A particular automatic sprinkler system has three different types of activation

Question

A particular automatic sprinkler system has three different types of activation devices for each sprinkler head. One type (A) has a reliability of 0.9; that is, the probability that will activate the sprinkler when it should is 0.9. Other types of activation devices, B and C, have reliability of 0.8 and 0.7, respectively. If at least one device is triggered, the sprinkler will activate. Each type of sprinkler operates independently. Suppose a fire starts near a sprinkler head. What is the probability that the sprinkler head will not be activated? What is the probability that the sprinkler head will be activated? What is the probability that all three activation devices will work properly? What is the probability that at least one device will not be activated? What is the probability that only A will work properly?

Explanation / Answer

a)P(the sprinker head will not be activated)=0.1*0.2*0.3=0.006 (As they are independent)

b)Probability that it will be activated =1- P(the sprinker head will not be activated)=0.994

c)Probability that all 3 will work=0.9*0.8*0.7=0.50

d)Probability that at least one will not be activated= 1- Probability that none will be actiavted=0.994

e)Probability that only A will work properly=0.9*0.2*0.3=0.054

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