Question

Three coins are tossed. Describe (i) Two events which are mutually exclusive. (ii) Three events which are mutually exclusive and exhaustive. (iii) Two events, which are not mutually exclusive. (iv) Two events which are mutually exclusive but not exhaustive. (v) Three events which are mutually exclusive but not exhaustive.

Answer 1

If three coins are tossed, the sample space, S is given by, S={ ( H,H,H ),( T,T,T ),( H,T,T ),( T,H,T ),( T,T,H ),( H,H,T ),( H,T,H ),( T,H,H ) }

(i)

Two events, A and B are mutually exclusive if, A∩B=ϕ.

Thus, two mutually exclusive events are,

A:getting all heads. A={ ( H,H,H ) }

B: getting all tails. B=( T,T,T )

(ii)

Three events, A, B and C are exhaustive if A∪B∪C=S.

Two events, A and B are mutually exclusive if, A∩B=ϕ.

Thus, three events which are mutually exclusive and exhaustive are,

A: getting no heads. A={ ( T,T,T ) }.

B: getting exactly one head. B={ ( H,T,T ),( T,H,T ),( T,T,H ) }.

C: getting at least two heads. C={ ( H,H,H ),( H,H,T ),( H,T,H ),( T,H,H ) }.

(iii)

Two events, which are not mutually exclusive are,

A: getting no heads. A={ ( T,T,T ) }.

B: getting at least twotail. B={ ( H,T,T ),( T,H,T ),( T,T,H ),( T,T,T ) }.

(iv)

Two events, A and B are mutually exclusive if, A∩B=ϕ.

Two events, A and B are exhaustive if, A∪B=S.

Two events, which are mutually exclusive but not exhaustive are,

A: getting all heads. A={ ( H,H,H ) }.

B: getting all tails. B=( T,T,T ).

A∩B={ ( H,H,H )∩( T,T,T ) } =ϕ A∪B={ ( H,H,H ),( T,T,T ) } ≠S

(v)

Two events, A and B are mutually exclusive if, A∩B=ϕ.

Three events, A, B and C are exhaustive if, A∪B∪C=S.

Three events are mutually exclusive but not exhaustive are,

A: getting all heads. A={ ( H,H,H ) }.

B: getting exactly one head. B={ ( H,T,T ),( T,H,T ),( T,T,H ) }.

C: getting one tail and two heads. C={ ( H,H,T ),( H,T,H ),( T,H,H ) }.