# Ex.15.1 Q5 Probability Solution - NCERT Maths Class 9

## Question

An organization selected \(2400\) families at random and surveyed them to determine a relationship between income level and the number of vehicles in a family. The information gathered is listed in the table below.

Suppose a family is chosen. Find the probability that the family chosen is

(i) earning \(\rm Rs 10000 – 13000\) per month and owning exactly \(2\) vehicles.

(ii) earning \(\rm Rs 16000\) or more per month and owning exactly \(1\) vehicle.

(iii) earning less than \(\rm Rs 7000\) per month and does not own any vehicle.

(iv) earning \(\rm Rs 13000 – 16000\) per month and owning more than \(2\) vehicles.

(v) owning not more than \(1\) vehicle.

## Text Solution

**What is known?**

Family monthly income and vehicles per family.

**What is unknown?**

Probability of family owning \(0\) or \(1\) or \(2\) or above \(2\) vehicles based on the incomes.

**Reasoning:**

The empirical probability \(P(E)\) of an event \(E\) happening, is given by:

\[\begin{align} & { = \frac{ {\text{Number of times }} {\text{heads occur}} }{{{\text{Total number of tosses}}}}} \\ &=\frac{72}{200} \\ &=\frac{9}{25} \end{align}\]

Use probability to derive the solution where

Probability (of family owning vehicle based on the earnings)

\[\begin{align} = \frac{ \begin{bmatrix} \text{No of vehicles owned} \\ \text{ based on earnings} \end{bmatrix} }{ \text{Total number} \\ \text{ of families} } \end{align}\]

**Steps:**

Total no of families \(= 2400\)

(i)

Probability \(P1\) (a family earning \(\rm Rs 13000\) to \(\rm Rs 10000\) per month and owning exactly \(2\) vehicles)

\[\begin{align} &=\!\frac{ \begin{bmatrix} \text{No of a families earning Rs 13000}\\\text{to Rs 10000 per month and} \\ \text{owning exactly 2 vehicles} \end{bmatrix} }{{{\text{Total number of families}}}}\\& = \,\frac{{{\rm{29}}}}{{{\rm{2400}}}}\end{align}\]

(ii)

Probability \(P2\) (a family earning \(\rm Rs 16000\) per month and owning exactly \(1\) vehicle)

\[\begin{align} &=\frac{ \begin{bmatrix} \text{No of a families earning}\\\text{ Rs $16000$ per month and }\\\text{owning exactly $1$ vehicle} \end{bmatrix} }{{{\text{Total number of families}}}}\\ \\ &= \frac{{{\rm{579}}}}{{{\rm{2400}}}}\, = \frac{{193}}{{800}}\end{align}\]

(iii)

Probability \(P3\) (a family earning less than \(\rm Rs 7000\) per month and does not own any vehicle)

\[ \begin{align}& =\! \frac{\begin{bmatrix} \text{No of a families earning} \\\text{less than Rs $7000$ per month} \\ \text{and does not own any vehicle} \end{bmatrix} }{{{\text{Total number of families}}}}\\ \\ &= \,\frac{{10}}{{{\rm{2400}}}}\\&= \frac{1}{{240}}\end{align}\]

(iv)

Probability \(P4\) (a family earning Rs \(13000\) to Rs \(16000\) per month and owning more than \(2\) vehicles)

\[\begin{align} &= \frac{ \begin{bmatrix} \text{No of a families earning } \\ \text{Rs $16000$ to Rs $13000$ per } \\ \text{month and owning more }\\\text{ than $2$ vehicles}\end{bmatrix} }{{{\text{Total number of families}}}}\\ & = \,\frac{{{\rm{25}}}}{{{\rm{2400}}}} = \frac{1}{{96}}\end{align}\]

(v)

Probability \(P5\) (a family owning \(0\) or \(1\) vehicle)

Total number of families

\(\begin{align} & = \,\frac{\begin{bmatrix} 10 + 0 + 1 + 2 + 1 + 160 \\ + 305 + 535 + 469 + 579 \end{bmatrix} }{{{\rm{2400}}}} \\ \\ &= \frac{{2062}}{{2400}} = \frac{{1031}}{{1200}}\end{align}\)