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?
Sample Space:
When three coins are tossed once,
S = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }
Given Events:
A = Three heads show = { HHH }
B = Two heads and one tail show = { HHT, HTH, THH }
C = Three tails show = { TTT }
D = A head shows on the first coin = { HHH, HHT, HTH, HTT }
(i) Mutually exclusive events
Events are mutually exclusive if they have no common outcomes.
A ∩ B = ∅, A ∩ C = ∅, B ∩ C = ∅
Hence, the mutually exclusive events are:
A and B, A and C, B and C
(ii) Simple events
A simple event contains only one outcome.
Event A = { HHH }
Event C = { TTT }
Hence, the simple events are:
A and C
(iii) Compound events
A compound event contains more than one outcome.
Event B = { HHT, HTH, THH }
Event D = { HHH, HHT, HTH, HTT }
Hence, the compound events are:
B and D
Final Summary:
Mutually exclusive events: A & B, A & C, B & C
Simple events: A, C
Compound events: B, D
A die is thrown. Describe the following events:
(i) \(A: a\) number less than \(7\)
(ii) \(B: a\) number greater than \(7\)
(iii) \(C: a\) multiple of \(3\)
(iv) \(D: a\) number less than \(4\)
(v) \(E: a\) even number greater than \(4\)
(vi) \(F: a\) number not less than \(\)\(3\)
Also, find \(A∪B, A∩B, B∪C, E∩F, D∩E, A-C, D-E, E∩F', F'\)
An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events: \(A:\) the sum is greater than \(8\), \(B:\)\(2\) occurs on either die \(C:\)The sum is at least \(7\), and a multiple of \(3\). Which pairs of these events are mutually exclusive?