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?
Sample Space:
When a pair of dice is rolled, outcomes are ordered pairs (i, j) where
i, j ∈ {1, 2, 3, 4, 5, 6}
Event A: The sum is greater than 8
Possible sums: 9, 10, 11, 12
A = { (3,6), (4,5), (5,4), (6,3), (4,6), (5,5), (6,4), (5,6), (6,5), (6,6) }
Event B: 2 occurs on either die
B = { (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (1,2), (3,2), (4,2), (5,2), (6,2) }
Event C: The sum is at least 7 and a multiple of 3
Possible sums ≥ 7 and multiples of 3 are: 9 and 12
C = { (3,6), (4,5), (5,4), (6,3), (6,6) }
Checking mutually exclusive pairs:
A and B:
A ∩ B = { (4,5)? no 2, (5,4)? no 2, (3,6)? no 2, (6,3)? no 2, (6,6)? no 2 }
Since no outcome contains 2,
A ∩ B = ∅
Hence, A and B are mutually exclusive.
A and C:
C ⊂ A
A ∩ C ≠ ∅
Hence, A and C are not mutually exclusive.
B and C:
No outcome in C contains the number 2.
B ∩ C = ∅
Hence, B and C are mutually exclusive.
Final Answer:
The mutually exclusive pairs are:
(A, B) and (B, C)
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 \(A∪B, A∩B, B∪C, E∩F, D∩E, A-C, D-E, E∩F', F'\)
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?