Ex 16.2, 6 - Two dice are thrown. The events A, B, C are - Algebra of events

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  1. Class 11
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Ex16.2, 6 Two dice are thrown. The events A, B and C are as follows: A: getting an even number on the first die. B: getting an odd number on the first die. C: getting the sum of the numbers on the dice ≤ 5 Describe the events If 2 dies are thrown then possible outcomes are 1, 2, 3, 4, 5, 6 on both dies 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: getting an even number on the first die A = (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),﷮(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), ﷮(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)﷯﷯ B: getting an odd number on the first die B = (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),﷮ (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),﷮ (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) ﷯﷯ C: getting the sum of the numbers on the dice ≤ 5 C = (1, 4)(2, 3)(3, 2)(4, 1)(1, 3)﷮ 2, 2﷯ 3, 1﷯ 1, 2﷯ 2, 1﷯(1, 1)﷯﷯ Ex16.2, 6 • A’ 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 = (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),﷮(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), ﷮(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)﷯﷯ A’ = S – A A’ = (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),﷮ (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),﷮(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), ﷯﷯ = Getting odd number on the first die = B Ex16.2,6 (ii) not B 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)}﷯﷯ B = (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),﷮ (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),﷮ (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) ﷯﷯ not B = S – B not B = (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),﷮(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), ﷮(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)﷯﷯ = getting even number of the first die = A Ex16.2,6 (iii) A or B A = (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),﷮(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), ﷮(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)﷯﷯ B = 1, 1﷯, 1, 2﷯, 1, 3﷯, 1, 4﷯, 1, 5﷯, 1, 6﷯,﷮ 3, 1﷯, 3, 2﷯, 3, 3﷯, 3, 4﷯, 3, 5﷯, 3, 6﷯﷯﷮ 5, 1﷯, 5, 2﷯, 5, 3﷯, 5, 4﷯, 5, 5﷯, 5, 6﷯﷯﷯ A or B = A ∪ B = (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)﷯﷯ = S Ex16.2, 6 (iv) A and B A = (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),﷮(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), ﷮(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)﷯﷯ B = 1, 1﷯, 1, 2﷯, 1, 3﷯, 1, 4﷯, 1, 5﷯, 1, 6﷯,﷮ 3, 1﷯, 3, 2﷯, 3, 3﷯, 3, 4﷯, 3, 5﷯, 3, 6﷯﷯﷮ 5, 1﷯, 5, 2﷯, 5, 3﷯, 5, 4﷯, 5, 5﷯, 5, 6﷯﷯﷯ A and B = A ∩ B = 𝛟 Ex16.2, 6 (v) A but not C A = (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),﷮(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), ﷮(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)﷯﷯ C = (1, 4)(2, 3)(3, 2)(4, 1)(1, 3)﷮ 2, 2﷯ 3, 1﷯ 1, 2﷯ 2, 1﷯(1, 1)﷯﷯ A but not C = A – C = 2, 4﷯, 2, 5﷯, 2, 6﷯,﷮ 4, 2﷯, 4, 3﷯, 4, 4﷯, 4, 5﷯, 4, 6﷯,﷯﷮ 6, 1﷯, 6, 2﷯, 6, 3﷯, 6, 4﷯, 6, 5﷯, 6 6﷯﷯﷯﷯ Ex16.2, 6 (vi) B or C B = 1, 1﷯, 1, 2﷯, 1, 3﷯, 1, 4﷯, 1, 5﷯, 1, 6﷯,﷮ 3, 1﷯, 3, 2﷯, 3, 3﷯, 3, 4﷯, 3, 5﷯, 3, 6﷯﷯﷮ 5, 1﷯, 5, 2﷯, 5, 3﷯, 5, 4﷯, 5, 5﷯, 5, 6﷯﷯,﷯ C = (1, 4)(2, 3)(3, 2)(4, 1)(1, 3)﷮ 2, 2﷯ 3, 1﷯ 1, 2﷯ 2, 1﷯(1, 1)﷯﷯ B or C = B ∪ C = 1, 1﷯, 1, 2﷯, 1, 3﷯, 1, 4﷯, 1, 5﷯, 1, 6﷯,﷮ 3, 1﷯, 3, 2﷯, 3, 3﷯, 3, 4﷯, 3, 5﷯, 3, 6﷯,﷯﷮ 5, 1﷯, 5, 2﷯, 5, 3﷯, 5, 4﷯, 5, 5﷯, 5, 6﷯,﷮ 2, 3﷯, 4, 1﷯, 2, 2﷯, (2, 1)﷯﷯ Ex16.2, 6 (vii) B and C B = 1, 1﷯, 1, 2﷯, 1, 3﷯, 1, 4﷯, 1, 5﷯, 1, 6﷯, ﷮ 3, 1﷯, 3, 2﷯, 3, 3﷯, 3, 4﷯, 3, 5﷯, 3, 6﷯﷮ 5, 1﷯, 5, 2﷯, 5, 3﷯, 5, 4﷯, 5, 5﷯, 5, 6﷯﷯﷯﷯ C = 1, 1﷯, 1, 2﷯, 1, 3﷯, 1, 4﷯, 2, 1﷯,﷮ 2, 2﷯, 2, 3﷯, 3, 1﷯, 3, 2﷯, 4, 1﷯﷯﷯ B and C = B ∩ C = { 1, 1﷯, 1, 2﷯, 1, 3﷯, 1, 4﷯, 3, 1﷯, 3, 2﷯,} Ex16.2, 6 (viii) A ∩ B’ ∩ C’ We know A & B’ (calculated in part(ii)) Finding C’ 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)﷯﷯ C = (1, 4)(2, 3)(3, 2)(4, 1)(1, 3)﷮ 2, 2﷯ 3, 1﷯ 1, 2﷯ 2, 1﷯(1, 1)﷯﷯ C’ = S – C = (1, 5), (1, 6),﷮(2, 4), (2, 5), (2, 6),﷮ (3, 3), (3, 4), (3, 5), (3, 6),﷮(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)﷯﷯ Now, A = 2, 1﷯, 2, 2﷯, 2, 3﷯, 2, 4﷯, 2, 5﷯, 2, 6﷯,﷮ 4, 1﷯, 4, 2﷯, 4, 3﷯, 4, 4﷯, 4, 5﷯, 4, 6﷯,﷮ 6, 1﷯, 6, 2﷯, 6, 3﷯, 6, 4﷯, 6, 5﷯, 6, 6﷯﷯﷯﷯ B’ = 2, 1﷯, 2, 2﷯, 2, 3﷯, 2, 4﷯, 2, 5﷯, 2, 6﷯,﷮ 4, 1﷯, 4, 2﷯, 4, 3﷯, 4, 4﷯, 4, 5﷯, 4, 6﷯,﷮ 6, 1﷯, 6, 2﷯, 6, 3﷯, 6, 4﷯, 6, 5﷯, 6, 6﷯﷯﷯﷯ C’ = (1, 5), (1, 6),﷮(2, 4), (2, 5), (2, 6),﷮ (3, 3), (3, 4), (3, 5), (3, 6),﷮(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 ∩ B’ ∩ C’ = 2, 4﷯, 2, 5﷯, 2, 6﷯,﷮ 4, 2﷯, 4, 3﷯, 4, 4﷯, 4, 5﷯, 4, 6﷯,﷮ 6, 1﷯, 6, 2﷯, 6, 3﷯, 6, 4﷯, 6, 5﷯, 6 6﷯﷯﷯

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