Straight Lines

Question
CBSEENMA11014629

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 event ‘a head shows on the first coin’, which events are: (i) multually exclusive? (ii) simple? (iii) compound?

Solution

Three coins are tossed. The sample space is:
          left curly bracket stack HHH comma with underbrace below space stack HHT comma space HTH comma space THH comma with underbrace below space stack TTH comma space THT comma space HTT comma with underbrace below TTT with underbrace below right curly bracket
Event A: three heads show.
rightwards double arrow            A = {HHH}
Event B: two heads and one tail shows.
rightwards double arrow            B = {HHT, HTH, THH}
Event C: three tails show.
rightwards double arrow            C= {TTT}
Event D: a head shows on the first coin.
rightwards double arrow            D = {HHH, HHT, HTH, HTT}
(i)            straight A intersection straight B space equals space left curly bracket HHH right curly bracket space intersection space left curly bracket HHT comma space THH comma space HTH right curly bracket space equals left curly bracket space right curly bracket equals space straight ϕ
∴  A and B are mutually exclusive.
              straight A intersection straight C space equals space left curly bracket HHH right curly bracket space intersection space left curly bracket TTT right curly bracket space equals space straight ϕ
∴ A and C are mutually exclusive.
              straight A intersection straight D space equals space left curly bracket HHH right curly bracket space intersection space left curly bracket HHH comma space HHT comma space HTH comma space HTT right curly bracket space equals space left curly bracket HHH right curly bracket space not equal to space straight ϕ
∴ A and D are not mutally exclusive
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∴ B and C are mutually exclusive.
              straight B intersection straight D space equals space left curly bracket HHT comma space HTH comma space THH right curly bracket space intersection space left curly bracket HHH comma space HHT comma space HTH comma space HTT right curly bracket
                       space space equals left curly bracket HHT comma space HTH right curly bracket space not equal to space straight ϕ
 ∴ B and D are not mutually exclusive.
              space space straight C intersection straight D space equals space left curly bracket TTT right curly bracket space intersection space left curly bracket HHH comma space HHT comma space HTH comma space HTT right curly bracket space equals space straight ϕ
∴ C and D are mutually exclusive.
(ii)                          A = {HHH}             rightwards double arrow                n (A) = 1
∴ A is a simple event.
                             C = {TTT}               rightwards double arrow                n(C) = 1
∴ C is a simple event.
(iii)                         B = {HHT, HTH, THH}
rightwards double arrow                      n(B) = 3 > 1
∴ B is a compound event.
                            D = {HHH, HHT, HTH, HTT}
rightwards double arrow                     n(D) = 4 > 1
∴ D is a compound event.