13. A particular automatic sprinkler system has two differ- ent types of activat

Question

13. A particular automatic sprinkler system has two differ- ent types of activation devices for each sprinkler head. One type has a reliability of 0.9; that is, the probabil- ity that it will activate the sprinkler when it should is 0.9. The other type, which operates independently of the first type, has a reliability of 0.8. If either device is triggered, the sprinkler will activate. Suppose a fire starts near a sprinkler head. a. What is the probability that the sprinkler head will be activated? b. What is the probability that the sprinkler head will c. What is the probability that both activation devices d. What is the probability that only the device with not be activated? will work properly? reliability 0.9 will work properly?

Explanation / Answer

a)P(sprinkler head activated)=1-P(none of device activated) =1-(1-0.9)*(1-0.8)=0.98

b)

P(sprinkler head not activated)=1-P(sprinkler head activated)=1-0.98 =0.02

c)

P(both devices work propoerly)=P(first device work properly)*P(second device wrk properly)

=0.9*0.8=0.72

d)

P(only device with probabiliy 0.9 work)=P(first device work)*P(second device not work)

=0.9*(1-0.8)=0.18

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