1. **State the problem:** We have a critical airline component with a failure probability of $0.0058$. The system uses triple modular redundancy, meaning there are three independent components working together. 2. **Formulas and rules:** - The probability all three components fail (disaster) is the product of their individual failure probabilities because failures are independent: $$P(\text{all fail}) = p^3$$ - The probability at least one component does not fail is the complement of all failing: $$P(\text{at least one works}) = 1 - P(\text{all fail})$$ 3. **Calculate the probability all three fail:** $$P(\text{all fail}) = (0.0058)^3 = 0.0058 \times 0.0058 \times 0.0058$$ Calculate stepwise: $$0.0058 \times 0.0058 = 0.00003364$$ $$0.00003364 \times 0.0058 = 0.000000195112$$ So, $$P(\text{all fail}) = 0.000000195112$$ 4. **Calculate the probability at least one does not fail:** $$P(\text{at least one works}) = 1 - 0.000000195112 = 0.999999804888$$ 5. **Final answers:** - Probability all three fail (disaster): $0.0000001951$ (approximately) - Probability at least one does not fail: $0.9999998$ (approximately) These results show that triple modular redundancy greatly reduces the chance of total failure.