Class 9 Maths Ganita Manjari Chapter 7 End of Chapter Exercises
NCERT Solutions for Class 9 Maths Ganit Manjari Chapter 7 End of Chapter Exercises provide detailed and step-by-step solutions to the questions from The Mathematics of Maybe: Introduction to Probability. These exercises help students revise and apply the key concepts of probability covered throughout the chapter, including possible outcomes, events and the likelihood of different outcomes.
The solutions are explained in a simple and easy-to-understand manner, helping students strengthen their problem-solving skills, verify their answers and prepare effectively for exams. A printable PDF is also available for convenient revision and practice.
Class 9 Maths Ganita Manjari Chapter 7 End of Chapter Exercises
The Mathematics of Maybe: Introduction to Probability
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The Mathematics of Maybe: Introduction to Probability
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The Mathematics of Maybe: Introduction to Probability
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The Mathematics of Maybe: Introduction to Probability
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The Mathematics of Maybe: Introduction to Probability
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The Mathematics of Maybe: Introduction to Probability
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The Mathematics of Maybe: Introduction to Probability
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The Mathematics of Maybe: Introduction to Probability
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The Mathematics of Maybe: Introduction to Probability
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The Mathematics of Maybe: Introduction to Probability
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The Mathematics of Maybe: Introduction to Probability
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The Mathematics of Maybe: Introduction to Probability
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The Mathematics of Maybe: Introduction to Probability
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The Mathematics of Maybe: Introduction to Probability
Medium
The Mathematics of Maybe: Introduction to Probability
Medium
The Mathematics of Maybe: Introduction to Probability
Medium
The Mathematics of Maybe: Introduction to Probability
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The Mathematics of Maybe: Introduction to Probability
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The Mathematics of Maybe: Introduction to Probability
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The Mathematics of Maybe: Introduction to Probability
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NCERT Solutions for Class 9 Maths Chapter 7 End-of-Chapter Exercises
The end-of-chapter exercises include probability scale, relative frequency, equally likely outcomes, sample space, cards, coins, dice, tables, surveys, tree diagrams and area-based probability.
Question 1. Fill in the blanks.
(i) The probability of an impossible event is _______.
Answer:
0
(ii) The set of all possible outcomes of a random experiment is called the _______.
Answer:
sample space
(iii) The probability of an event that is certain to happen is _______.
Answer:
1
(iv) Tossing a fair coin has a probability of _______ for getting heads.
Answer:
1/2
Question 2. In a survey of 50 students, 15 students said they liked football. The number of students who like football is 15, and the _______ is _______.
Answer:
The frequency is 15.
Relative frequency:
= 15/50
= 3/10
= 0.3
Final answer:
frequency = 15; relative frequency = 15/50 = 0.3
Question 3. Which of the following experiments have equally likely outcomes? Explain.
(i) A driver attempts to start a car. The car starts or does not start.
Answer:
The outcomes are not necessarily equally likely. A car in good condition is more likely to start.
Final answer:
Not equally likely
(ii) Tossing a fair coin once.
Answer:
Heads and tails are equally likely for a fair coin.
Final answer:
Equally likely
(iii) Rolling a fair 6-sided die.
Answer:
Each face has the same chance of appearing.
Final answer:
Equally likely
(iv) Choosing a marble randomly from a bag with 3 red and 7 blue marbles.
Answer:
Red and blue are not equally likely because there are more blue marbles.
Final answer:
Not equally likely by colour
(v) A baby is born. It is a boy or a girl.
Answer:
For simple school-level probability, this is often treated as approximately equally likely, though real-world biological data may not be exactly 50-50.
Final answer:
Approximately equally likely
Question 4. Write the sample space and calculate the probability.
(i) Two coins are tossed at the same time. What is the probability of getting at least one head?
Answer:
Sample space:
S = {HH, HT, TH, TT}
Event of at least one head:
E = {HH, HT, TH}
Probability:
= 3/4
Final answer:
3/4
(ii) Ten identical cards numbered 1 to 10 are placed in a box. One card is drawn. What is the probability of drawing an even number?
Answer:
Sample space:
S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Even numbers:
{2, 4, 6, 8, 10}
Probability:
= 5/10
= 1/2
Final answer:
1/2
(iii) A die is rolled once. What is the probability of getting a number greater than 4?
Answer:
Sample space:
S = {1, 2, 3, 4, 5, 6}
Numbers greater than 4:
{5, 6}
Probability:
= 2/6
= 1/3
Final answer:
1/3
(iv) A bag contains 3 red balls, 2 blue balls and 1 green ball. One ball is picked. What is the probability that it is not red?
Answer:
Total balls = 3 + 2 + 1 = 6
Not red balls = blue + green = 2 + 1 = 3
Probability:
= 3/6
= 1/2
Final answer:
1/2
(v) Three coins are tossed simultaneously. What is the probability of getting exactly two heads?
Answer:
Sample space:
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
Exactly two heads:
{HHT, HTH, THH}
Probability:
= 3/8
Final answer:
3/8
Question 5. A bag has 3 candies: strawberry, lemon and mint. One is picked at random. What is the probability of picking a strawberry candy?
Answer:
Total candies = 3
Strawberry candies = 1
Probability:
= 1/3
Final answer:
1/3
Question 6. A child has 2 shirts and 3 types of pants. List all possible combinations.
Answer:
| Shirt | Pants |
| Red shirt | Jeans |
| Red shirt | Khakis |
| Red shirt | Shorts |
| Blue shirt | Jeans |
| Blue shirt | Khakis |
| Blue shirt | Shorts |
Final answer:
There are 6 possible outfits.
Question 7. A tyre company records distances before replacement in 1000 cases.
| Distance | Less than 4000 | 4001 to 9000 | 9001 to 14000 | More than 14000 |
| Number of cases | 20 | 210 | 325 | 445 |
(i) Probability that a tyre lasts less than 4000 km
Answer:
Probability:
= 20/1000
= 0.02
Final answer:
0.02
(ii) Probability that a tyre lasts between 4000 and 14000 km
Answer:
Cases between 4000 and 14000 km:
= 210 + 325
= 535
Probability:
= 535/1000
= 0.535
Final answer:
0.535
(iii) Probability that a tyre lasts more than 14000 km
Answer:
Probability:
= 445/1000
= 0.445
Final answer:
0.445
Question 8. The letters of the word PEACE are placed on cards. Leela draws a card without looking.
(i) What is the probability that it is P, E or C?
Answer:
Letters:
P, E, A, C, E
Total cards = 5
Favourable cards: P, E, C
Since E appears twice, favourable cards are P, E, E, C = 4 cards.
Probability:
= 4/5
Final answer:
4/5
(ii) What is the probability that it is not an E?
Answer:
Cards not E:
P, A, C = 3 cards
Probability:
= 3/5
Final answer:
3/5
Question 9. A spinner has numbers 1 to 8 as equally likely outcomes. What is the probability that it points at:
(i) 8
Answer:
Probability = 1/8
Final answer:
1/8
(ii) An odd number
Answer:
Odd numbers = {1, 3, 5, 7}
Probability:
= 4/8
= 1/2
Final answer:
1/2
(iii) A number greater than 2
Answer:
Numbers greater than 2:
{3, 4, 5, 6, 7, 8}
Probability:
= 6/8
= 3/4
Final answer:
3/4
(iv) A number less than 9
Answer:
All numbers 1 to 8 are less than 9.
Probability:
= 8/8
= 1
Final answer:
1
(v) A multiple of 3
Answer:
Multiples of 3 from 1 to 8:
{3, 6}
Probability:
= 2/8
= 1/4
Final answer:
1/4
Question 10. A basket contains 4 red balls and 5 blue balls. One ball is drawn and kept aside, then a second ball is drawn.
(i) Probability of drawing a red ball and then a blue ball
Answer:
Total balls = 9
P(first red) = 4/9
After one red is removed, balls left = 8, blue balls = 5.
P(second blue after red) = 5/8
So:
P(red then blue) = 4/9 × 5/8
= 20/72
= 5/18
Final answer:
5/18
(ii) Probability of drawing 2 blue balls
Answer:
P(first blue) = 5/9
After one blue is removed, blue balls left = 4 and total balls left = 8.
P(second blue after first blue) = 4/8
So:
P(two blue balls) = 5/9 × 4/8
= 20/72
= 5/18
Final answer:
5/18
Question 11. I throw a pair of 6-sided dice. Write down an event that has probability 0 and an outcome that has probability 1.
Answer:
An event with probability 0:
Getting a sum of 13, because the maximum possible sum is 12.
An event with probability 1:
Getting a sum between 2 and 12 inclusive.
Final answer:
Probability 0: sum is 13. Probability 1: sum is between 2 and 12.
Question 12. Write the sample space and calculate the probability.
(i) Two dice are rolled. What is the probability that the sum is a prime number greater than 5?
Answer:
Total outcomes when two dice are rolled = 36
Prime sums greater than 5 possible:
7 and 11
Sum 7 outcomes:
(1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6
Sum 11 outcomes:
(5,6), (6,5) = 2
Total favourable outcomes = 8
Probability:
= 8/36
= 2/9
Final answer:
2/9
(ii) A bag contains 4 red, 3 green and 2 blue balls. Two balls are drawn without replacement. What is the probability that both are of different colours?
Answer:
Total balls = 9
Total ways to choose 2 balls:
= 9C2
= 36
Ways to choose two balls of same colour:
Red-red = 4C2 = 6
Green-green = 3C2 = 3
Blue-blue = 2C2 = 1
Total same-colour ways = 10
Different-colour ways:
= 36 - 10
= 26
Probability:
= 26/36
= 13/18
Final answer:
13/18
(iii) Three coins are tossed. What is the probability that the first coin shows heads and exactly two heads occur in total?
Answer:
Sample space:
{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
First coin heads and exactly two heads:
HHT, HTH
Favourable outcomes = 2
Total outcomes = 8
Probability:
= 2/8
= 1/4
Final answer:
1/4
(iv) A four-digit number is formed using digits 1, 2, 3 and 4 with no repetition. What is the probability that the number is even?
Answer:
Total four-digit numbers:
= 4!
= 24
For the number to be even, last digit must be 2 or 4.
Choices for last digit = 2
Remaining 3 digits can be arranged in 3! ways.
Favourable outcomes:
= 2 × 3!
= 2 × 6
= 12
Probability:
= 12/24
= 1/2
Final answer:
1/2
(v) A student takes a multiple-choice test with 3 questions, each having 4 options. What is the probability that the student guesses and gets exactly 2 answers correct?
Answer:
Probability of correct answer for one question = 1/4
Probability of wrong answer = 3/4
Exactly 2 correct out of 3 can happen in 3 ways.
Probability:
= 3 × (1/4)² × (3/4)
= 3 × 1/16 × 3/4
= 9/64
Final answer:
9/64
Question 13. A box contains 4 balls numbered 1 to 4. Record a sample space using a tree diagram.
(i) A ball is drawn, returned, and a second ball is drawn.
Answer:
With replacement, each draw has 4 possibilities.
Sample space:
S = {(1,1), (1,2), (1,3), (1,4),
(2,1), (2,2), (2,3), (2,4),
(3,1), (3,2), (3,3), (3,4),
(4,1), (4,2), (4,3), (4,4)}
Final answer:
There are 16 outcomes.
(ii) A ball is drawn and recorded. Without replacing it, a second ball is drawn.
Answer:
Without replacement, the same number cannot appear twice.
Sample space:
S = {(1,2), (1,3), (1,4),
(2,1), (2,3), (2,4),
(3,1), (3,2), (3,4),
(4,1), (4,2), (4,3)}
Final answer:
There are 12 outcomes.
(iii) What are the sizes of these sample spaces?
Answer:
With replacement: 4 × 4 = 16
Without replacement: 4 × 3 = 12
Final answer:
16 and 12
Question 14. List the elements of a sample space for simultaneous tossing of a coin and drawing a card from 6 cards numbered 1 through 6.
Answer:
Coin outcomes = {H, T}
Card outcomes = {1, 2, 3, 4, 5, 6}
Sample space:
S = {(H,1), (H,2), (H,3), (H,4), (H,5), (H,6), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6)}
Final answer:
S has 12 outcomes.
Question 15. Three coins are tossed, and the number of heads is recorded. Which list is a sample space?
Options:
(i) {1, 2, 3}
(ii) {0, 1, 2}
(iii) {0, 1, 2, 3, 4}
(iv) {0, 1, 2, 3}
Answer:
When three coins are tossed, the number of heads can be:
0, 1, 2 or 3
So the correct sample space is:
{0, 1, 2, 3}
Why others fail:
- {1, 2, 3} misses 0 heads.
- {0, 1, 2} misses 3 heads.
- {0, 1, 2, 3, 4} includes 4 heads, which is impossible with three coins.
Final answer:
Option (iv) {0, 1, 2, 3}
Question 16. A dye is dropped at random on a rectangular region of 3 m by 2 m. What is the probability that it lands inside the circle with diameter 1 m?
Answer:
Area of rectangle:
= 3 × 2
= 6 m²
Circle diameter = 1 m
Radius = 1/2 m
Area of circle:
= πr²
= π(1/2)²
= π/4 m²
Probability:
= area of circle / area of rectangle
= (Ï€/4)/6
= π/24
Final answer:
Ï€/24
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