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Chapter 6: Problem 51

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 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 were 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?

Short Answer

Expert verified
a) 0.81, b) 0.19 and 0.19, c) 0.0361 and 0.9639, d) It would increase, e) It would decrease.

Step by step solution

01

Probability of 1-2 subsystem working

Since components 1 and 2 form a series subsystem and they are independent, the probability of the subsystem working is the product of the individual probabilities of the components working: \(P_{1-2} = P(1) * P(2) = 0.9 * 0.9 = 0.81\)
02

Probability of 1-2 subsystem not working

The probability of the 1-2 subsystem not working (or failing) is simply one minus the probability of it working: \(P_{1-2}' = 1 - P_{1-2} = 1 - 0.81 = 0.19\)
03

Probability of 3-4 subsystem not working

Apply the same logic to 3-4 subsystem (which is identical to the 1-2 system in this case): \(P_{3-4}' = 1 - P_{3-4} = 1 - 0.81 = 0.19\)
04

Probability of the total system not working

The system fails only when both subsystems fail (they are in parallel). Since they are independent, one calculates the product of their probabilities of failing: \(P_{system}' = P_{1-2}' * P_{3-4}' = 0.19 * 0.19 = 0.0361\)
05

Probability of the system working

The probability of the whole system working is the complement to the system failing: \(P_{system} = 1 - P_{system}' = 1 - 0.0361 = 0.9639\)
06

Addition of 5-6 subsystem in parallel

If a 5-6 subsystem were added in parallel and works with probability 0.9, the probability that the overall system fails decreases (the complement event, the system works, increases its probability).
07

Change with three series components

If there were three components in series in each subsystem and each works with probability 0.9, the reliability of each subsystem would decrease (because they are in series, so the probability multiplies), thus the overall system's reliability would also decrease. The actual figures depend on the reliability of the new added component.

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Key Concepts

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

Understanding Probability Calculation
Probability is a fundamental concept in mathematics, particularly in the study of reliability. It helps us measure the likelihood of specific events occurring.
When we are dealing with systems, we often want to know the probability that a series of components will successfully work together.
This involves understanding two key concepts: independent events and how to multiply probabilities.

In essence, if you have a sequence of independent events, the overall probability of all events occurring is the product of the probabilities of each individual event.
  • For example, if Component 1 works with a probability of 0.9 and Component 2 also works with a probability of 0.9, then the probability of both components working is:
    \[P(1\text{ and }2) = P(1) \times P(2) = 0.9 \times 0.9 = 0.81\]
This multiplication rule is key for calculating the reliability of series systems, where every component must function for the system to succeed.
Understanding probability calculation is essential for analyzing more complex systems as well, where components might be arranged in parallel or a combination of series and parallel.
Series and Parallel Systems in Reliability
Systems can be organized in different configurations to achieve reliability goals. Knowing how these configurations affect overall system performance is vital.

  • Series Systems: These systems are like a chain of components where every part must be operational for the system to function. If any component fails, the entire system stops working.
    Therefore, the probability that a series system works is the product of the probabilities of individual components working.
  • Parallel Systems: In contrast to series systems, parallel systems consist of components that can independently ensure the system's operation. If one component fails, another can take over, keeping the system operational.
    A parallel system is more reliable, with its probability of working being higher, as it allows for redundancy.
In mixed systems, which combine series and parallel arrangements, you must account for both configurations when calculating the overall probability of system reliability. Carefully analyzing each part helps us understand how to ensure optimum system performance.
The Role of Independent Events in System Reliability
Independence among events plays a significant role when calculating the reliability of a system.
Two events are independent if the occurrence of one event has no impact on the occurrence of another.

This characteristic simplifies reliability calculations because it allows for the straightforward multiplication of probabilities.
  • For example, when components in a system operate independently, the failure or success of a component doesn鈥檛 affect the others.
    This means that their probabilities can be directly multiplied to determine the success of the system.
This independence impacts both series and parallel systems, but in different ways:
  • In a series system, every component needs to function. The failure of any part results in the whole system's failure.
  • In a parallel system, independence provides reliability; a component can fail without the entire system failing.
Recognizing and leveraging the independence of components helps engineers and analysts design systems that cater to specific reliability requirements, ensuring better performance and minimizing risks.

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Most popular questions from this chapter

Five hundred first-year students at a state university were classified according to both high school GPA and whether they were on academic probation at the end of their first semester. The data are $$ \begin{array}{lcccc} && {\text { High School GPA }} \\ & 2.5 \text { to } & 3.0 \text { to } & 3.5 \text { and } & \\ \text { Probation } & <3.0 & <3.5 & \text { Above } & \text { Total } \\ \hline \text { Yes } & 50 & 55 & 30 & 135 \\ \text { No } & 45 & 135 & 185 & 365 \\ \text { Total } & 95 & 190 & 215 & 500 \\ \hline \end{array} $$ a. Construct a table of the estimated probabilities for each GPA-probation combination. b. Use the table constructed in Part (a) to approximate the probability that a randomly selected first-year student at this university will be on academic probation at the end of the first semester. c. What is the estimated probability that a randomly selected first-year student at this university had a high school GPA of \(3.5\) or above? d. Are the two outcomes selected student has a bigh school GPA of \(3.5\) or above and selected student is on academic probation at the end of the first semester independent outcomes? How can you tell? e. Estimate the proportion of first-year students with high school GPAs between \(2.5\) and \(3.0\) who are on academic probation at the end of the first semester. f. Estimate the proportion of those first-year students with high school GPAs \(3.5\) and above who are on academic probation at the end of the first semester.

A radio station that plays classical music has a "by request" program each Saturday evening. The percentages of requests for composers on a particular night are as follows: $$ \begin{array}{lr} \text { Bach } & 5 \% \\ \text { Beethoven } & 26 \% \\ \text { Brahms } & 9 \% \\ \text { Dvorak } & 2 \% \\ \text { Mendelssohn } & 3 \% \\ \text { Mozart } & 21 \% \\ \text { Schubert } & 12 \% \\ \text { Schumann } & 7 \% \\ \text { Tchaikovsky } & 14 \% \\ \text { Wagner } & 1 \% \end{array} $$ Suppose that one of these requests is to be selected at random. a. What is the probability that the request is for one of the three B's (Bach, Beethoven, or Brahms)? b. What is the probability that the request is not for one of the two \(S^{\prime}\) s? c. All of the listed composers wrote at least one symphony except Bach and Wagner. What is the probability that the request is for a composer who wrote at least one symphony?

A shipment of 5000 printed circuit boards contains 40 that are defective. Two boards will be chosen at random, without replacement. Consider the two events \(E_{1}=\) event that the first board selected is defective and \(E_{2}=\) event that the second board selected is defective. a. Are \(E_{1}\) and \(E_{2}\) dependent events? Explain in words. b. Let \(n o t E_{1}\) be the event that the first board selected is not defective (the event \(E_{1}^{C}\) ). What is \(P\left(\right.\) not \(E_{1}\) )? c. How do the two probabilities \(P\left(E_{2} \mid E_{1}\right)\) and \(P\left(E_{2} \mid\right.\) not \(\left.E_{1}\right)\) compare? d. Based on your answer to Part (c), would it be reasonable to view \(E_{1}\) and \(E_{2}\) as approximately independent?

The events \(E\) and \(T_{j}\) are defined as \(E=\) the event that someone who is out of work and actively looking for work will find a job within the next month and \(T_{i}=\) the event that someone who is currently out of work has been out of work for \(i\) months. For example, \(T_{2}\) is the event that someone who is out of work has been out of work for 2 months. The following conditional probabilities are approximate and were read from a graph in the paper "The Probability of Finding a Job" (American Economic Review: Papers \& Proceedings [2008]: \(268-273\) ) $$ \begin{array}{ll} P\left(E \mid T_{1}\right)=.30 & P\left(E \mid T_{2}\right)=.24 \\ P\left(E \mid T_{3}\right)=.22 & P\left(E \mid T_{4}\right)=.21 \\ P\left(E \mid T_{5}\right)=.20 & P\left(E \mid T_{6}\right)=.19 \\ P\left(E \mid T_{7}\right)=.19 & P\left(E \mid T_{8}\right)=.18 \\ P\left(E \mid T_{9}\right)=.18 & P\left(E \mid T_{10}\right)=.18 \\ P\left(E \mid T_{11}\right)=.18 & P\left(E \mid T_{12}\right)=.18 \end{array} $$ a. Interpret the following two probabilities: i. \(\quad P\left(E \mid T_{1}\right)=.30\) ii. \(\quad P\left(E \mid T_{6}\right)=.19\) b. Construct a graph of \(P\left(E \mid T_{i}\right)\) versus \(i\). That is, plot \(P\left(E \mid T_{i}\right)\) on the \(y\) -axis and \(i=1,2, \ldots, 12\) on the \(x\) -axis. c. Write a few sentences about how the probability of finding a job in the next month changes as a function of length of unemployment.

A library has five copies of a certain textbook on reserve of which two copies ( 1 and 2 ) are first printings and the other three \((3,4\), and 5\()\) are second printings. \(\mathrm{A}\) student examines these books in random order, stopping only when a second printing has been selected. a. Display the possible outcomes in a tree diagram. b. What outcomes are contained in the event \(A\), that exactly one book is examined before the chance experiment terminates? c. What outcomes are contained in the event \(C\), that the chance experiment terminates with the examination of book 5 ?
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