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Question

# Three coins are tossed. Describe. (i) two events A and B which are mutually exclusive. (ii) three events A, B and C which are mutually exclusive and exhaustive. (iii) two events A and B which are not mutually exclusive. (iv) two events A and B which are mutually exclusive but not exhaustive.

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Solution

## When three coins are tossed, the sample space is given by S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} (i) The two events that are mutually exclusive are as follows: A: getting no heads B: getting no tails This is because sets A = {HHH} and B = {TTT} are disjoint. (ii) The three events that are mutually exclusive and exhaustive are as follows: A: getting no heads B: getting exactly one head C: getting at least two heads i.e. A = {TTT}, B = {HTT, THT, TTH} and C = {HHH, HHT, HTH, THH} This is because A ∩ B = B ∩ C = C ∩ A = Φ and A ∪ B ∪ C = S (iii) The two events that are not mutually exclusive a A: getting three heads B: getting at least 2 heads i.e. A = {HHH} and B = {HHH, HHT, HTH, THH} This is because A ∩ B = {HHH} ≠ Φ (iv) The two events which are mutually exclusive but not exhaustive are as follows: A: getting exactly one head B: getting exactly one tail i.e. A = {HTT, THT, TTH} and B = {HHT, HTH, THH} It is because, A ∩ B = Φ, but A ∪ B ≠ S

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