3 Homework 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 0.0057 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 \( 54.55 \%, 12 \) of 22 paints the flight? (b) What is the probability at least one of the components does not fail? (a) The probability is (Round to eight decimal places as needed.)

To solve the problem, we start by finding the required probabilities based on the given probability of failure for a single component, which is \( p = 0.0057 \). ### (a) Probability that all three components fail: The probability that a single component fails is \( p = 0.0057 \). Since there are three components, the probability that all three fail can be calculated as follows: \[ P(\text{all fail}) = p^3 = (0.0057)^3 \] Calculating that: \[ P(\text{all fail}) = 0.0057 \times 0.0057 \times 0.0057 \approx 0.0000001849 \] Now, rounding this to eight decimal places: \[ P(\text{all fail}) \approx 0.00000018 \] ### (b) Probability that at least one component does not fail: To find the probability that at least one does not fail, we can use the complement rule. First, we find the probability that all three components fail (which we calculated) and subtract that from 1: \[ P(\text{at least one does not fail}) = 1 - P(\text{all fail}) \] Using our previously calculated value: \[ P(\text{at least one does not fail}) = 1 - 0.0000001849 \] Calculating that: \[ P(\text{at least one does not fail}) \approx 0.9999998151 \] So, rounding this to eight decimal places, we have: ### Final Answers: (a) The probability is approximately \( 0.00000018 \). (b) The probability of at least one component not failing is approximately \( 0.99999982 \).