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 \(D\) denotes the event a head shows on the first coin”. Which events are (I) mutually exclusive? (ii) simple? (iii) Compound?

Answer 1

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

\( B = \{HHT, HTH, THH\} \)
\(C = \{TTT\} \)
\(D = \{HHH, HHT, HTH, HTT\} \)
We now observe that,
\(A ∩ B =Φ, A ∩ C =Φ, A ∩ D = \{HHH\} ≠ Φ\)
\(B ∩ C =Φ, B ∩ D = \{HHT, HTH\} ≠ Φ\)
\(C ∩ D = Φ\)
(i) Events \(A \) and \(B;\) Events A and \(C;\) Events \(B \) and \(C;\) and Events \(C\) and \(D\) are all mutually exclusive. 
(ii) If an event has only one sample point of a sample space, it is called a simple event. Thus, \(A\) and \(C\) are simple events. 
(iii) If an event has more than one sample point of a sample space, it is called a compound event. Thus, \(B\) and \(D\) are compound events.