91Ó°ÊÓ

Consider a system consisting of four components, as pictured in the following diagram: Components 1 and 2 form a series subsystem, as do Components 3 and 4 . The two subsystems are connected in parallel. Suppose that \(P(1\) works \()=.9\), \(P(2\) works \()=.9, P(3\) works \()=.9\), and \(P(4\) works \()=.9\) and that these four outcomes are independent (the four components work independently of one another). a. The \(1-2\) subsystem works only if both components work. What is the probability of this happening? b. What is the probability that the \(1-2\) subsystem doesn't work? that the \(3-4\) subsystem doesn't work? c. The system won't work if the \(1-2\) subsystem doesn't work and if the \(3-4\) subsystem also doesn't work. What is the probability that the system won't work? that it will work? d. How would the probability of the system working change if a \(5-6\) subsystem was added in parallel with the other two subsystems? e. How would the probability that the system works change if there were three components in series in each of the two subsystems?

Step by step solution

01

Calculate the probability of the \(1-2\) subsystem working

The \(1-2\) subsystem works only if both Component 1 and Component 2 work. Since these two components are independent, the probability that both work is the product of their individual probabilities. Thus, \(P(1-2\) works \() = P(1\) works \() * P(2\) works \() = .9 * .9 = .81.

02

Calculate the probability of the \(1-2\) subsystem not working

The probability of an event's complement is 1 minus the probability of the event itself. So, \(P(1-2\) doesn't work \() = 1 - P(1-2\) works \() = 1 - .81 = .19.

03

Calculate the probability of the \(3-4\) subsystem not working

This calculation is similar to Step 2, because Components 3 and 4 are identical to Components 1 and 2. So, \(P(3-4\) doesn't work \() = 1 - P(3-4\) works \() = 1 - .81 = .19.

04

Calculate the probability that the system won't work

For the entire system to not work, neither the \(1-2\) subsystem nor the \(3-4\) subsystem can work. Since these are independent, this probability is the product of their individual probabilities. Thus, \(P(system\) doesn't work \() = P(1-2\) doesn't work \() * P(3-4\) doesn't work \() = .19 * .19 = .0361.

05

Calculate the probability that the system will work

The probability that the system will work is the complement of the system not working. So, \(P(system\) works \() = 1 - P(system\) doesn't work \() = 1 - .0361 = .9639.

06

Calculate the effect of adding a \(5-6\) subsystem in parallel

If a \(5-6\) subsystem is added in parallel, valid with the same properties as the others, and it fails with probability .19, then the system fails only when all three subsystems fail. Thus, \(P(system\) doesn't work with \(5-6\) \() = .19^3 = .006859. As a result, the system working increases to \(P(system\) works with \(5-6\) \() = 1 - .006859 = .993141.

07

Calculate the effect of having three series components in each subsystem

If there were three components in series in each subsystem, with a .9 chance of each working, then each subsystem would now fail with a probability of .1^3 = .001. The system will fail when both subsystems fail, which is \(P(system\) doesn't work = .001^2 = .000001. So, \(P(system\) works = 1 - .000001 = .999999.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Independent Events

In probability theory, a core concept is the idea of independent events. When events are considered independent, the occurrence of one event does not affect the probability of another. In simpler terms, each event happens without influencing the other. This is a crucial assumption in problems involving multiple components or situations.

For instance, suppose we have four components in a system that need to function without depending on each other. Component 1 working is independent of Component 2, 3, or 4 working. The probability of all components working together is the product of their individual probabilities. If each component independently works with a probability of 0.9, then for the series part consisting of Component 1 and Component 2 to work, the probability is:

• \( P(1\) and \(2 \text{ work}) = P(1 \text{ works}) \times P(2 \text{ works}) = 0.9 \times 0.9 = 0.81 \)

This multiplication process extends to other independent events, providing a powerful method to analyze components in systems and many other scenarios involving probability.

Series and Parallel Systems

Understanding the designs of series and parallel systems is essential in determining a system's functioning probability. A series system consists of multiple components arranged in sequence, where every component must function for the entire system to work. Hence, the overall probability of success is the product of the individual component probabilities.

Take an example of a simple two-component series system with both components having a working probability of 0.9:

• \( P(1\text{-}2 \text{ works}) = 0.9 \times 0.9 = 0.81 \)

In contrast, a parallel system has components arranged side-by-side, so the system works if at least one component works. For parallel subsystems like the one shown above with 1-2 and 3-4:

• \( P(\text{system doesn't work}) = P(1\text{-}2 \text{ doesn't work}) \times P(3\text{-}4 \text{ doesn't work}) = 0.19 \times 0.19 = 0.0361 \)
• \( P(\text{system works}) = 1 - 0.0361 = 0.9639 \)

Parallel arrangements improve reliability since multiple pathways can lead to success.

Complementary Probability

The concept of complementary probability helps determine the likelihood of an event not occurring. It is based on the principle that the sum of the probabilities of all possible outcomes equals one. If one knows the probability of an event, the probability of its complement is straightforward to calculate.

In a probability context, the probability of an event not happening is 1 minus the probability of the event itself. For instance, if the probability of a series system not functioning is 0.81, then the probability of it failing is:

• \( P(1\text{-}2 \text{ doesn't work}) = 1 - 0.81 = 0.19 \)

Similarly, if a parallel system only fails if every component or subsystem fails, the complementary probability rule becomes useful:

• Overall failure probability in such combined systems can be calculated and adjusted by subtracting from one.

Using complementary probabilities simplifies the task of understanding complex systems by focusing on the few possible reasons for their failure rather than success.