Question

A particular fault is supposed to trigger five(5) redundant alarms. These alarms fail independently. If the probabilities that Alarms 1 and 2 will not trigger when this fault occurs are 0.05 and 0.1, respectively, and that Alarms 3-5 will trigger when this fault occurs are 0.90 for each one. What is the probability that at least one alarm will trigger when a fault occurs? What is the probability that only one will trigger when a fault occurs?

Answer 1

If A is the event of triggering alarm then P(A) is the probability that the alarm will trigger when the fault occurs. And if P(A') is the probability that the alarm will not trigger when the fault occurs then, P(A') = 1 - P(A)

Now, P(1) = 1-0.05 = 0.95

P(2) = 1-0.1 = 0.9

P(3) = P(4) = P(5) = 0.9

Question 1:

P(no alarm will trigger when fault occurs) = P(1') * P(2') * P(3') * P(4') * P(5') = 0.05 * 0.1 * 0.1 * 0.1 * 0.1

So, P(at lease one alarm will trigger when fault occurs) = 1 -  0.05 * 0.1 * 0.1 * 0.1 * 0.1 = 0.999995

Question 2:

P(only one alarm will trigger when a fault occurs) = P(1) * P(2') * P(3') * P(4') * P(5') + P(1') * P(2) * P(3') * P(4') * P(5') + P(1') * P(2') * P(3) * P(4') * P(5') + P(1') * P(2') * P(3') * P(4) * P(5') + P(1') * P(2') * P(3') * P(4') * P(5) = 2.75e-04

** If the answers do not match and of there is any confusion please comment.