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?

Answer:
(a) The probability that all three components fail is \( (0.0072)^3 = 0.000000373248 \) or approximately \( 0.0000373\% \).
(b) The probability that at least one component does not fail is \( 1 - (0.0072)^3 = 1 - 0.000000373248 \approx 0.999999626752 \) or approximately \( 99.99996\% \).

Explanation:
(a) To find the probability that all three components fail, we multiply the probability of failure for one component by itself three times, since the failures are independent.

(b) To find the probability that at least one component does not fail, we can use the complement rule. We first calculate the probability that all three components fail, then subtract that from 1.

Steps:

1. Calculate the probability of failure for one component: \( P(\text{failure}) = 0.0072 \).
2. Calculate the probability that all three components fail:

1. Calculate the probability that at least one component does not fail: