefore, the probability that a family has 3 children is \[\dfrac{{37}}{{200}}\], or \[0.185\].
(ii)
First, we will find the number of favourable outcomes and total outcomes.
The total number of families is 400.
Therefore, the total number of outcomes is 400.
Next, we will use the table to find the number of families that have 2 children.
We can observe that 182 families have 2 children.
Therefore, the number of favourable outcomes is 182.
Finally, we will use the formula for probability of an event to calculate the probability that a family has 2 children.
Let \[F\] be the event that a family has 2 children.
Substituting 182 for the number of favourable outcomes, and 400 for the number of total outcomes in the formula for probability, we get
\[ \Rightarrow P\left( F \right) = \dfrac{{182}}{{400}}\]
Simplifying the expression, we get
\[ \Rightarrow P\left( F \right) = \dfrac{{91}}{{200}}\]
Therefore, the probability that a family has 2 children is \[\dfrac{{91}}{{200}}\], or \[0.455\].

Note: The table given is a frequency distribution table. A frequency distribution table shows the various observations in a series, and their respective frequencies. A frequency is the number of times an observation appears in a series.