anyone can help? To minimize the possible false alarm, the building fire alarm s
ID: 3361771 • Letter: A
Question
anyone can help?
To minimize the possible false alarm, the building fire alarm system is designed as a k-out- of-N system, that is, at least k detectors identify the fire occurrence for trigging the building fire alarm signal. All the detectors are identical and independently functioning of each other Two failures modes of detectors may be considered i) Fail-to-danger: The detector is designed to detect an actually present fire but fails to send the fire signal to the building fire alarm system. Assume the probability that each detector being failed to identify the fire occurrence is 0.05 ii) False alarm: The detector sends out a fire signal in absence of fire. Assume the probability that each detector gives a false alarm in absence of fire is 0.09 As the designer, you are required to design the system (i.e. find out the value of k and N) to meet the following requirement a) The number of detectors, N, must be as small as possible to optimize the system cost. How will you design the building fire alarm system so that the probability of the system being failed to detect an actually present fire could be less than 0.0001? The false alarm probability for the building fire alarm system is also required to be less than 0.0001. How will you design the building fire alarm system so that both of the two types of failures could be less than 0.0001? b)Explanation / Answer
During fire
probability of an alarm's failure (false negative) is 0.05
probability of an alarm's success (true positive) is 0.95
During non-fire
probability of an alarm going on (false positive) is 0.09
probability of an alarm being silent (true negative) is 0.91
probability of k alarms not going on during fire= nCk p^k (1-p)^(n-k)
where p = 0.05
probability of k alarms going on during non-fire= nCk p^k (1-p)^(n-k)
where p = 0.09
nCk 0.05^k 0.95^(n-k) - equation 1
nCk 0.09^k 0.91^(n-k) - equation 2
divide both equations by nCk and both the equations should be less than 0.0001
Take natural logarithms on both sides for the 2 equations.
k ln(0.09/0.91) - n ln(0.91) = ln(0.0001)
k ln(0.05/0.95) - n ln(0.95) = ln(0.0001)
solving both the equations we get
k = 2.49
n = 36.75
there fore the minimum number of n and k are
n = 37 and k = 3
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