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

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 extra 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?

Key Concepts

System Reliability and Redundancy

System reliability often benefits from redundancy, where multiple components perform similar functions. However, redundancy can introduce complexities such as increased load or stress on the remaining components if one fails. Analyzing reliability in such systems requires accounting for both the backup provided by redundancy and the potential for increased failure rates under dependent conditions.

Marginal and Joint Probabilities

Marginal probability refers to the overall probability of a single event occurring, while joint probability is the probability that two or more events occur simultaneously. Understanding both is crucial for analyzing system failures, as it helps in determining individual component reliability and the likelihood of multiple components failing together.

Dependent Events

In many systems, especially those with interacting components, the occurrence of one event can influence the likelihood of another event. Here, dependency means that the failure of one component may increase the risk of failure of a related component, as opposed to the events being independent.

Conditional Probability

This is a measure of the likelihood that an event will occur given that another event has already occurred. In reliability analysis, conditional probability helps us understand how the failure of one component can affect the probability of failure of another component within a system.

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