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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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Answer:

(a) 0.000000373248

(b) 0.999999626752

Step-by-step explanation:

The aircraft works when at least one of its components works

Probability of failure, f=0.0072

Probability of success, s=1-0.0072=0.9928

(a)

P(all three components fail)=[tex]f^{3}[/tex]

probability=[tex]0.0072^{3}[/tex]=0.000000373248

(b)

Probability (at least one of the components does not​ fail)=1-P(all three components fail)

Probability=[tex]1-0.0072^{3}[/tex]

Probability=1-0.000000373248=0.999999626752

fichoh

The probability the all three components fail and that atleast one of the component does not fail is :

  • P(all 3 fail) = 0.000000373248

  • P(Atleast 1 does not fail) = 0.999999626752

The probability of failure of each component is given as :

  • P(failure) = 0.0072

The probability that all 3 components fail can be expressed thus :

P(all 3 fail) = P(failure) × P(Failure) × P(Failure)

P(all 3 fail) = 0.0072 × 0.0072 × 0.0072 = 0.000000373248

The probability that atleast one of the components does not fail is :

P(Atleast 1 does not fail) = 1 - P(all 3 fail)

P(Atleast 1 does not fail) = 1 - 0.000000373248

P(Atleast 1 does not fail) = 0.999999626752

Therefore, the probabilities that all 3 fail and that atleast one does not fail are 0.000000373248 and 0.999999626752

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