**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 consists of rolling a die.

∴ 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 ≠ ⏀

⇒ 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)** 8: 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 consists of rolling a die.

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

**(i)** A: a number less than 7 = {1, 2, 3, 4, 5, 6}

**(ii)** B: a number less 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)∩{l, 2}= ⌽

**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 consists of rolling a pair of dice.

∴ Sample space consists 6 x 6 = 62 = 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 above shows that A and B; B and C are mutually exclusive events.

**Question 4.**

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

**(i)** Mutually exclusive?

**(ii)** Simple?

**(iii)** Compound?

**Solution:**

An experiment consists of tossing threecoins:

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

**A : **Three heads show = {HHH}**B : **Two heads and one tail show = {HHT, HTH, THH}**C : **Three tail show = {TTT}**D : **A head show 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.

**Question 5.**

**Three coins are tossed. Describe**

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

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

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

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

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

**Solution:**

An experiment consists of tossing three coins then the sample space S is S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

**(i) **Two events A and B which are mutually exclusive are

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

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

A : “getting atleast 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 almost two heads”

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

A : “getting atleast two heads” and B: “getting atleast three tails”

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

A : “getting atleast three tails”

B : “getting atleast three heads”

C : “getting exactly two 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 on the dice ≤ 5.

**Describe the events**

**(i) **A’**(ii)** not B**(iii)** A or B**(iv)** A and B**(v)** A but bot C**(vi)** B or C**(vii)** B and C**(viii)** A ∩ B’ ∩C’

**Solution:**

An experiment consists of rolling two dice Sample space consists 6 x 6 = 36 outcomes.

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

**(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 = BuC = {(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), B or C = BuC = {(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= **{(l, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (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)}

**Question 7.**

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

**(i)** A and B are mutually exclusive

**(ii)** A and 6 are mutually exclusive and exhaustive

**(iii)** A = B’

**(iv)** A and C are mutually exclusive

**(v)** A and S’ 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 is 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.**

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

**(v) False.**

Since B’ = A [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.