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Example 7 - Two dice are thrown, sum of numbers which come up - Mutually Exculsive and Exhaustive EventsSlide14.JPGSlide15.JPGSlide16.JPG


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Example 7 Two dice are thrown and the sum of the numbers which come up on the dice is noted. Let us consider the following events associated with this experiment A: ‘the sum is even’. B: ‘the sum is a multiple of 3’. C: ‘the sum is less than 4’. D: ‘the sum is greater than 11’. Which pairs of these events are mutually exclusive? If two dice are thrown then possible outcomes are 1, 2, 3, 4, 5 & 6 on both dies Hence S = ﷐﷐﷐1, 1﷯,﷐1, 2﷯,﷐1, 3﷯,﷐1, 4﷯,﷐1, 5﷯,﷐1, 6﷯﷮﷐2, 1﷯,﷐2, 2﷯,﷐2, 3﷯,﷐2, 4﷯,﷐2, 5﷯,﷐2, 6﷯﷮﷐3, 1﷯,﷐3, 2﷯,﷐3, 3﷯,﷐3, 4﷯,﷐3, 5﷯,﷐3, 6﷯﷮﷐4, 1﷯,﷐4, 2﷯,﷐4, 3﷯,﷐4, 4﷯,﷐4, 5﷯,﷐4, 6﷯﷮﷐5, 1﷯,﷐5, 2﷯,﷐5, 3﷯,﷐5, 4﷯,﷐5, 5﷯,﷐5, 6﷯﷮﷐6, 1﷯,﷐6, 2﷯,﷐6, 3﷯,﷐6, 4﷯,﷐6, 5﷯,﷐6 6﷯,﷯﷯ A : the sum is even So, Sum can be 2, 4, 6, 8, 10, 12 A = ﷐﷐﷐1, 1﷯,﷐1, 3﷯, ﷐1, 5﷯﷮﷐2, 2﷯, ﷐2, 4﷯, ﷐2, 6﷯﷮﷐3, 1﷯,﷐3, 3﷯,﷐3, 5﷯,﷮﷐4, 2﷯, ﷐4, 4﷯, ﷐4, 6﷯﷮﷐5, 1﷯,﷐5, 3﷯,﷐5, 5﷯, ﷮﷐6, 2﷯,﷐6, 4﷯,﷐6, 6﷯,﷯﷯ B: The Sum is multiple of 3 the multiple of 3 are 3, 6, 9, 12 B = ﷐﷐(1,2),(2,1),(1,5),(5,1),(3,6),(2,4),﷮(4,2),(3,6),(6,3),(4,5),(5,4),(6,6)﷯﷯ C: Sum is less than 4 Hence sum possible are 1, 2 and 3 C = ﷐(1,1),(2,1),(1,2)﷯ D: ‘the sum is greater than 11’. So, sum can be 12 D = {(6,6)} Now, A = ﷐﷐﷐1, 1﷯,﷐1, 3﷯, ﷐1, 5﷯﷮﷐2, 2﷯, ﷐2, 4﷯, ﷐2, 6﷯﷮﷐3, 1﷯,﷐3, 3﷯,﷐3, 5﷯,﷮﷐4, 2﷯, ﷐4, 4﷯, ﷐4, 6﷯﷮﷐5, 1﷯,﷐5, 3﷯,﷐5, 5﷯, ﷮﷐6, 2﷯,﷐6, 4﷯,﷐6 6﷯,﷯﷯ B = ﷐﷐(1,2),(2,1),(1,5),(5,1),(3,3),(2,4),﷮(4,2),(3,6),(6,3),(4,5),(5,4),(6,6)﷯﷯ C = ﷐(1,1),(2,1),(1,2)﷯ D = {(6,6)} If 2 elements are Mutually exclusive , then there should not be any common element A ∩ B ≠ φ A ∩ C ≠ φ, A ∩ D ≠ φ, B ∩ C ≠ φ and B ∩ D ≠ φ. But C ∩ D = φ So C and D are mutually exclusive events.

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can contact him here.
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