There are two fire alarm devices in a house A and B These tw

There are two fire alarm devices in a house: A and B. These two alarm system share the circuit system so their performance may be correlated. It is found that if A works alone (B is turned off), the probability that A works well is 0.90; if B works alone (A is turned off), the probability of B working well is 0.85. If A and B work together and A has been found malfunctioned, the probability of B still working well is 0.8. Answer the following question. Please show your definition of each event in your solution. (a) Given that A and B are working together, what is the probability that at least one alarm works well? (b) If B has been found malfunctioned, what is the probability that A still works well? (c) What is the probability that BOTH alarms work well when they work together?

Solution

a)  Given that A and B are working together, what is the probability that at least one alarm works well

Prob of atleast one alarm works well = P(A works well) or P(B works well ) OR Prob (Both work well)

Prob A is not malfunctioned and B working well =0.92 by using (1-0.08)

P(A or B works well) = P(A works well not B) + P(B works well not A) + P(Both work well)

=0.10+0.15+0.10(0.92) = 0.10+0.15+0.092 = 0.342

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b) P (B malfunctioned and A works well ) = 0.10

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c) Both work well when working together = 0.092