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

(vi) are mutually exclusive and exhaustive.

Here,
S = {(x,y): 1 x,y 6}
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)}
(vi) X and Y are mutually...

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

(v) and are mutually exclusive.

X and Y are mutually exclusive if and only if X Y =
A B' = A A = A (From (iii))
A B’
Hence A and B' not mutually exclusive. FALSE

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

(iv) A and C are mutually exclusive

Here,
S = {(x,y): 1 x,y 6}
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)}
C = {(1, 1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (3,1), (3,2), (4,1)}
(iv) X and Y are mutually exclusive if and only if X Y =
A C = {(2,1), (2,2), (2,3), (4,1)} ,
Hence, A and B are not mutually exclusive. FALSE

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

(iii)

Here,
S = {(x,y): 1 x,y 6}
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)}
(iii) Therefore, B' = S -B = {(2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4),...

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

(ii) A and B are mutually exclusive and exhaustive

Here,
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)}
(ii) X and Y are mutually exclusive if and only if X Y =
A B = , since A and B have no common element amongst them.
Hence, A...

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

(i) A and B are mutually exclusive

Here,
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)}
(i) X and Y are mutually exclusive if and only if X Y =
A B = , since A and B have no common element amongst them.
Hence, A...

**6.(viii) **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

Describe the events

(viii)

Sample space when two dice are thrown:
S = {(x,y): 1 x,y 6}
A: getting an even number on the first die = {(a,b): a {2,4,6} and 1 b 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)}
B: getting an odd number on the first die = {(a,b): a {1,3,5} and 1 b 6}
= {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6),...

**6.(vii) **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

Describe the events

(vii) B and C

Sample space when two dice are thrown:
S = {(x,y): 1 x,y 6}
B: getting an odd number on the first die = {(a,b): a {1,3,5} and 1 b 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)}
C: getting the sum of the numbers on the dice 5
The possible sum are 2,3,4,5
C = {(a,b): 2 a + b 5} = {(1, 1), (1,2), (1,3),...

**6.(vi) **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

Describe the events

(vi) B or C

**6.(v) **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

Describe the events

(v) A but not C

Sample space when two dice are thrown:
S = {(x,y): 1 x,y 6}
A: getting an even number on the first die = {(a,b): a {2,4,6} and 1 b 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)}
C: getting the sum of the numbers on the dice 5
The possible sum are 2,3,4,5
C = {(a,b): 2 a + b 5} = {(1, 1), (1,2),...

**6.(iv) ** 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

Describe the events

(iv) A and B

**6.(iii) **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 .

Describe the events

(iii) A or B

**6.(ii)** 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 .

Describe the events

(ii) not B

Sample space when two dice are thrown:
S = {(x,y): 1 x,y 6}
B: getting an odd number on the first die = {(a,b): a {1,3,5} and 1 b 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)}
(ii) Therefore, 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.(i) **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 .

Describe the events

(i)

Sample space when two dice are thrown:
S = {(x,y): 1 x,y 6}
A: getting an even number on the first die = {(a,b): a {2,4,6} and 1 b 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)}
(i) Therefore, 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. **Three coins are tossed. Describe

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

Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]
S = {HHH, HHT, HTH, HTT, THH, TTH, THT, TTT}
Let ,
A = Getting exactly one tail = {HHT, HTH, THH}
B = Getting exactly two tails = {HTT, TTH, THT}
C = Getting exactly three tails = {TTT}
Clearly, A B = ; B C = ; C A =
Since (A and B), (B and C) and (A and C) are mutually exclusive
Therefore A,...

**5. **Three coins are tossed. Describe

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

Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]
S = {HHH, HHT, HTH, HTT, THH, TTH, THT, TTT}
Let ,
A = Getting exactly one head = {HTT, THT, TTH}
B = Getting exactly one tail = {HHT, HTH, THH}
Clearly, A B =
Hence, A and B are mutually exclusive.
Also, A B S
Hence, A and B are not exhaustive.

**5. **Three coins are tossed. Describe

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

Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]
S = {HHH, HHT, HTH, HTT, THH, TTH, THT, TTT}
Let ,
A = Getting at least one head = {HHH, HHT, HTH, THH, TTH}
B = Getting at most one head = {TTH, TTT}
Clearly, A B = {TTH}
Hence, A and B are two events which are not mutually exclusive.

**5. ** Three coins are tossed. Describe

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

Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]
S = {HHH, HHT, HTH, HTT, THH, TTH, THT, TTT}
Let ,
A = Getting no tails = {HHH}
B = Getting exactly one tail = {HHT, HTH, THH}
C = Getting at least two tails = {HTT, THT, TTH}
Clearly, A B = ; B C = ; C A =
Since (A and B), (B and C) and (A and C) are mutually exclusive
Therefore A, B and C are...

**5. ** Three coins are tossed. Describe

(i) Two events which are mutually exclusive.

Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]
S = {HHH, HHT, HTH, THH, TTH, TTT}
(i)
A = Event that three heads show up = {HHH}
B = Event that three tails show up = {TTT}
A B = {HHH} {TTT} = ; Hence A and B are mutually exclusive.

**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

(iii) Compound?

Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]
S = {HHH, HHT, HTH, THH, TTH, TTT}
Now,
A = Event that three heads show up = {HHH}
B = Event that two heads and one tail show up = {HHT, HTH, THH}
C = Event that three tails show up = {TTT}
D = Event that a head shows on the first coin = {HHH, HHT, HTH, HTT}
(iv). If an event has more than one sample...

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