# Ex.15.1 Q8 Probability Solution - NCERT Maths Class 9

## Question

The distance (in km) of \(40\) engineers from their residence to their place of work were found as follows: What is the empirical probability that an engineer lives:

\(5\) | \(3\) | \(10\) | \(20\) | \(25\) | \(11\) | \(13 \) | \(7\) | \(12\) | \(31\) |

\(19\) | \(10\) | \(12\) | \(17\) | \(18\) | \(11\) | \(32\) | \(17\) | \(16\) | \(2\) |

\(7\) | \(9\) | \(7\) | \(8\) | \(3\) | \(5\) | \(12\) | \(15\) | \(18\) | \(3\) |

\(12\) | \(14\) | \(2\) | \(9\) | \(6\) | \(15\) | \(15\) | \(7\) | \(6\) | \(12\) |

(i) less than \(7 \,\rm km\) from her place of work?

(ii) more than or equal to \(7\,\rm km\) from her place of work?

(iii) within \(½\,\rm km\) from her place of work?

## Text Solution

**What is known?**

No of engineers and distance of their work place to the residence.

**What is unknown?**

Probability of distance of work place of engineers from their residence.

**Reasoning:**

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

\(\begin{align}{P}({E})=\frac{ \begin{pmatrix} \text { Number of trials in which }\\ \text{the event happened } \end{pmatrix} }{ \text { The total number of trials } }\end{align}\)

Use probability to derive the solution where

Probability (no of engineers travelling certain distance from residence to their place of work)

\[\begin{align}&\\ &=\frac{\text { Number of engineers }}{\text { Total number of engineers }}\end{align}\]

**Steps:**

Total no of engineers \(= 40\)

By arranging the numbers in increasing order

\(2, 2, 3, 3, 3, 5, 5, 6, 6, 7, 7, 7,\\ 7, 8, 9, 9, 10, 10, 11, 11, 12, \\ 12, 12, 12, 12, 13, 14, 15, 15, \\ 15, 16,17, 17, 18, 18, 19, 20, \\ 25, 31, 32.\)

No. of engineers who live less than \(7\,\rm km\) from their place of work \(= 9\)

No. of engineers who live more than or equal to \(7\,\rm km\) from their work place \(= 31\)

No. of engineers who live within \(½ \,\rm km\) from their place of work\(=0\)

(i)

Probability \(P1\)(An engineer lives less than \(7\,\rm km\) from their work place)

\(\begin{align}\\&=\frac{ \begin{pmatrix} \text { Number of engineers who} \\ \text{ live less than $7$ km } \\ \text{from their work place } \end{pmatrix} }{\text { Total number of engineers }}\\ &=\frac{9}{40}\end{align}\)

(ii)

Probability \(P2\)(An engineer lives more than or equal to \(7\,\rm km\) from their work place)

\(\begin{align}\\&=\frac{ \begin{pmatrix} \text { Number of engineers who} \\ \text{ live more than or equal to} \\ 7 \text{ km from their work place } \end{pmatrix} }{\text { Total number of engineers }}\\ &=\frac{31}{40}\end{align}\)

(iii)

Probability \(P3\)(An engineer lives within \(1/2\,\rm km\) from their work place)

\(\begin{align}\\&=\frac{ \begin{pmatrix} \text { Number of engineers who }\\ \text{ live within 1/2 km } \\ \text{from their work place } \end{pmatrix}}{\text { Total number of engineers }}\\ &=\frac{0}{40}=0\end{align}\)