QQuestionMathematics
QuestionMathematics
In airline applications, failure of a component can result in catastrophe. As a result, many airline components utilize something called triple modular redundancy. This means that a critical component has two backup components that may be utilized should the initial component fail.
Suppose a certain critical airline component has a probability of failure of 0.0072 and the system that utilizes the component is part of a triple modular redundancy.
(a) Assuming each component's failure/success is independent of the others, what is the probability all three components fail, resulting in disaster for the flight?
(b) What is the probability at least one of the components does not fail?
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Step 1:I'll solve this problem step by step using the specified LaTeX formatting guidelines:
Step 2:: Understand the Given Information
- Probability of a single component failing: $$p = 0.0072
- Number of components in triple modular redundancy: 3 - We need to calculate the probability of all components failing
Step 3:: Calculate Probability of All Components Failing
- $$p^{3} = (0.0072)^{3}
- Since the failures are independent, we multiply the individual failure probabilities
Step 4:: Calculate the Probability of All Components Failing
- $$p^{3} = 0.0072^{3}
- p^{3} = 3.7 \times 10^{- 7}
Step 5:: Calculate Probability of At Least One Component Not Failing
- $$0.9999996
- This is the complement of all components failing
Final Answer
(a) Probability of all components failing: 3.7 \times 10^{- 7} (b) Probability of at least one component not failing: 0.9999996
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