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**Exercise 16.2**

**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?**

**Ans. **When a die is rolled, then S = {1, 2, 3, 4, 5, 6}

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

Now, E F = {4}

Therefore, E and F are not mutually exclusive events.

**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**

**Ans. **When a die is thrown, then 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}

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

BC = {3, 6}EF = {6}

DE = A – C = {1, 2, 4, 5}

D – E = {1, 2, 3}EF’ =

F’ = {1, 2}

**3. An experiment involves rolling a pair of dice and recording the numbers 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?**

**Ans. **When a pair of die is rolling then

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

AB = therefore, A and B are mutually exclusive events.

BC = therefore, B and C are mutually exclusive events.

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

therefore, A and C are not mutually exclusive events.

**4. Three coins are tossed once. Let A denotes the event “three heads show”, B denotes the event “two heads and one tail show”, C denotes the event “three tails show” and D denotes the event “a head shows on the first coin”. Which events are (i) mutually exclusive, (ii) simple, (iii) compound?**

**Ans. **When three coins are tossed then S = {HHH, HHT, HTH, THH, TTH, HTT, TTT}

**A:** three heads shows = {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) **AB = therefore, A and B are mutually exclusive events.

BC = therefore, B and C are mutually exclusive events.

AD = {HHH} therefore, A and D are not mutually exclusive events.

BD = {HHT, HTH} therefore, B and D are not mutually exclusive events.

CD = therefore, C and D are mutually exclusive events.

**(ii) **A = {HHH}, Here therefore, A is a simple event.

C = {TTT}, Here therefore, C is a simple event.

**(iii) **B = {HHT, HTH, THH}, Here therefore, B is a compound event.

D = {HHH, HHT, HTH, HTT} Here therefore, B is a compound event.

**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.**

**Ans. **When three coins are tossed then S = {HHH, HHT, HTH, THH, TTH, HTT, TTT}

**(i) **A: getting at least two heads = {HHH, HHT, HTH, THH}

B: getting at least two tails = {HTT, THT, TTH, TTT}

AB = therefore, A and B are mutually exclusive events.

**(ii) **A: getting at least two heads = {HHH, HHT, HTH, THH}

B: getting exactly one head = {HTT, THT, TTH}

C: getting no head = {TTT}

AB = therefore, A and B are mutually exclusive events.

BC = therefore, B and C are mutually exclusive events.

AC = therefore, A and C are mutually exclusive events.

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

Therefore, A, B, C are mutually exclusive events.

**(iii) **A: getting at least two heads = {HHH, HHT, HTH, THH}

B: getting exactly two heads = {HHT, HTH, THH}

AB = {HHT, HTH, THH}

Therefore, A and B are not mutually exclusive events.

**(iv) **A: getting three heads = {HHH}

B: getting three tails = {TTT}

AB = therefore, A and B are mutually exclusive events.

AB = {HHH, TTT} S therefore, A and B are not exhaustive.

**(v) **A: getting three heads = {HHH}

B: getting exactly two heads = {HHT, HTH, THH}

C: getting three tails = {TTT}

AB = therefore, A and B are mutually exclusive events.

BC = therefore, B and C are mutually exclusive events.

AC = therefore, A and C are mutually exclusive events.

Also ABC = {HHH, HHT, HTH, THH, TTT} S

Therefore, A, B, C are not mutually exclusive events.

**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’**

**Ans. **When two dice are thrown, then

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 number 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’ = {(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)} = B

**(ii) **not 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)} = A

**(iii) **A or B = {(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)} = S

**(iv) **A and B = AB =

**(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 = BC = {(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 = BC = {(1, 1), (1, 2), (1, 3), (1, 4), (3, 1), (3, 2)}

**(viii) **AB’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)}

**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’**

**Ans. **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 number 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) **AB = therefore, A and B are mutually exclusive events. True

**(ii) **AB = and AB = S

Therefore, A and B are mutually exclusive and exhaustive events. True

**(iii) **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)} = A True

**(iv) **AC = {(2, 1), (2, 2), (2, 3), (4, 1)} =

Therefore, A and C are not mutually exclusive events. False

**(v) **AB’ = A Therefore, A and B’ are not mutually exclusive events. False

**(vi) **Since A’ = B and B’ = A, AB = , BC = {(1, 1), (1, 2), (1, 3), (1, 4), (3, 1), (3, 2)

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

Therefore, A’, B’ and C are not mutually exclusive. False