Explanation Formula used: Principle of inclusion-exclusion: | A 1 ∪ A 2 ∪ ... ∪ A n | = ∑ 1 ≤ i ≤ n | A i | − ∑ 1 ≤ i < j ≤ n | A i ∩ A j | + ∑ 1 ≤ i < j < k ≤ n | A i ∩ A j ∩ A k | − .... + ( − 1 ) n + 1 | A 1 ∩ A 2 ∩ .... ∩ A n | Calculation: Let the five events be A 1 , A 2 , A 3 , A 4 , A 5 No four of the events can occur at the same time, thus we know for every possible set of distinct integers i , j , k , l ∈ { 1 , 2 , 3 , 4 , 5 } | A i ∩ A j ∩ A k ∩ A l | = 0 | A 1 ∩ A 2 ∩ A 3 ∩ A 4 ∩ A 5 | = 0 The principle of inclusion-exclusion: | A 1 ∪ A 2 ∪ A 3 ∪ A 4 ∪ A 5 | = ∑ 1 ≤ i ≤ n | A i | − ∑ 1 ≤ i < j ≤ n | A i ∩ A j | + ∑ i ≤ i < j < k ≤ n | A i ∩ A j ∩ A k | − .... + ( − 1 ) n + 1 ⋅ | A 1 ∩ A 2 ∩ A 3 ∩ .... ∩ A n | = ∑ 1 ≤ i ≤ n | A i | − ∑ 1 ≤ i < j ≤ n | A i ∩ A j | + ∑ i ≤ i < j < k ≤ n | A i ∩ A j ∩ A k | − 0 + 0 = ∑ 1 ≤ i ≤ n | A i | − ∑ 1 ≤ i < j ≤ n | A i ∩ A j | + ∑ i ≤ i < j < k ≤ n | A i ∩ A j ∩ A k | = | A 1 | + | A 2 | + | A 3 | + | A 4 | + | A 5 | − | A 1 ∩ A 2 | − | A 1 ∩ A 3 | − | A 1 ∩ A 4 | − | A 1 ∩ A 5 | − | A 2 ∩ A 3 | − | A 2 ∩ A 4 | − | A 2 ∩ A 5 | − | A 3 ∩ A 4 | − | A 3 ∩ A 5 | − | A 4 ∩ A 5 | + | A 1 ∩ A 2 ∩ A 3 | + | A 1 ∩ A 2 ∩ A 4 | + | A 1 ∩ A 2 ∩ A 5 | + | A 1 ∩ A 3 ∩ A 4 | + | A 1 ∩ A 3 ∩ A 5 | + | A 1 ∩ A 4 ∩ A 5 | + | A 2 ∩ A 3 ∩ A 4 | + | A 2 ∩ A 3 ∩ A 5 | + | A 2 ∩ A 4 ∩ A 5 | + | A 3

Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991

8th Edition

ISBN: 9781259676512

Author: Kenneth H Rosen

Publisher: McGraw-Hill Education

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Chapter 8.5, Problem 29E

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A formula for the probability of the union of five events in a sample space if no four of them can occur at the same time.

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Chapter 8.5, Problem 28E

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Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991

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A formula for the probability of the union of five events in a sample space if no four of them can occur at the same time.

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A formula for the probability of the union of five events in a sample space if no four of them can occur at the same time.

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