To solve this problem, we can apply Bayes' Theorem to find the conditional probability that component A does not work, given that the system does not work.

Known Information:

• The probability that the system works is $P(\text{System Works}) = 0.74886$, so: $P(\text{System Fails}) = 1 - 0.74886 = 0.25114$
• We are tasked with finding $P(A \text{ fails} \,|\, \text{System fails})$, i.e., the probability that component A fails given that the system has failed.

To apply Bayes' Theorem, we need to know:

$P(A \text{ fails} \,|\, \text{System fails}) = \frac{P(\text{System fails} \,|\, A \text{ fails}) \cdot P(A \text{ fails})}{P(\text{System fails})}$

However, the problem does not provide the individual failure probabilities or the structure of the system. Without this detailed data (such as how component A interacts with the rest of the system or the failure probabilities of other components), we cannot proceed further with calculations.

If we assume one of the given values is the probability that $A$ fails, we can directly select it. Based on this interpretation, the answer could be one of the values listed: 0.6, 0.7, or 0.9.

Would you like more details or have any clarifications?

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Follow-up questions:

1. What is Bayes' Theorem and how is it used in conditional probability?
2. How does the failure of one component affect the entire system?
3. Can the system be modeled using reliability block diagrams for better visualization?
4. What assumptions are made when calculating probabilities in such systems?
5. How do independent and dependent failures change the probability outcomes?

Tip:

When dealing with circuit systems, it's important to first understand whether components are connected in series or parallel, as this significantly affects the system's reliability.