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# A system consists of two identical pumps, #1 and #2. If

ISBN: 9780321629111 32

## Solution for problem 52E 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

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Problem 52E

A system consists of two identical pumps, #1 and #2. If one pump fails, the system will still operate. However, because of the added strain, the remaining pump is now more likely to fail than was originally the case. That is, ?r ?= ?P?(#2 fails ??#1 fails) > ?P?(#2 fails) = ?q. ?If at least one pump fails by the end of the pump design life in 7% of all systems and both pumps fail during that period in only 1%, what is the probability that pump #1 will fail during the pump design life?

Step-by-Step Solution:

Answer: Step 1 of 1 Let A be the event that Pump #1 fails and B be the event that Pump #2 fails. We are given that: P(at least one pump fails) = 0.07 P(A or B) = 0.07 P(both pumps fail) = 0.01 P(A and B) = 0.01 Since both pumps are identical, we can assume that P(A) = P(B). Recall the following identity: P(A or B) = P(A) + P(B) - P(A and B) P(A or B) = P(A) + P(A) - P(A and B) P(A or B) = 2[P(A)] - P(A and B) P(A or B) + P(A and B) = 2[P(A)] P(A) = (1/2)[P(A or B) + P(A and B)] P(A) = (1/2)[0.07 + 0.01] = (1/2)(0.08) = 0.04 So the probability that pump #1 will fail during the pump design life is 4%.

Step 2 of 1

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A system consists of two identical pumps, #1 and #2. If