NCERT Solutions for Class 9 Maths Ganit Manjari Chapter 7 The Mathematics of Maybe: Introduction to Probability

Probability is a number between 0 and 1 that measures how likely an event is to happen. In Class 9 Maths Chapter 7, students learn how randomness, sample space, events, experimental probability, theoretical probability and tree diagrams help measure uncertainty.

NCERT Solutions for Class 9 Maths Ganit Manjari Chapter 7 help students solve textbook questions from The Mathematics of Maybe: Introduction to Probability in the new Ganita Manjari Class 9 Maths textbook. This chapter introduces random experiments, outcomes, probability scale, subjective and objective probability, relative frequency, theoretical probability, experimental probability, sample spaces, events and tree diagrams.

Coin tosses, dice rolls, lucky draws, surveys, sweets, cards, pens and spinners make probability easier to connect with everyday situations and CBSE 2026-27 question patterns. The exercise-wise solutions guide students in listing sample spaces, identifying events, calculating probabilities and using tree diagrams for multi-step experiments.

NCERT Solutions for Class 9 Maths Chapter 7 PDF Download

The NCERT Solutions for Class 9 Maths Chapter 7 PDF helps students revise probability questions with sample spaces and stepwise calculations for CBSE 2026-27. It is useful for practising coin, dice, card, survey and tree-diagram questions.

Students can use the PDF to revise:

1. Meaning of probability
2. Random experiments and outcomes
3. Probability scale from 0 to 1
4. Impossible, equally likely and certain events
5. Experimental probability
6. Relative frequency
7. Theoretical probability
8. Sample space
9. Events
10. Tree diagrams
11. Multi-step experiments
12. Probability from tables and data

Access Exercise-Wise NCERT Solutions for Class 9 Maths Chapter 7

NCERT Solutions for Class 9 Maths Chapter 7 Exercise Set 7.1

Exercise Set 7.1 focuses on ranking events on the probability scale from 0 to 1. Students classify events as impossible, less likely, equally likely, more likely or certain.

Question 1. Rank the following events on a scale from 0 to 1. Label each event as impossible, less likely, equally likely, more likely or certain. Give reasons.

(i) The next Monday will come after Sunday.

Answer:
Monday always comes after Sunday in the weekly calendar.

So, the event is certain.

Probability = 1

Final answer:
Certain; probability = 1

(ii) It will snow in Mumbai in July.

Answer:
Mumbai has a tropical coastal climate. Snowfall in Mumbai in July is practically impossible.

Probability = 0

Final answer:
Impossible; probability = 0

(iii) An elephant will walk through your classroom today.

Answer:
This is extremely unlikely in a normal school setting, but not mathematically impossible in every possible situation.

Final answer:
Less likely; probability is close to 0

(iv) You will greet at least one friend at school tomorrow.

Answer:
If it is a normal school day and the student attends school, greeting at least one friend is highly likely.

Final answer:
More likely; probability is close to 1

NCERT Solutions for Class 9 Maths Chapter 7 Exercise Set 7.2

Exercise Set 7.2 focuses on experimental probability, theoretical probability, data-based estimation and relative frequency.

Question 1. A teacher mixes a large bag of sweets of different colours and randomly selects a sample of 30 sweets: 10 red, 8 green, 7 yellow and 5 blue.

(i) Calculate the probability that a randomly picked sweet from the sample is green.

Answer:
Number of green sweets = 8
Total sweets in sample = 30

Experimental probability of green sweet:

= 8/30
= 4/15

Final answer:
4/15

(ii) If there are 600 sweets in total in the large bag, estimate how many are likely to be yellow.

Answer:
Number of yellow sweets in sample = 7
Total sweets in sample = 30

Experimental probability of yellow sweet:

= 7/30

Estimated number of yellow sweets in 600:

= 7/30 × 600
= 7 × 20
= 140

Final answer:
About 140 yellow sweets

Question 2. A random sample of 40 students is asked about their favourite club: 14 Science Club, 11 Arts Club, 9 Sports Club and 6 Debate Club. Assume there are 800 students in the school.

(i) What is the probability that a randomly chosen student from the sample prefers the Arts Club?

Answer:
Number of students preferring Arts Club = 11
Total students in sample = 40

Probability:

= 11/40
= 0.275

Final answer:
11/40 or 0.275

(ii) Estimate how many students in the whole school are likely to prefer the Sports Club.

Answer:
Number of students preferring Sports Club = 9
Total sample = 40

Probability of Sports Club preference:

= 9/40

Estimated number in 800 students:

= 9/40 × 800
= 9 × 20
= 180

Final answer:
About 180 students

Question 3. Toss a coin 20 times and record the result each time.

(i) How many times did you get heads?

Answer:
This answer depends on the actual experiment.

Final answer:
Record the number of heads from your 20 tosses.

(ii) How many times did you get tails?

Answer:
This answer depends on the actual experiment.

If heads = H, then:

Tails = 20 - H

Final answer:
Tails = 20 - number of heads

(iii) Calculate the experimental probability of getting heads.

Answer:
Experimental probability:

= Number of heads / Total tosses
= H/20

Final answer:
Experimental probability of heads = H/20

(iv) If you toss the coin once more, what is the probability of getting tails?

Answer:
For a fair coin, each toss is independent.

The theoretical probability of tails is:

= 1/2

Final answer:
1/2

Question 4. Toss a paper cup into the air 100 times. Record whether the cup lands on its bottom, upside down on its top or on its side. Assign probabilities using experimental probability.

Answer:
Let:

Number of times cup lands on bottom = B
Number of times cup lands on top = T
Number of times cup lands on side = S

Total trials = 100

Experimental probabilities:

P(bottom) = B/100
P(top) = T/100
P(side) = S/100

Final answer:
Use the recorded frequencies: B/100, T/100 and S/100.

Question 5. What is the probability of getting an even number when rolling a fair 6-sided die?

Answer:
Sample space:

S = {1, 2, 3, 4, 5, 6}

Even outcomes:

{2, 4, 6}

Number of favourable outcomes = 3
Total outcomes = 6

Probability:

= 3/6
= 1/2

Final answer:
1/2

Question 6. Suppose you roll a 6-sided die 12 times and get a 3 three times.

(i) What is the experimental probability of rolling a 3?

Answer:
Number of times 3 occurred = 3
Total rolls = 12

Experimental probability:

= 3/12
= 1/4

Final answer:
1/4

(ii) What is the theoretical probability of rolling a 3?

Answer:
A fair die has 6 equally likely outcomes.

Only one outcome is 3.

Theoretical probability:

= 1/6

Final answer:
1/6

(iii) Why might these probabilities be different? What would happen if you roll the die 60, 600 or 6000 times?

Answer:
Experimental probability is based on actual results. Theoretical probability is based on equally likely outcomes.

With only 12 rolls, experimental results may differ from theoretical probability.

As the number of rolls increases, the experimental probability is expected to get closer to 1/6.

Final answer:
They differ because of random variation. With more trials, experimental probability usually gets closer to theoretical probability.

NCERT Solutions for Class 9 Maths Chapter 7 Exercise Set 7.3

Exercise Set 7.3 focuses on sample space and events. A sample space lists all possible outcomes, while an event is a subset of the sample space.

Question 1. When a single 6-sided die is rolled, what is the total number of possible outcomes in the sample space?

Answer:
Sample space:

S = {1, 2, 3, 4, 5, 6}

Number of possible outcomes:

n(S) = 6

Final answer:
6

Question 2. For the following experiments, write down the sample space S.

(i) Rolling a die and tossing a coin together.

Answer:
Die outcomes = {1, 2, 3, 4, 5, 6}
Coin outcomes = {H, T}

Sample space:

S = {(1, H), (1, T), (2, H), (2, T), (3, H), (3, T), (4, H), (4, T), (5, H), (5, T), (6, H), (6, T)}

Final answer:
S = {(1, H), (1, T), (2, H), (2, T), (3, H), (3, T), (4, H), (4, T), (5, H), (5, T), (6, H), (6, T)}

(ii) Choosing a random integer between -5 and +5.

Answer:
Integers from -5 to +5 are:

S = {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}

Final answer:
S = {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}

(iii) A box contains 5 green and 7 red balls. One ball is drawn at random.

Answer:
If we record only the colour, the sample space is:

S = {Green, Red}

Final answer:
S = {Green, Red}

Question 3. In a village fair, there are 3 popular snacks: Samosa, Pakora and Bhaji. For drinks, villagers can choose either Chai or Lassi.

(i) List the sample space of all possible snack and drink combinations.

Answer:
Snacks = {Samosa, Pakora, Bhaji}
Drinks = {Chai, Lassi}

Sample space:

S = {(Samosa, Chai), (Samosa, Lassi), (Pakora, Chai), (Pakora, Lassi), (Bhaji, Chai), (Bhaji, Lassi)}

Final answer:
S = {(Samosa, Chai), (Samosa, Lassi), (Pakora, Chai), (Pakora, Lassi), (Bhaji, Chai), (Bhaji, Lassi)}

(ii) List the event “Selecting Samosa as a snack.”

Answer:
The event includes all combinations where the snack is Samosa.

E = {(Samosa, Chai), (Samosa, Lassi)}

Final answer:
E = {(Samosa, Chai), (Samosa, Lassi)}

NCERT Solutions for Class 9 Maths Chapter 7 Exercise Set 7.4

Exercise Set 7.4 focuses on tree diagrams for multi-step experiments.

Question 1. There are two fruit baskets A and B. Basket A has one apple and two oranges. Basket B has one banana and one mango. You randomly pick one fruit from each basket.

(i) Draw a tree diagram showing all possible pairs of fruits.

Answer:
Basket A outcomes:

Apple, Orange, Orange

Basket B outcomes:

Banana, Mango

Tree structure:

Apple → Banana, Mango
Orange → Banana, Mango
Orange → Banana, Mango

Since the two oranges are identical in fruit type, the fruit-type pairs are:

Apple-Banana, Apple-Mango, Orange-Banana, Orange-Mango

Final answer:
The tree has branches from Basket A to Apple or Orange, and from each branch to Banana or Mango.

(ii) List the sample space.

Answer:
If only fruit types are recorded:

S = {(Apple, Banana), (Apple, Mango), (Orange, Banana), (Orange, Mango)}

If individual oranges are treated separately, there are 6 equally likely outcomes.

Final answer:
S = {(Apple, Banana), (Apple, Mango), (Orange, Banana), (Orange, Mango)}

(iii) What is the probability of picking one apple and one banana?

Answer:
Probability of apple from Basket A:

= 1/3

Probability of banana from Basket B:

= 1/2

So:

P(apple and banana) = 1/3 × 1/2
= 1/6

Final answer:
1/6

Question 2. A box contains 3 red pens, 4 black pens and 2 green pens. You pick a pen and put it back. Then your friend does the same.

(i) What are the possible outcomes of the pen colours? Draw a tree diagram.

Answer:
Possible colours:

Red, Black, Green

Since the pen is replaced, both picks have the same possible colour outcomes.

Sample space:

S = {(R, R), (R, B), (R, G), (B, R), (B, B), (B, G), (G, R), (G, B), (G, G)}

Tree structure:

Red → Red, Black, Green
Black → Red, Black, Green
Green → Red, Black, Green

Final answer:
S = {(R, R), (R, B), (R, G), (B, R), (B, B), (B, G), (G, R), (G, B), (G, G)}

(ii) Use the tree diagram to guess the probability that both you and your friend pick pens of the same colour.

Answer:
Total pens = 3 + 4 + 2 = 9

P(Red) = 3/9 = 1/3
P(Black) = 4/9
P(Green) = 2/9

Since the pen is replaced, the events are independent.

Probability of same colour:

= P(R, R) + P(B, B) + P(G, G)

= (3/9 × 3/9) + (4/9 × 4/9) + (2/9 × 2/9)

= 9/81 + 16/81 + 4/81
= 29/81

Final answer:
29/81

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:

Final answer:
There are 6 possible outfits.

Question 7. A tyre company records distances before replacement in 1000 cases.

(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. {1, 2, 3} misses 0 heads.
2. {0, 1, 2} misses 3 heads.
3. {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
= (π/4)/6
= π/24

Final answer:
π/24

Topics Covered in NCERT Solutions for Class 9 Maths Chapter 7

1. Probability
2. Randomness
3. Random experiment
4. Outcomes
5. Probability scale
6. Impossible event
7. Certain event
8. Less likely and more likely events
9. Experimental probability
10. Relative frequency
11. Theoretical probability
12. Equally likely outcomes
13. Statistical probability
14. Sampling
15. Law of Large Numbers
16. Gambler’s Fallacy
17. Sample space
18. Sample size
19. Events
20. Tree diagrams
21. Multi-step experiments
22. Probability using cards, coins and dice

Important Formulas and Rules in NCERT Solutions for Class 9 Maths Chapter 7

NCERT Class 9 Maths Ganita Manjari 2026 Chapter Solutions