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#### 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 exclusive if and only if X  Y =

A'  B' = B  A =  (from (iii) and (i))

Hence A' and B' are mutually exclusive.

Again,

B'  C = A  C   (from (iv))

Hence B' and C are not mutually exclusive.

Hence, A', B' and C are not mutually exclusive and exhaustive. FALSE

#### 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 =

B' = A  A = A   (From (iii))

A  B’

Hence A and B' not mutually exclusive. FALSE

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#### 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 =

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

TRUE

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#### 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 =

B =  , since A and B have no common element amongst them.

Hence, A and B are mutually exclusive.

Also,

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

Hence, A and B are exhaustive.

TRUE

#### 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 =

B =  , since A and B have no common element amongst them.

Hence, A and B are mutually exclusive. TRUE

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#### 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), (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), (1,4), (2,1), (2,2), (2,3), (3,1), (3,2), (4,1)}

(viii) A  B'  C' = A  A  C'  (from (ii))

= A  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)}

#### 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), (1,4), (2,1), (2,2), (2,3), (3,1), (3,2), (4,1)}

(vi) B and C = B  C = {(1, 1), (1,2), (1,3), (1,4), (3,1), (3,2)}

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#### 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

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), (1,4), (2,1), (2,2), (2,3), (3,1), (3,2), (4,1)}

(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)}

#### 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), (1,3), (1,4), (2,1), (2,2), (2,3), (3,1), (3,2), (4,1)}

(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)}