The primary cooling unit in a nuclear power plant has a reliability of 0.95. There is alsoa back-up cooling unit to substitute for the primary unit when it fails. The reliability ofthe back-up unit is 0.85. The cooling system of the plant is considered reliable if eitherone of the systems is working. Assuming that the two systems are independent, thereliability of the cooling system of the power plant is:

Question

BlackTom AI · Accepted Answer

Because the system is successful whenever at least one unit works, the total reliability equals 1 minus the probability that both units fail.

1) Relevant concepts:
- Reliability of a component is the probability it works.
- For independent components, the probability both fail is the product of their individual failure probabilities.
- For a parallel (redundant) system that works if either component works, system reliability = 1 − P(both fail) = P(A works ∪ B works).

2) Step-by-step reasoning:
- Primary unit reliability: P(A works) = 0.95 → failure probability P(A fails) = 1 − 0.95 = 0.05.
- Backup unit reliability: P(B works) = 0.85 → failure probability P(B fails) = 1 − 0.85 = 0.15.
- Independence implies P(both fail) = P(A fails) × P(B fails) = 0.05 × 0.15.

3) Calculations (all intermediate steps):
- P(both fail) = 0.05 × 0.15 = 0.0075.
- System reliability = 1 − 0.0075 = 0.9925.
(Alternative check using union formula: P(A ∪ B) = P(A) + P(B) − P(A)P(B) = 0.95 + 0.85 − (0.95×0.85) = 1.80 − 0.8075 = 0.9925.)

4) Why this answer is correct:
- The system is redundant (parallel). It only fails if both components fail simultaneously. Using the independence assumption and basic probability rules yields 1 − 0.0075 = 0.9925, so the cooling system is 99.25% reliable.

5) Why the other options are incorrect:
- a. 0.85: This is just the backup unit’s reliability, not the combined system.
- d. 0.95: This is just the primary unit’s reliability, not the combined system.
- c. 0.8075: This equals 0.95 × 0.85, the probability both units work simultaneously (useful but not the event “at least one works”); it undercounts cases where exactly one unit works.

6) Conclusion:
The correct reliability for the cooling system is 0.9925, matching answer b. 0.9925.

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