Final answer:
The overall reliability of the product is calculated by multiplying the reliabilities of components A, B, and the combined reliability of C and its redundant D. The combined unit's reliability is found by subtracting the probability of both C and D failing from 1. The final overall reliability for the system is approximately 0.640635.
Step-by-step explanation:
To calculate the overall reliability of a product with multiple components, each having its own reliability, we need to consider that the failure of any component can cause the entire system to fail. The reliability of components A, B, and C are given as 0.85, 0.81, and 0.77, respectively. Since the failure of component C can cause safety issues, it has a redundant backup D with a reliability of 0.70. To find the overall reliability of the system, we calculate the reliabilities of components A, B, and the combined system of C and D.
The reliability of component C and its redundant D is calculated by considering that the combined unit fails only if both C and D fail. The probability that C fails is 1 - 0.77 = 0.23, and the probability that D fails is 1 - 0.70 = 0.30. The probability that both C and D fail is their probabilities multiplied together, (0.23)(0.30) = 0.069. Therefore, the reliability of the combined unit C and D is 1 - 0.069 = 0.931.
Finally, the overall system reliability is the product of the reliabilities of A, B, and the combined unit C and D: (0.85)(0.81)(0.931) = 0.640635.