Mutually Exculsive and Exhaustive Events

Chapter 16 Class 11 Probability
Concept wise

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### Transcript

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 has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.