Question

Three coins are tossed once. Let A denote the event ‘three heads show”, B denote the event “two heads and one tail show”. C denote the event “three tails show” and D denote the event ‘a head shows on the first coin”. Which events are (i) mutually exclusive? (ii) simple? (iii) compound?

Answer 1

In the given experiment, three coins are tossed. A coin has two faces, head (H) and tail (T).

Let, A be the event “three heads show”.

A=( H,H,H )

Let, B be the event “two heads and one tail show”.

B={ ( H,H,T ),( H,T,H ),( T,H,H ) }

Let, C be the event “three tails show”.

C=( T,T,T )

Let, D be the event “a head shows on the first coin”.

D={ ( H,H,H ),( H,H,T ),( H,T,H ),( H,T,T ) }

(i)

For event A and B,

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

Thus, A and B are mutually exclusive.

For event A and C,

A∩C={ ( H,H,H )∩( T,T,T ) } =ϕ

Thus, A and C are mutually exclusive.

For event A and D,

A∩D={ ( H,H,H )∩{ ( H,H,H ),( H,H,T ),( H,T,H ),( H,T,T ) } } =( H,H,H )

Thus, A and D are not mutually exclusive.

For event B and C,

B∩C={ { ( H,H,T ),( H,T,H ),( T,H,H ) }∩( T,T,T ) } =ϕ

Thus, B and C are mutually exclusive.

For event B and D,

B∩D={ { ( H,H,T ),( H,T,H ),( T,H,H ) }∩{ ( H,H,H ),( H,H,T ),( H,T,H ),( H,T,T ) } } ={ ( H,H,T ),( H,T,T ) }

Thus, B and D are not mutually exclusive.

For event C and D,

C∩D={ ( T,T,T )∩{ ( H,H,H ),( H,H,T ),( H,T,H ),( H,T,T ) } } =ϕ

Thus, C and D are mutually exclusive.

(ii)

A simple event is one which has only a single sample point of a sample space.

Event A and C have only a singlesample point. Thus, the events A and C are simple events.

(iii)

A compound event is one having more than one sample point of a sample space.

Since events B and C have more than one sample points, thus they are compound events.