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Question
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?
Solution
When two dice are thrown, the total number of possible come up is
= 6 × 6
= 36
A = The sum of the numbers obtained is more than 8.
= {(3, 6), (4, 5), (5, 4), (6, 3), (4, 6), (5, 5), (6, 4), (5, 6), (6, 5), (6, 6)}
B = number 2 occurs on at least one dice
= {(1, 2), (2, 2), (3, 2), (4, 2), (5, 2), (6, 2), (2, 1), (2, 3), (2, 4), (2, 5), (2, 6)}
C = The sum of the numbers revealed is at least 7 and is a multiple of 3.
= The sum of the numbers occurs is 9 and 12 which is a multiple of 3.
= {(3, 6), (6, 3), (4, 5), (5, 4), (6, 6)}
A ∩ C = {(3, 6), (4, 5), (5, 4), (6, 3), (4, 6), (5, 5), (6, 4), (5, 6), (6, 5), (6, 6)} ∩ {(3, 6), (6, 3), (5, 4), (6, 6)}
= {(3, 6), (6, 3), (4, 5), (5, 4), (6, 6)}
A ∩ B = {(3, 6), (6, 3), (4, 5), (5, 4), (4, 6), (6, 4), (5, 5), (5, 6), (6, 5), (6, 6) ∩ {(1, 2), (3, 2), (2, 1), (2, 3), (4, 2), (2, 4), (5, 2), (2, 5), (2, 6), (6, 2)}
= ϕ
B ∩ C = {(1, 2), (2, 1), (2, 2), (2, 3), (3, 2), (2, 4), (4, 2), (2, 5), (5, 2), (2, 6), (6, 2)} ∩ {(3, 6), (6, 3), (4, 5), (5, 4), (6, 6)}
= ϕ
A ∩ B = ϕ , B ∩ C = ϕ i.e. A and B, B and C are mutually exclusive.
But A ∩ C ≠ ϕ,
Hence, A and C are not mutually exclusive.
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