Two dice are rolled, A is the event that the sum of the number shown o

1.	Two dice are rolled, A is the event that the sum of the number shown on the two dice is 5 and B is the event that at least one of the dice shown up 3. Are the two events A and B (i) mutually exclusive? (ii) exhaustive?
Answer» When two dice are rolled, sample space is– 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 = {(1, 4}, (2, 3), (3, 2), (4, 1)} B = {(1, 3), (2, 3), (3, 1), (3, 2) (3, 3), (3, 4), (3, 5), (3, 6), (4, 3), (5, 3), (6, 3)} A ∩ B = {(2, 3), (3, 2)} and A ∪ B ≠ S. (i) A and B are not mutually exclusive. (ii) A and B are not exhaustive.

Two dice are rolled, A is the event that the sum of the number shown on the two dice is 5 and B is the event that at least one of the dice shown up 3. Are the two events A and B (i) mutually exclusive? (ii) exhaustive?

Answer»

When two dice are rolled, sample space is–

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

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

A ∩ B = {(2, 3), (3, 2)} and A ∪ B ≠ S.

(i) A and B are not mutually exclusive.

(ii) A and B are not exhaustive.

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