In order to increase the reliability of a process, a critical process variable is measured on-line using two sensors.
Sensor A is available 95% of the time, while Sensor B is available 90% of the time. Suppose that the two sensors operate independently and that their periods of unavailability occur randomly.What is the probability that neither sensor is available at any arbitrarily selected time?

Question

Accepted Answer

Let A denote the event that SensorAis not available and B denote the event that Sensor B is not available. The event that neither Sensor is available can be expressed as(A ∪ B)′.Then from Eqs. F-1 and F-2,

P(A′) = 1 − P(A) (F-1)

P(A ∪ B) = P(A) + P(B) − P(A ∩ B) (F-2)

P(A ∪ B)′ = 1 − P(A ∪ B)
P(A ∪ B)′ = 1 − [P(A) + P(B) − P(A ∩ B)]
P(A ∪ B)′ = 1 − [0.95 + 0.90 − (0.95)(0.90)] = 0.005

Question F.2: In order to increase the reliability of a process, a critica...

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