91Ó°ÊÓ

A device has five sensors connected to an alarm system. The alarm is given if three or more of the sensors trigger a switch. If a dangerous condition is present, each of the switches has high (but not unit) probability of activating; if the dangerous condition does not exist, each of the switches has low (but not zero) probability of activating (falsely). Suppose \(D=\) the event of the dangerous condition and \(A=\) the event the alarm is activated. Proper operation consists of \(A D \bigvee A^{c} D^{c}\). Suppose \(E_{i}=\) the event the ith unit is activated. Since the switches operate independently, we suppose \(\left\\{E_{1}, E_{2}, E_{3}, E_{4}, E_{5}\right\\} \operatorname{ci} \mid D\) and ci \(\mid D^{c}\) Assume the conditional probabilities of the \(E_{1},\) given \(D,\) are \(0.91,0.93,0.96,0.87,0.97,\) and given \(D^{c},\) are \(0.03,0.02,0.07,0.04,0.01,\) respectively. If \(P(D)=0.02,\) what is the probability the alarm system acts properly? Suggestion. Use the conditional independence and the procedure ckn.