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Chapter 2: Problem 74

Two pumps connected in parallel fail independently of each other on any given day. The probability that only the older pump will fail is 10 , and the probability that only the newer pump will fail is \(.05\). What is the probability that the pumping system will fail on any given day (which happens if both pumps fail)?

Short Answer

Expert verified
The probability that both pumps fail is 0.15.

Step by step solution

01

Understand the Problem

We need to find the probability that both pumps fail on the same day given certain probabilities for individual failures.
02

Define Relevant Probabilities

Let the probability that the older pump fails be \( P(O) \) and the probability that the newer pump fails be \( P(N) \). The problem specifies \( P(O \text{ fails and } N \text{ does not fail}) = 0.10 \) and \( P(N \text{ fails and } O \text{ does not fail}) = 0.05 \).
03

Find Probability of Older Pump Failing

The probability that the older pump fails, \( P(O) \), can be found by considering that it is a part of two scenarios: it fails alone or it fails together with the new pump. Hence, \( P(O) = P(O \text{ fails and } N \text{ does not fail}) + P(O \text{ and } N) \).
04

Find Probability of Newer Pump Failing

Similarly, \( P(N) = P(N \text{ fails and } O \text{ does not fail}) + P(O \text{ and } N) \).
05

Express Probability of Both Failing

Express the probability of both failing \( P(O \text{ and } N) \) in terms of unknowns: \( P(O \text{ and } N) = P(O) + P(N) - P(O \text{ fails and } N \text{ does not fail}) - P(N \text{ fails and } O \text{ does not fail}) \).
06

Calculate Probability of Both Failing

We propose that \( P(O \text{ and } N) = x \). We substitute in the given values: \[ x = (0.10 + x) + (0.05 + x) - 0.10 - 0.05 \]. Solve it to find \( x = 0.15 \).
07

Conclude the Solution

Thus, the probability that both pumps fail, and thus the whole system fails, is \( P(O \text{ and } N) = 0.15 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Independent Events
In probability theory, independent events are those whose outcomes do not affect each other. This means that knowing the outcome of one event provides no information about the other.
In our exercise, the failure of the older pump is independent of the newer pump's failure. The probability that one pump fails does not change the probability of the other pump failing. Understanding this concept is essential, as it allows us to reason about the events without having to consider complex interactions between them.
For independent events, the joint probability of both events occurring is simply the product of their individual probabilities. This property is key when analyzing scenarios like the pumping system in the exercise.
Parallel Systems
A parallel system is designed such that multiple components perform the same function. If at least one component works, the system continues to function. However, in our problem, the parallel pumps mean that the system fails only if both pumps fail. This setup is typical in engineering because it increases reliability, as the failure of one component doesn't immediately lead to system failure. In our exercise, a question was posed about the failure of the entire system, highlighting the situation in which both pumps aren't operational. Parallel systems thus offer enhanced system reliability by spreading the risk of failure across multiple components.
System Reliability
System reliability refers to the probability that a system will perform its intended function under stated conditions for a specified period. Reliability is crucial in systems where failure can lead to costly or dangerous outcomes. In the exercise, the reliability of the pumping system relates directly to whether at least one pump continues to operate. If we consider each pump as a separate entity with its own failure probability, we can determine the reliability of the entire system. By calculating the joint probability of both pumps failing, we say the system's reliability is affected directly. Improving system reliability often involves incorporating redundant components like our parallel pumps, to ensure system functionality even when parts fail.
Joint Probability
Joint probability is a statistical measure that calculates the likelihood of two events happening at the same time. When dealing with independent events in a system, joint probabilities are essential in determining the likelihood of a combined outcome.In the exercise, the joint probability relates to both pumps failing together. We denote this as \( P(O \text{ and } N) \), and the task involves calculating this value knowing individual probabilities. By understanding the contribution of each event to the joint outcome, we used known values and a little algebra.The final solution \( P(O \text{ and } N) = 0.15 \) shows the scenario where both pumps fail, giving insight into the potential risks facing the pump system.

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Most popular questions from this chapter

Fifteen telephones have just been received at an authorized service center. Five of these telephones are cellular, five are cordless, and the other five are corded phones. Suppose that these components are randomly allocated the numbers 1 , \(2, \ldots, 15\) to establish the order in which they will be serviced. a. What is the probability that all the cordless phones are among the first ten to be serviced? b. What is the probability that after servicing ten of these phones, phones of only two of the three types remain to be serviced? c. What is the probability that two phones of each type are among the first six serviced?

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