**NCERT Solutions for Class 11 Maths Chapter 14 Probability Ex 14.1**

**Ex 14.1 Class 11 Maths 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?

**Solution:**

An experiment involves rolling a die.

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

E: die shows 4 = {4}

F: die shows an even number = {2, 4, 6}

∴ E ∩ F = {4} ⇒ E ∩ F ≠ ⏀

Therefore, E and F are not mutually exclusive.

**Ex ****14.1**** Class 11 Maths Question 2.**

### 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’.

**Solution:**

An experiment involves rolling a die.

∴ Sample space, S = {1, 2, 3, 4, 5, 6}**(i)** A: a number less than 7 = {1, 2, 3, 4, 5, 6}**(ii)** B: a number greater than 7 = ⏀**(iii)** C: a multiple of 3 = {3, 6}**(iv)** D: a number less than 4 = {1, 2, 3}**(v)** E: an even number greater than 4 = {6}**(vi)** F: a number not less than 3 = {3, 4, 5, 6}

A ∪ B = {1, 2, 3, 4, 5, 6) ∪ ⏀

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

A ∩ B = {1, 2, 3, 4, 5, 6) ∩ ⏀ = ⏀

B ∪ C = ⏀ ∪ {3, 6} = {3, 6}

E ∩ F = {6} ∩ {3, 4, 5, 6) = {6}

D ∩ E = {1, 2, 3} ∩ (6} = ⏀

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

D – E = {1, 2, 3} – {6} = {1, 2, 3}

F’ = {1, 2, 3, 4, 5, 6) – {3, 4, 5, 6) = {1, 2)

E ∩ F’={6} ∩ {1, 2} = ⏀

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**Ex ****14.1**** Class 11 Maths 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?

**Solution:**

An experiment involves rolling a pair of dice.

∴ Sample space = 6 × 6 = 6^{2} = 36 possible outcomes.

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

Now,

A: the sum is greater than 8

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

B: 2 occurs on either die = {(1, 2), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 2), (4, 2), (5, 2), (6, 2)}

C: The sum is at least 7 and a multiple of 3 = {(3, 6), (4, 5), (5, 4), (6, 3), (6, 6)}

A ∩ B =⏀, B ∩ C = ⏀

Thus, the above relations show that A and B; B and C are mutually exclusive events.

**Ex ****14.1**** Class 11 Maths 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?

**Solution:**

An experiment involves tossing three coins:

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

The given events are:

A: Three heads show = {HHH}

B: Two heads and one tail show = {HHT, HTH, THH}

C: Three tails show = {TTT}

D: A head shows on the first coin = {HHH, HHT, HTH, HTT}**(i)** Since A ∩ B = ⏀, A ∩ C = ⏀, B ∩ C = ⏀, C ∩ D = ⏀.

⇒ A and B; A and C; B and C; C and D are mutually exclusive events.**(ii)** A and C are simple events.**(iii)** B and D are compound events.

** **

**Ex ****14.1**** Class 11 Maths Question 5.**

### Three coins are tossed. Describe

**(i)** Two events which are mutually exclusive.

**(ii)** Three events which are mutually exclusive and exhaustive.

**(iii)** Two events, which are not mutually exclusive.

**(iv)** Two events which are mutually exclusive but not exhaustive.

**(v)** Three events which are mutually exclusive but not exhaustive.

**Solution:**

An experiment involves tossing three coins.

The sample space, S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}**(i)** Two events A and B which are mutually exclusive are:

A: “getting at most one head” and B: “getting at most one tail”

**(ii)** Three events A, B and C which are mutually exclusive and exhaustive are:

A: “getting at least two heads”

B: “getting exact two tails” and C: “getting exactly three tails”

**(iii)** Two events A and B which are not mutually exclusive are:

A: “getting exactly two tails” and B: “getting at most two heads”

**(iv)** Two events A and B which are mutually exclusive but not exhaustive are:

A: “getting at least two heads” and B: “getting at least three tails”

**(v)** Three events A, B and C which are mutually exclusive but not exhaustive are:

A: “getting at least three tails”

B: “getting at least three heads”

C: “getting exactly two tails”

** **

**Ex ****14.1**** Class 11 Maths Question 6.**

### 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’

**Solution:**

An experiment involves rolling two dice.

The sample space, S = 6 × 6 = 36 outcomes.

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

A: getting an even number on the first die = {(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: getting an odd number on the first die = {(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: getting the sum of the numbers on the dice ≤ 5 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1)}**(i)** A’: getting an odd number on the first die = B**(ii)** not B: getting an even number on the first die = A**(iii)** A or B = A ∪ B = S

∴ A ∪ B = {(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), (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)}**(iv)** A and B = A ∩ B = ⏀**(v)** A but not 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 = {(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 = {(1, 1), (1, 2), (1, 3), (1, 4), (3, 1), (3, 2)}**(viii)** A: getting an even number on the first die = B’

B’: getting an even number on the first die = {(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)}

C’: getting the sum of numbers on two dice > 5 = {(1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5) (3, 6), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}

∴ 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)}

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**Ex ****14.1**** Class 11 Maths Question 7.**

### 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.

**Solution:****(i) True.**

A = getting an even number on the first die.

B = getting an odd number on the first die.

There are no common elements in A and B.

⇒ A ∩ B = ⏀

∴ A and B are mutually exclusive.

**(ii) True.**

From (i), A and B are mutually exclusive.

A ∪ B = {(1, 1), (1, 2), … (1, 6), (2, 1), (2, 2), … (2, 6),…, (6, 1), (6, 2), …, (6, 6)} = S

∴ A ∪ B is mutually exhaustive.

**(iii) True.**

B = getting an odd number on the first die.

B’ = getting an even number on first die = A.

∴ A = B’

**(iv) False.**

We have, A ∩ C = {(2, 1), (2, 2), (2, 3), (4, 1)}

Since A ∩ C ≠ ⏀, therefore, A and C are not mutually exclusive.

**(v) False.**

Since A = B’ [from (iii)]

∴ A ∩ B’= A ∩ A = A ≠ ⏀

**(vi) False.**

Since A’ = B and B’ = A, A’ ∩ B’ = ⏀

B ∩ C = {(1, 1), (1, 2), (1, 3), (1, 4), (3, 1), (3, 2)} ≠ ⏀

A ∩ C = {(2, 1), (2, 2), (2, 3), (4, 1)} ≠ ⏀

Thus, A’, B’ and C are not mutually exclusive.

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