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

Probability of 1-2 subsystem working

The probability that both components of the 1-2 subsystem work is the product of their respective probabilities since they work independently. This is calculated as \(P(1-2) = P(1) \times P(2) = 0.9 \times 0.9 = 0.81) .

02

Probability of 1-2 subsystem not working

The probability that the 1-2 subsystem doesn't work is 1 minus the probability that it works. This can be calculated as \(P'(1-2) = 1 - P(1-2) = 1 - 0.81 = 0.19\) .

03

Probability of 3-4 subsystem not working

Following the same rules as in Step 1 and Step 2, the probability of the 3-4 subsystem not working can be calculated as \(P'(3-4) = 1 - P(3) \times P(4) = 1 - 0.9 \times 0.9 = 0.19\) .

04

Probability of system not working

The system won't work if both the 1-2 and 3-4 subsystems don't work. Since they work independently, this can be calculated as \(P(\text{System Not Working}) = P'(1-2) \times P'(3-4) = 0.19 \times 0.19 = 0.0361\) .

05

Probability of the system working

The probability of the system working is 1 minus the probability that it doesn't work. Hence, \(P(\text{System Working}) = 1 - P(\text{System Not Working}) = 1 - 0.0361 = 0.9639\) .

06

Adding 5-6 subsystem

If a 5-6 subsystem with the same probabilities as other components were added in parallel, it would increase the system's reliability since now the system would work if any of the three subsystems worked. However, exact calculation can't be performed without knowing the values of \(P(5)\) and \(P(6)\).

07

Three components in a subsystem

If there were three components in series in each of the two subsystems, then the subsystems would be less reliable since to work, all three components would need to work. The probability of subsystem working would be now \(P(1) \times P(2) \times P(3)\) and so the system's reliability would decrease.

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Key Concepts

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

Independent Events

In probability, independent events mean that the outcome of one event does not affect the outcome of another. For the problem at hand, the functioning of each component in the system is independent of the others. This independence allows us to calculate the probability of multiple events occurring simultaneously by multiplying their individual probabilities. Imagine flipping a coin twice; each flip doesn't affect the other. Similarly, if Component 1 operates with a probability of 0.9, this probability isn't influenced by whether Component 2 works or not. Hence, their combined functioning is calculated as the product of their probabilities. Thus, independence helps simplify complex systems into manageable parts.

Series and Parallel Systems

Series and parallel systems describe different ways components in a system are connected. In a series system, all components must work for the system to work. It's like old Christmas lights; if one bulb goes out, the whole string may not light. For the given exercise:

• Components 1 and 2 form a series subsystem. Both must function for the subsystem to work with a combined probability of 0.81.
• In contrast, parallel systems allow for more reliability since only one of the parallel paths needs to function for the whole system to work. This setup ensures higher reliability, as the given problem combines two series subsystems in parallel: only one series needs to function for the whole system to be operational.

Reliability Engineering

Reliability engineering focuses on ensuring systems continue to perform under expected conditions. In our problem, reliability means estimating how often the system will function correctly. This is crucial for applications like safety-critical systems, where failures might cause severe consequences. Understanding reliability involves:

• Recognizing the system's configuration (like series or parallel).
• Calculating the probability of each component's success and failure.
• Using these calculations to predict overall system performance success or failure rates.

It's all about maximizing efficiency and minimizing downtime, which is why adding more parallel paths (as explored with the hypothetical 5-6 subsystem) enhances reliability.

Probability Calculations

Probability calculations allow us to quantify the likelihood of various outcomes. In reliability engineering, this means determining how likely a system is to succeed or fail. For example, calculating the probability of a Series component working involves multiplying the individual probabilities. Steps include:

• Finding probabilities of individual components, like 0.9 in our exercise for each component working.
• Using these to calculate series probabilities by multiplying: e.g., for subsystem 1-2, it's 0.9 x 0.9 = 0.81.
• Determining parallel probabilities by understanding that the system will work if at least one path works, using complementary probabilities to find out potential failure rates.

Therefore, operations like these help us determine the overall reliability and guide improvements or changes to system design.