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Consider four independent events A1, A2, A3, and A4, and

Probability and Statistics for Engineers and the Scientists | 9th Edition | ISBN: 9780321629111 | Authors: Ronald E. Walpole; Raymond H. Myers; Sharon L. Myers; Keying E. Ye ISBN: 9780321629111 32

Solution for problem 112E Chapter 2

Probability and Statistics for Engineers and the Scientists | 9th Edition

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Probability and Statistics for Engineers and the Scientists | 9th Edition | ISBN: 9780321629111 | Authors: Ronald E. Walpole; Raymond H. Myers; Sharon L. Myers; Keying E. Ye

Probability and Statistics for Engineers and the Scientists | 9th Edition

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Problem 112E

Consider four independent events ?A?1, ?A?2, ?A?3, and ?A?4, and let pi = P(A) for I = 1,2,3,4. Express the probability that at least one of these four events occurs in terms of the ?p?is, and do the same for the probability that at least two of the events occur.

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Step 1 of 3

Problem 112E Answer: Step1: We have four independent events A , A , A , A and 1 2 3 4 Let p =i(A) , i =i,2,3,4. We need to Express the probability that at least one of these four events occurs in terms of the p is, and do the same for the probability that at least two of the events occur. Step2: Consider, the probability that none of these events occurs is q = 1 - p = P(A ) i i 1i Let “x” denotes the event that at least one of the A events occurs. i Now, P(x) = 1 - P(x ) 1 = 1 - P(A A 1A A 2 13 14 = 1 - P(A )•P(A )•P(A )•P(A ) 1 1 1 2 3 4 = 1 - q •q •q •q 1 2 3 4 Therefore, the probability that at least one of the A events occurs is P(x) = 1 - q •q •q •q . i 1 2 3 4 Step3: Let Y denote the event that at least two of the A events occurs. i Now, P(Y) = 1 - P(Y ) 1 = 1 - [P(x ) + P(Exactly one A)] i 4 = 1 - P (x ) -[ P(only Ai occurs )] ………(1) i=1 Where, 4 1 1 1 1 1 1 1 1 P(only Ai occurs ) = {[P(A A A A )]+[P(A1 A A2 A ]+[3(A A4A 1 2 3 4 1 2 3 i=1 1 1 1 1 A ]4P(A A A1A ]} 2 3 4 = [(p 1q •q2 •q 3+(p4q •q2•q )1(p 3 4 3 •q 1q •q2 )+(4 •q •4 •q1)] 2 3 Substitute these into above equation (1) we get, 1 P(Y) = 1 - P(x ) - (p •q •q1 •q 2-(p3•q •4 •q 2-(p1•q •3 •q4-(p3•q •1 •q2 ) 4 4 1 2 3

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Chapter 2, Problem 112E is Solved
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Textbook: Probability and Statistics for Engineers and the Scientists
Edition: 9
Author: Ronald E. Walpole; Raymond H. Myers; Sharon L. Myers; Keying E. Ye
ISBN: 9780321629111

This textbook survival guide was created for the textbook: Probability and Statistics for Engineers and the Scientists, edition: 9. This full solution covers the following key subjects: Events, Probability, least, let, express. This expansive textbook survival guide covers 18 chapters, and 1582 solutions. Since the solution to 112E from 2 chapter was answered, more than 415 students have viewed the full step-by-step answer. Probability and Statistics for Engineers and the Scientists was written by and is associated to the ISBN: 9780321629111. The full step-by-step solution to problem: 112E from chapter: 2 was answered by , our top Statistics solution expert on 05/06/17, 06:21PM. The answer to “Consider four independent events ?A?1, ?A?2, ?A?3, and ?A?4, and let pi = P(A) for I = 1,2,3,4. Express the probability that at least one of these four events occurs in terms of the ?p?is, and do the same for the probability that at least two of the events occur.” is broken down into a number of easy to follow steps, and 50 words.

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Consider four independent events A1, A2, A3, and A4, and