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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...
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
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
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),...
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...
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...
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),...
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),...
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),...
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),...
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),...
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),...
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),...
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),...
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,...
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.
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.
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...
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.
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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