For two components in parallel, what is the general formula for system reliability?

• A. R1 × R2
• B. R1 + R2
• C. max(R1, R2)
• D. 1 − (1−R1)(1−R2)

Answer: (D)

In a parallel arrangement, the system works as long as at least one path is functioning. The only way the system fails is if both components fail. If failures are independent, the probability both fail is (1 − R1)(1 − R2). So the overall reliability is 1 minus that probability: Rsys = 1 − (1 − R1)(1 − R2). This captures the redundancy benefit of parallel setups—the reliability improves as either component becomes more reliable, and the system remains operational as long as one component works.

This differs from a series case, where both components must work (R1 × R2), and from simpler add-ons like R1 + R2 or max(R1, R2), which don’t correctly model the probability that both paths could fail or work together.