NCERT Solutions for Class 9 Maths Chapter 15 - Probability

NCERT Solutions for Class 9 Maths Chapter 15, "Probability," is based on the concept of probability and its applications in various fields. The chapter consists of the following exercises:

Exercise 15.1: This exercise discusses the theoretical approach to probability and how it is calculated.

Access NCERT Solutions for Class 9 Mathematics Chapter 15 – Probability

Exercise – 15.1

1. In a cricket match, a batswoman hits a boundary 6 times out of 30 balls she plays. Find the probability that she did not hit a boundary.

Ans: No. of time she hit boundary: $ 6$ 

Total No. of balls played: $ 30$ 

$ \therefore $ No. of times she does not hit boundary: $ 30 - 6 = 24$ 

Probability that she does not hit the ball: 

$ \dfrac{{24}}{{30}} = \dfrac{4}{5} = 0.8$ 

2. 1500 families with 2 children were selected randomly, and the following data were recorded:

Compute the probability of a family, chosen at random, having

(i) 2 girls

(ii) 1 girl

(iii) No girl

Also check whether the sum of these probabilities is 1.

Ans: Total number of families = $ 1500$ 

1. P1 (family with $ 2$  girls)

$ \dfrac{{475}}{{1500}} = \dfrac{{19}}{{60}} = 0.317$ 

2. P2 (family with $ 1$  girl)

$ \dfrac{{814}}{{1500}} = \dfrac{{407}}{{750}} = 0.54$ 

3. P3 (family with $ 0$  girls)

$ \dfrac{{211}}{{1500}} = 0.14$ 

Sum of Probabilities:

$ ~~\frac{475}{1500}+\frac{814}{1500}+\frac{211}{1500}=\frac{1500}{1500}\text{ }=~1$ 

3. In a particular section of Class IX, $ {40}$  students were asked about the months of their birth and the following graph was prepared for the data so obtained:

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Ans: Total no. of students: $ 40$ 

No. of students born in august: $ 6$ 

Probability of a student born in august: 

$ \dfrac{6}{{40}} = \dfrac{3}{{20}} = 0.15$ 

4. Three coins are tossed simultaneously $ {200}$ times with the following frequencies of different outcomes:

If the three coins are simultaneously tossed again, compute the probability of 2 heads coming up.

Ans: Total number of tosses: $ 200$ 

Number of times we obtained $ 2$  heads: $ 72$ 

Probability of obtaining $ 2$  heads again: $ \dfrac{{72}}{{200}} = \dfrac{9}{{25}} = 0.36$ 

5. 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:

1. Earning Rs. $ {10000 - 13000}$  per month and owning exactly 2 vehicles.

Ans: Total number of families surveyed: $ 2400$ 

Number of families earning $ 10000 - 13000$  and own exactly $ 2$ vehicles: $ 29$ 

Probability of choosing this family: $ \dfrac{{29}}{{2400}}$ 

2. Earning Rs. $ {16000}$ or more per month and owning exactly 1 vehicle. 

Ans: Number of families earning Rs. $ 16000$ or more per month and owning exactly $ 1$  vehicle: $ 579$ 

Probability of choosing this family:$ \dfrac{{579}}{{2400}} = \dfrac{{193}}{{800}}$ 

3. Earning less than Rs. $ {7000}$ per month and does not own any vehicle.

Ans: Number of families earning less than Rs. 7,000 per month and does not own any vehicle: $ 10$  

Probability of choosing this family: $ \dfrac{{10}}{{2400}} = \dfrac{1}{{240}}$ 

4. Earning Rs. $ {13000 - 16000}$ per month and owning more than 2 vehicles.

Ans: Number of families earning Rs. $ 13000 - 16000$ per month and owning more than 2 vehicles: $ \dfrac{{25}}{{2400}} = \dfrac{1}{{96}}$ 

5. Owning not more than 1 vehicle.

Ans: Number of families owning not more than 1 vehicle:$ \dfrac{{25 + 27 + 29 + 59 + 82 + 2 + 1 + 25 + 88}}{{2400}} = \dfrac{{338}}{{2400}} = \dfrac{{169}}{{1200}}$ 

6. A teacher wanted to analyse the performance of two sections of students in a mathematics test of 100 marks. Looking at their performances, she found that a few students got under $ {20}$  marks and a few got $ {70}$  marks or above. So she decided to group them into intervals of varying sizes as follows:\[{\mathbf{0}} - {\mathbf{20}},{\text{ }}{\mathbf{20}} - {\mathbf{30}},{\text{ }} \ldots ,{\text{ }}{\mathbf{60}} - {\mathbf{70}},{\text{ }}{\mathbf{70}} - {\mathbf{100}}\]. Then she formed the following table:

1. Find the probability that a student obtained less than 20% in the mathematics test. 

Ans: Total number of students:90

 $\text{No}\text{. of students obtained less than 20 }\!\!%\!\!\text{  marks =}$ is 7

Probability of required event: $ \dfrac{7}{{90}}$ 

2. Find the probability that a student obtained marks 60 or above.

Ans: No. of students who obtained marks 60 or above: $ 15 + 8 = 23$ 

Probability of required event: $ \dfrac{{23}}{{90}}$ 

7. To know the opinion of the students about the subject statistics, a survey of $ {200}$  students was conducted. The data is recorded in the following table.

Find the probability that a student chosen at random

i. Likes statistics

Ans: Total number of students: $ 200$ 

Number of students who like the subject: $ 135$ 

Required probability: $ \dfrac{{135}}{{200}} = \dfrac{{27}}{{40}}$ 

ii. Does not like it

Ans: Number of students who does not like the subject: $ 65$ 

Required Probability: $ \dfrac{{65}}{{200}} = \dfrac{{13}}{{40}}$ 

8. 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:

i. Less than 7 km from her place of work

Ans: Total number of engineers: $ 40$ 

Number of engineers who reside within less than 7km from their work: $ 9$ 

Probability of choosing one of them: $ \dfrac{9}{{40}}$ 

ii. More than or equal to 7 km from her place of work?

Ans: Probability of choosing an engineer who resides more than or equal to 7km from their work: $ 1 - \dfrac{9}{{40}} = \dfrac{{31}}{{40}}$ 

iii. within ½ km from her place of work?

Ans: Number of engineers who reside within $ \dfrac{1}{2}$ km from their work: 0

Probability of choosing one such engineer: $ \dfrac{0}{{40}} = 0$ 

9. Activity 

Note the frequency of two – wheeler, three-wheeler and four-wheelers going past during a time – interval in front of the school gate. Find the probability that any one vehicle out of the total vehicles you have observed is a two-wheeler.

Ans: Let the number of two-wheeler vehicles you observed be: x

Number of three-wheeler vehicles be y

And number of four-wheeler vehicles be z

Now, probability of any vehicle be a two-wheeler is: $ \dfrac{x}{{x + y + z}}$ 

Note: One has to physically observe vehicles and note the values of x, y and z.

10. Activity 

Ask all the students in your class to write a 3-digit number. Choose any student from the room at random. What is the probability that the number written by him/her is divisible by 3? Remember that a number is divisible by 3, if the sum of it’s digits is divisible by 3.

Ans: A dataset is provided below:$ 561,\text{ }891,\text{ }352,\text{ }658,\text{ }735,\text{ }187,\text{ }\!\!~\!\!\text{  }286,\text{ }374,\text{ }537,\text{ }385,\text{ }165,\text{ }537,\text{ }438,\text{ }682,\text{ }658,\text{ }572,\text{ }647,\text{ }901,\text{ }374,\text{ }977,\text{ }531,$ 

$ \text{ }909,\text{ }549,\text{ }286,\text{ }\!\!~\!\!\text{  }520,\text{ }170,\text{ }102,\text{ }328,\text{ }856\text{ }$ 

Total numbers: \[29\]

Numbers divisible by 3: \[11\]

Probability that a number is divisible by 3: $ \dfrac{{11}}{{29}}$ 

11. Eleven bags of wheat flour, each marked 5 kg, actually contained the following weights of flour (in kg): 

\[{\mathbf{4}}.{\mathbf{97}},{\text{ }}{\mathbf{5}}.{\mathbf{05}},{\text{ }}{\mathbf{5}}.{\mathbf{08}},{\text{ }}{\mathbf{5}}.{\mathbf{03}},{\text{ }}{\mathbf{5}}.{\mathbf{00}},{\text{ }}{\mathbf{5}}.{\mathbf{06}},{\text{ }}{\mathbf{5}}.{\mathbf{08}},{\text{ }}{\mathbf{4}}.{\mathbf{98}},{\text{ }}{\mathbf{5}}.{\mathbf{04}},{\text{ }}{\mathbf{5}}.{\mathbf{07}},{\text{ }}{\mathbf{5}}.{\mathbf{00}}\]

Find the probability that any of these bags chosen at random contains more than 5 kg of flour.

Ans: Total number of bags:  $ 11$ 

Number of bags which contain more than 5kg wheat: $ 7$ 

Hence the required probability is: $ \dfrac{7}{{11}}$ 

12. The below frequency distribution table represents the concentration of Sulphur dioxide in the air in parts per million of a certain city for \[{\mathbf{30}}\]days. Using this table, find the probability of the concentration of Sulphur dioxide in the interval \[{\mathbf{0}}.{\mathbf{12}} - {\mathbf{0}}.{\mathbf{16}}\]on any of these days.

Ans: Total number of days, data is recorded: $ 30$ 

Number of days found with concentration\[0.120.16\]: 2

Hence the required probability is: $ \dfrac{2}{{30}} = \dfrac{1}{{15}}$ 

13. The below frequency distribution table represents the blood groups of \[{\mathbf{30}}\] students of a class. Use this table to determine the probability that a student of this class, selected at random, has blood group AB.

Ans: Total number of students: $ 30$ 

Number of students with blood group AB: $ 3$ 

Probability of choosing a student with blood group AB: $ \dfrac{3}{{30}} = \dfrac{1}{{10}}$ 

Important Points to Remember before Solving NCERT Class 9 Maths Chapter 15 - Probability

• Probability is the possibility of event occurrence. 

• Probability of occurrence of an event can be calculated as

Probability = P (E) =  Number of trials in which the event happenedTotal number of trials

P(E) represents the probability of event E to occur.

• Probability of the event lies between 0 and 1. 0 represents that event will not occur whilst 1 represents the event will definitely occur.

Types of Questions Asked in Probability Class 9 Maths NCERT 

Below given is the list of types of questions asked under Chapter 15 of NCERT Maths for Class 9. 

• Related to coins.

• Related to rolling of dice.

• Related to a pack of cards.

• Related to the number system, i.e., find the probability of any digit of a number whether it is an even/ odd/ divisible by 3 or 4, etc. 

• Event occurrence questions.

Probability: Class 9 Math Chapter 15 NCERT Solutions Summary

Class 9 is completely a new level for the students. They have entered the board level and it can be quite fascinating, as well as intimidating for the students. The syllabus suddenly becomes huge. All subjects, especially mathematics, introduce you to new concepts. All these chapters and concepts are quite crucial to learn to help you build a strong foundation. By using the Class 9 Probability NCERT Solutions, this foundation of knowledge will help you build your confidence when you move ahead to an advanced class. This foundation will be made stronger with the help of NCERT Solutions for Class 9 Probability.

You Can Find the Solutions of All the Maths Chapters Below.

NCERT Solutions for Class 9 Maths

• Chapter 1 - Number System

• Chapter 2 - Polynomials 

• Chapter 3 - Coordinate Geometry

• Chapter 4 - Linear Equations in Two Variables

• Chapter 5 - Introductions to Euclid's Geometry

• Chapter 6 - Lines and Angles

• Chapter 7 - Triangles

• Chapter 8 - Quadrilaterals

• Chapter 9 - Areas of Parallelogram and Triangles

• Chapter 10 - Circles

• Chapter 11 - Constructions

• Chapter 12 - Heron's formula

• Chapter 13 - Surface area and Volumes

• Chapter 14 - Statistics

• Chapter 15 - Probability

NCERT Solutions Class 9 Mathematics Chapter 15 have been designed by the experienced teachers of Vedantu. Every question in the NCERT Solutions for Class 9 Maths Probability has been simplified and solved by the teachers in an elaborate manner. They are aware that the students will find Maths Class 9 Chapter 15 a little intimidating. It is new in their academic syllabus and is very important for the concept development of Mathematics. Chapter 15 Class 9 Maths has been added to the syllabus to make the students think of how probable incidents can be calculated by analyzing the provided information. These questions are designed to challenge the intellect of a student. This is why the students will need Class 9 Maths Chapter 15 NCERT Solutions is essential as students can know how to tackle particular question formats.

You can download and use the NCERT Solutions Class 9 Maths Probability to understand how the different patterns of questions can be answered. Class 9 Maths Ch 15 is quite challenging for the students. By using the NCERT Solutions for Class 9 Maths Chapter 15 Probability, these events will be analyzed by the students to calculate the possible outcomes. The Class 9 Chapter 15 Maths will teach them how to generally approach and calculate these outcomes of a particular event. Every event will deliver conditional clues to check and develop your concept. Chapter 15 Maths Class 9 NCERT Solutions will help you to develop this concept.

Why Should You Prefer Using Class 9 Maths NCERT Solutions Chapter 15?

The preparation of Class 9th Maths Chapter 15 Probability will need the assistance of the NCERT Class 9 Maths Chapter 15 Solutions. This solution to the NCERT Class 9 Maths Chapter 15 exercises will help you in the following ways. Have a look at the questions that are covered in Exercise 15.1 .

Simple Ways to Solve the Questions

The Class 9 Chapter 15 Maths NCERT Solutions will help you find the simplest solutions to all the questions.

Preparation

Chapter 15 Probability Class 9 NCERT Solutions are used for completing the preparation of Chapter 15 Maths Class 9 right on time.

Clearing Doubts 

You can use the Class 9 Maths Chapter 15 Solutions to clarify your doubts on your own.

Concept Development 

NCERT Solutions for Class 9 Maths Chapter 15 can be utilized for understanding the concept and to learn how to answer the questions.

Learning Question Types 

The question types of this chapter can be understood using Probability Class 9 NCERT Solutions. 

As you can see, Chapter 15 Class 9 Maths NCERT Solutions will help you prepare the chapter in so many different ways. Download Class 9 Maths Ch 15 NCERT Solutions and use them offline to make your preparation better.

You Can Find the Solutions of All the Maths Chapters Below

NCERT Solutions for Class 9 Maths

• Chapter 1 - Number System

• Chapter 2 - Polynomials 

• Chapter 3 - Coordinate Geometry

• Chapter 4 - Linear Equations in Two Variables

• Chapter 5 - Introductions to Euclid's Geometry

• Chapter 6 - Lines and Angles

• Chapter 7 - Triangles

• Chapter 8 - Quadrilaterals

• Chapter 9 - Areas of Parallelogram and Triangles

• Chapter 10 - Circles

• Chapter 11 - Constructions

• Chapter 12 - Heron's formula

• Chapter 13 - Surface area and Volumes

• Chapter 14 - Statistics

• Chapter 15 - Probability

Conclusion

The NCERT Solutions for Class 9 Maths Chapter 15 - Probability, provided by Vedantu, is a valuable tool for Class 9 students. It helps introduce Maths concepts in an accessible manner. The provided solutions and explanations simplify complex ideas, making it easier for Class 9 Students to understand the material. By using Vedantu's resources, Students can develop a deeper understanding of NCERT concepts. These solutions are a helpful aid for grade 9 students, empowering them to excel in their studies and develop a genuine appreciation for “Probability”.