Gujarat Board GSEB Textbook Solutions Class 11 Maths Chapter 16 Probability Ex 16.2 Textbook Questions and Answers.

## Gujarat Board Textbook Solutions Class 11 Maths Chapter 16 Probability Ex 16.2

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:

When we throw a die, it can result in any one of the six numbers, 1, 2, 3, 4, 5, 6 and

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 ≠ ϕ.

⇒ E and F are not mutually exclusive.

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, E ∪ F, D ∩ E, A – C, D – E, F’ and E ∩ F’.

Solution:

When we throw a die, it can result in any one of the six numbers 1, 2, 3, 4, 5, 6 and

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 = { } = ϕ

(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} ∪ ϕ = {1, 2, 3, 4, 5, 6}

A ∩ B = (1, 2, 3, 4, 5, 6} ∩ ϕ = ϕ

E ∪ F = {6} ∪ (3, 4, 5, 6} = (3, 4, 5, 6}

D ∩ E = {1, 2, 3} ∩ {6} = ϕ

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} = 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.

Also, find A ∩ B, B ∩ C and A ∩ C.

Are

- A and B mutually exclusive?
- B and C mutually exclusive?
- A and C mutually exclusive?

Solution:

When two dice are thrown, there are 6 × 6 = 6^{2} possible outcomes and

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 : the sum is greater than 8 = {(3, 6), (4, 5), (5, 4), (6, 3), (4, 6), (5, 5), (6, 4), (5, 6), (6, 5), (6, 6)}.

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

C : the sum is at least 7 and a multiple of 3

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

A ∩ B = ϕ, B ∩ C = ϕ

and A ∩ C = {(3, 6), (6, 3), (5, 4), (4, 5), (6, 6)}.

- Since A ∩ B = ϕ, so A and B are mutually exclusive.
- Since B ∩ C = ϕ so B and C are mutually exclusive.
- Since A ∩ C ≠ ϕ, so A and C are not mutually exclusive.

Question 4.

Three coins are tossed once. Let A denotes the event “three heads show”, B denotes the event “two heads and one tail shows”, C denotes the event “three tails show” and D denote the event “a head shows on the first coin”.

Which events are

- mutually exclusive?
- simple?
- compound?

Solution:

When three coins are tossed, then the simple space S is

= {HHH, HHT, HTH, HIT, THH, THT, TTH, TTT}

∴ A : Three heads show = {HHH}

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

C : A three tails show = {TTT}

D : A head show on the first coin = {HHH, HHT, HTH, HTT}

- Since A ∩ B = ϕ, A ∩ C = ϕ, B ∩ C = ϕ, C ∩ D = ϕ, A ∩ B ∩ C = ϕ.

⇒ A and B; A and C; B and C; C and D and A, B and C are mutually exclusive. - A and C are simple events.
- B and D are compound events.

Question 5.

Three coins are tossed. Describe:

- two events which are mutually exclusive.
- three events which are mutually exclusive and exhaustive.
- two events which are not mutually exclusive.
- two events which are mutually exclusive but not exhaustive.
- three events which are mutually exclusive but not exhaustive.

Solution:

When three coins are tossed, then the samole space S is

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

1. two events A and B which are mutually exclusive are

A : “getting at least two heads” and

B : “getting at least two tails”.

2. Three events A, B and C which are mutually exclusive and exhaustive are

A : “getting at most one head”

B : “getting exactly two heads” and

C : “getting exactly three heads”

Alternatively

Getting no head,

Getting exactly one head Getting at least two heads

3. Two events A and B which are not mutually exclusive are

A : “getting at most two tails” and

B : “getting exactly two heads” or “getting exactly two tails”

4. Two events A and B which are mutually exclusive but not exhaustive are

A : “getting exactly one head” and

B : “getting exactly two heads”.

5. Three events A, B and C which are mutually exclusive but not exhaustive are

A : “getting exactly one tail”

B : “getting exactly two tails” and

C : “getting exactly three tails”.

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 of 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:

When two dice are thrown, there are 6 × 6 = 36 possible outcomes and

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 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 = getting an even number on the first die

B : getting an odd number on the first die.

A or B = A ∪ B = S = {1, 2, 3, 4, 5, 6} appear on the first die as well as on the second die.

A ∪ 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), (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)} = S.

(iv) A and B = A ∩ B = ϕ

(v) A but not C

A : getting are even number on the first die.

not C : getting the sum of numbers on two dice > 5.

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

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

(vi) B : getting an odd number of first die.

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

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’ = getting an even number on the first die.

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

Question 7.

Refer to question 6 above, state true or false:

(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.

Since A : getting an even number on the first die.

B = getting an odd number on the first die.

There is no common element in A and B.

⇒ A ∩ B = ϕ.

∴ A and B are mutually exclusive.

(ii) True. A and B are mutually exclusive.

A ∪ B = {1, 2, 3, 4, 5, 6} × (1, 2, 3, 4, 5, 6} = {(1, 1), (1, 2), …, (1, 6), (2, 1), (2, 2), …, (2, 6) … (6, 1), (6, 2), …, (6, 6)}

∴ A ∪ B is 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. Since A ∩ C = {(2, 1), (2, 2), (2, 3), (4,1)} ≠ ϕ.

(v) False. Since B’ = A, So, A ∩ B’ = A ∩ A = A ≠ ϕ.

(vi) False. Since A’and B’are mutually exclusive.

A’ ∩ B’ = ϕ ⇒ A’ ∩ B’ ∩ C = ϕ.

But A’ ∩ C ≠ ϕ, B’ ∩ C ≠ ϕ ⇒ A’, B’ and C are not mutually exclusive.