Which of the following events is certain?

The sun rises in the west tomorrow

The number 7 showing on a standard die

Assertion (A): After rolling a fair die 5 times without getting a 6, the probability of getting a 6 on the next roll is higher than 1/6. Reason (R): Each roll of a die is independent of the previous rolls.

Both A and R are true and R is the correct explanation of A.

Both A and R are true but R is not the correct explanation of A.

A die is rolled 200 times and a '6' appears 38 times. Calculate the experimental probability of rolling a 6. Based on this, do you think the die is fair? Briefly justify.

Experimental probability = 0.19. The die may not be perfectly fair because the experimental probability (0.19) is somewhat higher than the theoretical probability (approximately 0.167).

A coin is tossed three times. Write the sample space for this experiment.

{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

Define the term 'sample space' in probability. Give one example to illustrate your definition.

The sample space of a random experiment is the set of all possible outcomes. Example: When a fair coin is tossed, the sample space is {Heads, Tails}.

In a class of 40 students, 25 students like football, 20 like cricket, and 10 like both football and cricket. A student is chosen at random from the class. (i) Draw a Venn diagram to represent the number of students who like football, cricket, both, or neither. (ii) What is the probability that the selected student likes neither football nor cricket? (iii) Find the probability that the student likes exactly one of the two sports. (iv) What is the probability that the student likes football but not cricket? Show all your calculations.

(i) Venn diagram: Intersection = 10; Football only = 25-10=15; Cricket only = 20-10=10; Outside (neither) = 40-(15+10+10)=5. (ii) P(neither) = 5/40 = 1/8 = 0.125. (iii) P(exactly one) = (15+10)/40 = 25/40 = 5/8 = 0.625. (iv) P(football but not cricket) = 15/40 = 3/8 = 0.375.

A fair six-sided die is rolled once. (i) How many possible outcomes are there? (ii) Find the probability of getting a number less than 3. (iii) Find the probability of getting a number that is not a factor of 6. (iv) Are the events in (ii) and (iii) complementary? Give a reason for your answer.

(i) 6 possible outcomes. (ii) 1/3. (iii) 1/3. (iv) No, they are not complementary because the outcomes of the two events do not together cover the entire sample space (the number 3 is neither less than 3 nor a non-factor of 6; it is a factor of 6 but not less than 3).

Class 9 Maths Chapter 7: The Mathematics of Maybe: Introduction to Probability — Important Questions & Sample Paper

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Yes — this page has 44+ original Class 9 Mathematics Chapter 7 (“The Mathematics of Maybe: Introduction to Probability”) important questions with answers (Multiple Choice (MCQ), Assertion–Reason, Short Answer, Short Answer, Long Answer, Case Study). Practise them free, or generate a full CBSE board-pattern sample paper (80 marks) and export it to PDF or Word — in English & Hindi, for 2026-27.

Chapter 7 of Class 9 NCERT Mathematics, titled "The Mathematics of Maybe: Introduction to Probability" in the Ganita Manjari textbook, introduces students to the fascinating world of chance and uncertainty. This chapter lays the groundwork for understanding probability as a measure of likelihood, ranging from 0 (impossible) to 1 (certain). Key concepts include random experiments, outcomes, sample space (the set of all possible outcomes), and events. Students learn to distinguish between theoretical probability, based on equally likely outcomes, and experimental probability, derived from actual trials. The chapter emphasizes calculating probabilities for simple experiments like tossing coins, rolling dice, drawing cards, and spinning wheels. For example, finding the probability of getting at least two tails in three coin tosses or determining the chance of drawing a multiple of 3 or 5 from a set of numbered cards. It also covers complementary events, where P(not A) = 1 − P(A). Exam questions often test these concepts through multi-step problems: picking a slip of paper and then a letter to find vowel probability, or comparing experimental and theoretical probabilities to assess fairness. The chapter builds intuition for data-driven predictions, asking students to compute expected frequencies in repeated trials. By mastering sample space construction and probability formulas, students tackle a variety of word problems that blend logical reasoning with basic counting.

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Mathematics — The Mathematics of Maybe: Introduction to Probability

Class 9Time: 3 hrsMax Marks: 80

SECTION A

1.
Two dice are thrown. What is the probability that the sum of the numbers on the two dice is 8?
(a) 5/36(b) 1/6(c) 1/12(d) 1/9
1
2.
A bag contains 4 red, 5 blue, and 6 green balls. One ball is drawn. What is the probability that the ball is not red?
(a) 4/15(b) 11/15(c) 1/3(d) 2/5
1
3.
Which of the following events is certain?
(a) The sun rises in the west tomorrow(b) A fair coin landing on heads(c) The number 7 showing on a standard die(d) A day having 24 hours
1

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Marks distribution & blueprint

In a CBSE exam, this chapter typically contributes questions across the following types. The last column shows how many original questions of each type we have ready in our bank for this chapter:

Question type	Marks each	In our bank
Multiple Choice (MCQ)	1 mark	13
Assertion–Reason	1 mark	6
Short Answer	2 marks	8
Short Answer	3 marks	6
Long Answer	5 marks	5
Case Study	4 marks	6

44 original, exam-style questions in our bank for this chapter — with answers.

Important & sample questions (with answers)

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Q1. Two dice are thrown. What is the probability that the sum of the numbers on the two dice is 8?
1 mark
Multiple Choice (MCQ)
(A) 5/36(B) 1/6(C) 1/12(D) 1/9
▸ Answer▾ Answer
5/36
Q2. A bag contains 4 red, 5 blue, and 6 green balls. One ball is drawn. What is the probability that the ball is not red?
1 mark
Multiple Choice (MCQ)
(A) 4/15(B) 11/15(C) 1/3(D) 2/5
▸ Answer▾ Answer
11/15
Q3. Which of the following events is certain?
1 mark
Multiple Choice (MCQ)
(A) The sun rises in the west tomorrow(B) A fair coin landing on heads(C) The number 7 showing on a standard die(D) A day having 24 hours
▸ Answer▾ Answer
A day having 24 hours
Q4. A box contains 100 cards numbered 1 to 100. One card is drawn. What is the probability that the number on the card is a multiple of 2 or 5?
1 mark
Multiple Choice (MCQ)
(A) 3/5(B) 1/2(C) 2/5(D) 7/10
▸ Answer▾ Answer
3/5
Q5. Assertion (A): After rolling a fair die 5 times without getting a 6, the probability of getting a 6 on the next roll is higher than 1/6. Reason (R): Each roll of a die is independent of the previous rolls.
1 mark
Assertion–Reason
(A) Both A and R are true and R is the correct explanation of A.(B) Both A and R are true but R is not the correct explanation of A.(C) A is true but R is false.(D) A is false but R is true.
▸ Answer▾ Answer
A is false but R is true.
Q6. A die is rolled 200 times and a '6' appears 38 times. Calculate the experimental probability of rolling a 6. Based on this, do you think the die is fair? Briefly justify.
2 marks
Short Answer
▸ Answer▾ Answer
Experimental probability = 0.19. The die may not be perfectly fair because the experimental probability (0.19) is somewhat higher than the theoretical probability (approximately 0.167).
Q7. A coin is tossed three times. Write the sample space for this experiment.
2 marks
Short Answer
▸ Answer▾ Answer
{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
Q8. A bag contains 7 green marbles and 5 purple marbles. One marble is drawn at random. What is the probability that it is green?
3 marks
Short Answer
▸ Answer▾ Answer
7/12
Q9. Define the term 'sample space' in probability. Give one example to illustrate your definition.
3 marks
Short Answer
▸ Answer▾ Answer
The sample space of a random experiment is the set of all possible outcomes. Example: When a fair coin is tossed, the sample space is {Heads, Tails}.
Q10. In a class of 40 students, 25 students like football, 20 like cricket, and 10 like both football and cricket. A student is chosen at random from the class. (i) Draw a Venn diagram to represent the number of students who like football, cricket, both, or neither. (ii) What is the probability that the selected student likes neither football nor cricket? (iii) Find the probability that the student likes exactly one of the two sports. (iv) What is the probability that the student likes football but not cricket? Show all your calculations.
5 marks
Long Answer
▸ Answer▾ Answer
(i) Venn diagram: Intersection = 10; Football only = 25-10=15; Cricket only = 20-10=10; Outside (neither) = 40-(15+10+10)=5. (ii) P(neither) = 5/40 = 1/8 = 0.125. (iii) P(exactly one) = (15+10)/40 = 25/40 = 5/8 = 0.625. (iv) P(football but not cricket) = 15/40 = 3/8 = 0.375.
Q11. A fair six-sided die is rolled once. (i) How many possible outcomes are there? (ii) Find the probability of getting a number less than 3. (iii) Find the probability of getting a number that is not a factor of 6. (iv) Are the events in (ii) and (iii) complementary? Give a reason for your answer.
5 marks
Long Answer
▸ Answer▾ Answer
(i) 6 possible outcomes. (ii) 1/3. (iii) 1/3. (iv) No, they are not complementary because the outcomes of the two events do not together cover the entire sample space (the number 3 is neither less than 3 nor a non-factor of 6; it is a factor of 6 but not less than 3).
Q12. A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. One marble is drawn at random from the bag. All marbles are identical in size and shape.
4 marks
Case Study
1. (i) What is the probability of drawing a red marble?2 marks
2. (ii) What is the probability of drawing a marble that is NOT blue?2 marks
▸ Answer▾ Answer
(i) 1/2, (ii) 7/10

Frequently asked questions

What is a sample space in probability? Give an example.

The sample space is the set of all possible outcomes of a random experiment. For instance, when tossing two fair coins, the sample space S = {HH, HT, TH, TT}.

How is experimental probability different from theoretical probability?

Theoretical probability assumes that all outcomes are equally likely and is calculated as (favorable outcomes)/(total outcomes). Experimental probability is based on actual results from an experiment, computed as (number of times the event occurs)/(total number of trials). The two values can differ due to randomness, especially in small samples.

What is the complement of an event, and how do you find its probability?

The complement of an event A, denoted A', is the set of all outcomes in the sample space that are not in A. Its probability is given by P(A') = 1 – P(A). For example, if the probability of rain is 0.45, the probability of no rain is 0.55.

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