(20 pts) A particular automatic sprinkler system has two different types of acti

Question

(20 pts) A particular automatic sprinkler system has two different types of activation devices for each sprinkler head. One type has reliability of 0.95; that is, the probability that it will activate when it should is 0.95. The other type, which operates independently of the first type, has a reliability of 0.85. If either device is triggered, then the sprinkler will activate. Suppose a fire starts near a sprinkler head. a. What is the probability that the sprinkler head will be activated? qs, it125-.gs. a S 9913 b. What is the probability that the sprinkler head will not be activated? c. What is the probability that both activation devices will work properly? d. What is the probability that only the device with reliability 0.95 will work properly? P(ini) g ) Pc In) 4

Explanation / Answer

a)P(sprinkler activated) =P(at least one device is triggered) =1-P(none of device triggered) =1-P(1c)*P(2c)

=1-(1-0.95)*(1-0.85)

=0.9925

b)

probability no be actiivated =P(none of device triggered) =P(1c)*P(2c) =(1-0.95)*(1-0.85) =0.0075

c)

probability that both work properly =P(1)*P(2) =0.95*0.85=0.8075

d)

P(1) =P(1)-P(1 & 2) =0.95-0.95*0.85=0.1425

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