Three coins are tossed. Describe

When three coins are tossed, the sample space is given by
S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
 

(i) Two events that are mutually exclusive can be

A: getting no heads and

B: getting no tails This is because sets

A = {TTT} and B = {HHH} are disjoint.
 

(ii) Three events that are mutually exclusive and exhaustive can be

A: getting no heads

B: getting exactly one head

C: getting at least two heads

A = {TTT}

B = {HTT, THT, TTH}

C = {HHH, HHT, HTH, THH}

This is because A ∩ B = B ∩ C =C ∩ A = φ and A U B U C = S

(iii) Two events that are not mutually exclusive can be A: getting three heads B: getting at least 2 heads i.e.,

A = {HHH}

B = {HHH, HHT, HTH, THH}

This is because A ∩ B = (HHH) ≠ φ

(iv) Two events which are mutually exclusive but not exhaustive can be

A: getting exactly one head

B: getting exactly one tail

A = {HTT, THT, TTH}

B = {HHT, HTH, THH}

This is because A ∩ B = φ, but A U B ≠ S

(v) Three events that are mutually exclusive but not exhaustive can be

A: getting exactly three heads

B: getting one head and two tails

C: getting one tail and two heads

A = {HHH}

B = {HTT, THT, THH}

C = {HHT, HTH, THH}

This is because A ∩ B = B ∩ C = C ∩ A= φ, but A U B U C ≠ S