NCERT Solutions for Class 11 Maths Chapter 14 – Probability Exercise 14.1

Class 11 Maths Chapter 14 – Probability, Exercise 14.1 introduces the foundational concepts of probability — sample spaces, events, and their types such as mutually exclusive, simple, and compound events. In this exercise, students learn how to define outcomes of any random experiment using set notation and logically describe relationships between different events — such as union, intersection, and complement. These concepts are clearly explained through classic experiments like throwing a die, tossing coins, and rolling a pair of dice.

Probability is an important chapter in the CBSE Class 11 syllabus that forms the base for advanced probability theory in Class 12. Questions from this chapter appear regularly in board exams — especially on mutually exclusive events, exhaustive events, and event descriptions. Probability is also a high-weightage topic in JEE Main and JEE Advanced, and the concepts covered in Exercise 14.1 are the first step towards that preparation.

NCERT Solutions for Class 11 Maths Chapter 14 – Probability Exercise 14.1

Probability
Easy
Q.
In each of the following Exercises 1 to 7, describe the sample space for the indicated experiment.

1. A coin is tossed three times.
2. A die is thrown two times.
3. A coin is tossed four times.
4. A coin is tossed and a die is thrown.
5. A coin is tossed and then a die is rolled only in case a head is shown on the coin.
6. 2 boys and 2 girls are in Room X, and 1 boy  and 3 girls in Room Y. Specify the sample space for the experiment in which a room is selected and then a person.
7. One die of red colour, one of white colour and one of blue colour are placed in a bag. One die is selected at random and rolled, its colour and the number on its uppermost face is noted. Describe the sample space.
Probability
Easy
Q.
An experiment consists of tossing a coin and then throwing it second time if a head occurs. If a tail occurs on the first toss, then a die is rolled once. Find the sample space.
Probability
Easy
Q.
A coin is tossed. If the outcome is a head, a die is thrown. If the die shows up an even number, the die is thrown again. What is the sample space for the experiment?
Probability
Easy
Q.
A die is thrown repeatedly until a six comes up. What is the sample space for this experiment?
Probability
Easy
Q.
A die is rolled. Let E be the event “die shows 4” and F be the event “die shows even number”. Are E and F mutually exclusive?
Probability
Easy
Q.
A die is thrown. Describe the following events:

(i) A: a number less than 7
(ii) B: a number greater than 7
(iii) C: a multiple of 3
(iv) D: a number less than 4
(v) E: an even number greater than 4
(vi) F: a number not less than 3

Also find A ∪ B,   A ∩ B,  B ∪ C,  E ∩ F,  D ∩ E,  A – C,  D – E,  E ∩ F′,  F′
Probability
Medium
Q.
An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events:

A: the sum is greater than 8,
B: 2 occurs on either die
C: the sum is at least 7 and a multiple of 3.

Which pairs of these events are mutually exclusive?
Probability
Medium
Q.
Three coins are tossed once. Let A denote the event ‘three heads show”, B denote the event “two heads and one tail show”, C denote the event” three tails show and D denote the event ‘a head shows on the first coin”.
Which events are    (i) mutually exclusive?   (ii) simple?   (iii) Compound?
Probability
Medium
Q.
Two dice are thrown. The events A, B and C are as follows:

A: getting an even number on the first die.
B: getting an odd number on the first die.
C: getting the sum of the numbers on the dice ≤ 5.

Describe the events
(i) A′                    (ii) not B ,
(iii) A or B            (iv) A and B       
(v) A but not C      (vi) B or C        
(vii) B and C         (viii) A ∩ B′∩C′
Probability
Easy
Q.
Refer to question 6 above, state true or false: (Give reason for your answer)

(i) A and B are mutually exclusive
(ii) A and B are mutually exclusive and exhaustive  
(iii) A = B′
(iv) A and C are mutually exclusive
(v) A and B′ are mutually exclusive.
(vi) A′, B′, C are mutually exclusive and exhaustive.
Probability
Medium
Q.
A die is thrown; find the probability of following events:

(i) A prime number will appear,
(ii) A number greater than or equal to 3 will appear,
(iii) A number less than or equal to one will appear,
(iv) A number more than 6 will appear,
(v) A number less than 6 will appear.
Probability
Medium
Q.
Three coins are tossed once. Find the probability of getting

(i) 3 heads
(ii) 2 heads
(iii) at least 2 heads
(iv) at most 2 heads
(v) no head (vi) 3 tails
(vii) exactly two tails
(viii) no tail
(ix) at most two tails

NCERT Solutions for Class 11 Maths Chapter 14 – Probability Exercise 14.1

Question 1. A die is rolled. Let E be the event "die shows 4" and F be the event "die shows even number". Are E and F mutually exclusive?

Answer:

Sample space when a die is thrown: S = {1, 2, 3, 4, 5, 6}

Given: E = {4} F = {2, 4, 6}

E ∩ F = {4} ∩ {2, 4, 6} = {4}

Since E ∩ F ≠ φ, E and F are not mutually exclusive.

Question 2. A die is thrown. Describe the following events:

Answer:

Sample space: S = {1, 2, 3, 4, 5, 6}

(i) A: a number less than 7 All numbers on a die are less than 7. A = {1, 2, 3, 4, 5, 6}

(ii) B: a number greater than 7 No number on a die is greater than 7. B = φ

(iii) C: a multiple of 3 Multiples of 3 between 1 and 6: 3 and 6 C = {3, 6}

(iv) D: a number less than 4 D = {1, 2, 3}

(v) E: an even number greater than 4 Only 6 is even and greater than 4. E = {6}

(vi) F: a number not less than 3 F = {3, 4, 5, 6}

Also find: A ∪ B, A ∩ B, B ∪ C, E ∩ F, D ∩ E, D – E, A – C, E ∩ F′, F′

A ∪ B = {1,2,3,4,5,6} ∪ φ = {1, 2, 3, 4, 5, 6}

A ∩ B = {1,2,3,4,5,6} ∩ φ = φ

B ∪ C = φ ∪ {3,6} = {3, 6}

E ∩ F = {6} ∩ {3,4,5,6} = {6}

D ∩ E = {1,2,3} ∩ {6} = φ

D – E = {1,2,3} – {6} = {1, 2, 3}

A – C = {1,2,3,4,5,6} – {3,6} = {1, 2, 4, 5}

F′ = S – F = {1,2,3,4,5,6} – {3,4,5,6} = {1, 2}

E ∩ F′ = {6} ∩ {1,2} = φ

Question 3. An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events: A: the sum is greater than 8 B: 2 occurs on either die C: the sum is at least 7 and a multiple of 3 Which pairs of these events are mutually exclusive?

Answer:

Sample space S has 36 outcomes (all ordered pairs from (1,1) to (6,6)).

A = {(3,6), (4,5), (4,6), (5,4), (5,5), (5,6), (6,3), (6,4), (6,5), (6,6)}

B = {(2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (1,2), (3,2), (4,2), (5,2), (6,2)}

C = {(3,6), (6,3), (6,6)}

Checking mutual exclusivity:

(i) A ∩ B: A has no element with 2 on either die. A ∩ B = φ ∴ A and B are mutually exclusive.

(ii) B ∩ C: C has no element with 2 on either die. B ∩ C = φ ∴ B and C are mutually exclusive.

(iii) A ∩ C: A ∩ C = {(3,6), (6,3), (6,6)} ≠ φ ∴ A and C are NOT mutually exclusive.

Question 4. Three coins are tossed once. Let A denote the event 'three heads show', B denote the event 'two heads and one tail show', C denote the event 'three tails show' and D denote the event 'a head shows on the first coin'. Which events are (i) Mutually exclusive? (ii) Simple? (iii) Compound?

Answer:

Sample space: S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

A = {HHH} B = {HHT, HTH, THH} C = {TTT} D = {HHH, HHT, HTH, HTT}

(i) Mutually exclusive:

A ∩ B = φ → A and B are mutually exclusive A ∩ C = φ → A and C are mutually exclusive A ∩ D = {HHH} ≠ φ → A and D are NOT mutually exclusive B ∩ C = φ → B and C are mutually exclusive B ∩ D = {HHT, HTH} ≠ φ → B and D are NOT mutually exclusive C ∩ D = φ → C and D are mutually exclusive

(ii) Simple events: A simple event has only one outcome. A = {HHH} → Simple C = {TTT} → Simple

(iii) Compound events: A compound event has more than one outcome. B = {HHT, HTH, THH} → Compound D = {HHH, HHT, HTH, HTT} → Compound

Question 5. Three coins are tossed. Describe:

Answer:

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

(i) Two events which are mutually exclusive: Let A = getting all heads = {HHH} Let B = getting all tails = {TTT} A ∩ B = φ → A and B are mutually exclusive.

(ii) Three events which are mutually exclusive and exhaustive: Let P = exactly two tails = {HTT, TTH, THT} Let Q = at least two heads = {HHT, HTH, THH, HHH} Let R = three tails = {TTT}

P ∩ Q ∩ R = φ → mutually exclusive. P ∪ Q ∪ R = S → exhaustive.

(iii) Two events which are not mutually exclusive: Let A = at least two heads = {HHH, HHT, HTH, THH} Let B = all heads = {HHH} A ∩ B = {HHH} ≠ φ → Not mutually exclusive.

(iv) Two events which are mutually exclusive but not exhaustive: Let P = {HHH}, Let Q = {TTT} P ∩ Q = φ → mutually exclusive. P ∪ Q = {HHH, TTT} ≠ S → Not exhaustive.

(v) Three events which are mutually exclusive but not exhaustive: Let X = {HHH}, Y = {TTT}, Z = {HHT, HTH, THH} X ∩ Y ∩ Z = φ → mutually exclusive. X ∪ Y ∪ Z = {HHH, TTT, HHT, HTH, THH} ≠ S → Not exhaustive.

Question 6. Two dice are thrown. Events A, B and C are: A: getting an even number on the first die. B: getting an odd number on the first die. C: getting the sum of the numbers on the dice ≤ 5.

Describe the events:

Answer:

A = {(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}

B = {(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)}

C = {(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(3,1),(3,2),(4,1)}

(i) A′: A′ = B = {(1,1),(1,2), … ,(5,6)} (all odd first-die outcomes)

(ii) Not B: Not B = A = {(2,1),(2,2), … ,(6,6)} (all even first-die outcomes)

(iii) A or B (A ∪ B): A ∪ B = S (all 36 outcomes)

(iv) A and B (A ∩ B): A ∩ B = φ (a number cannot be both even and odd)

(v) A but not C (A – C): A – C = {(2,4),(2,5),(2,6),(4,2),(4,3),(4,4),(4,5),(4,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}

(vi) B or C (B ∪ C): B ∪ C = {(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)}

(vii) B and C (B ∩ C): B ∩ C = {(1,1),(1,2),(1,3),(1,4),(3,1),(3,2)}

(viii) A ∩ B′ ∩ C′: Since B′ = A: A ∩ B′ ∩ C′ = A ∩ A ∩ C′ = A ∩ C′ = A – C A ∩ B′ ∩ C′ = {(2,4),(2,5),(2,6),(4,2),(4,3),(4,4),(4,5),(4,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}

Question 7. Refer to question 6 above, state true or false (give reason for your answer):

Answer:

(i) A and B are mutually exclusive A ∩ B = φ → True.

(ii) A and B are mutually exclusive and exhaustive A ∩ B = φ and A ∪ B = S → True.

(iii) A = B′ A = even first-die outcomes; B′ = complement of odd = even first-die outcomes. → True.

(iv) A and C are mutually exclusive A ∩ C = {(2,1),(2,2),(2,3),(4,1)} ≠ φ → False.

(v) A and B′ are mutually exclusive A ∩ B′ = A ∩ A = A ≠ φ → False.

(vi) A′, B′, C are mutually exclusive and exhaustive A′ ∩ B′ = φ ✓ but B′ ∩ C = {(2,1),(2,2),(2,3)} ≠ φ → Not mutually exclusive. A′ ∪ B′ ∪ C = S → Exhaustive. False (exhaustive but not mutually exclusive).

FAQs – Chapter 14 Probability Exercise 14.1

Q1. What are mutually exclusive events? Give an example.

When two events cannot occur at the same time — that is, their intersection is empty — they are called mutually exclusive events. For example, getting a 3 and getting a 4 on a single die throw are mutually exclusive, since both cannot happen simultaneously.

Q2. What is the difference between simple and compound events?

A simple event has exactly one outcome, such as {HHH}. A compound event has more than one outcome, such as {HHT, HTH, THH} (two heads and one tail).

Q3. What does exhaustive events mean?

If the union of certain events equals the entire sample space S, those events are called exhaustive. It means at least one of those events will definitely occur in any trial.

Q4. How do you define a sample space in Exercise 14.1?

For a single die, S = {1,2,3,4,5,6}; for two dice, S has 36 ordered pairs; for three coins, S has 8 outcomes from HHH to TTT. The sample space consists of all possible outcomes of the experiment.

Q5. Is it necessary for mutually exclusive events to be exhaustive?

No — for mutual exclusivity, only the intersection needs to be empty. Exhaustiveness is a separate condition. For example, {1} and {2} are mutually exclusive on a die, but their union {1,2} ≠ S, so they are not exhaustive.