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

(vi) ${A}',{B}',C$ 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) $A$ and ${B}'$ 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) $A=B{}'$

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  $\leq 5$

Describe the events

(viii) $A\cap {B}'\cap {C}'$

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)} 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  $\leq 5$

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  $\leq 5$

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),...

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  $\leq 5$

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  $\leq 5$

Describe the events

(iv) A and B

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.(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  $\leq 5$.

Describe the events

(iii) A or B

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.(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  $\leq 5$.

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  $\leq 5$.

Describe the events

(i) $A{}'$

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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