A city water supply system involves three pumps, the failure...

A city water supply system involves three pumps, the failure of any one of which crashes the system. The probabilities of failure for each pump in a given year are $0.025,0.034,$ and o.02, respectively. Assuming the pumps operate independently of each other, what is the probability that the system does crash during the year? (A) $0.025+0.034+0.02$ (B) $1-(0.025+0.034+0.02)$ (C) $1-(0.025)(0.034)(0.02)$ (D) $(1-0.025)(1-0.034)(1-0.02)$ (E) $1-(1-0.025)(1-0.034)(1-0.02)$

A city water supply system involves three pumps, the failure of any one of which crashes the system. The probabilities of failure for each pump in a given year are $0.025,0.034,$ and
o.02, respectively. Assuming the pumps operate independently of each other, what is the probability that the system does crash during the year?
(A) $0.025+0.034+0.02$
(B) $1-(0.025+0.034+0.02)$
(C) $1-(0.025)(0.034)(0.02)$
(D) $(1-0.025)(1-0.034)(1-0.02)$
(E) $1-(1-0.025)(1-0.034)(1-0.02)$

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Key Concepts

Complement Rule

The complement rule in probability states that the probability of an event occurring is equal to one minus the probability that the event does not occur. This rule is particularly useful when calculating the probability of at least one occurrence among several independent events by first considering the probability that none occur and then subtracting that value from one.

Series System Reliability

Series system reliability refers to systems in which the failure of any individual component results in the failure of the entire system. In such systems, the overall reliability is determined by considering the combined probability that all individual components function correctly. Conversely, the probability of a system failure is calculated as one minus the probability that all components operate successfully.

Independent Events

In probability theory, independent events are those whose outcomes do not affect each other. When events are independent, the probability of all events occurring simultaneously is the product of their individual probabilities. This principle is essential for calculating the combined probabilities in systems where each component operates without influence from the others.

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