The system shown below is a segment of an electrical circuit

The system shown below is a segment of an electrical circuit which has three relays. The current will flow from point (a) to
point (b), that is system is working, if there is at least one close path when the relays are switched to \"close\". However, the relays may malfunction. Suppose when the switch is thrown, the probability that relay 1 closes properly is .3, relay 2 closes
properly is .5 and relay 3 closes properly is .8, where relays are working independent of each others.

. Find the probability that system is working (that is, current is flowing from point a to point b). That is find , where W
denotes the event that system is working

Find the (conditional) probability that the component 1 is not working given that we know that system is working. That is find P(A\'1/W}

Solution

P( relay 1 closes) = 0.3
P( relay 2 closes) = 0.5
P( relay 3 closes) = 0.8

a)
P(W) = P( System is working fine) = P(A1)*P(A2)*P(A3) = 0.3*0.5*0.8 = 0.12

b)
P(A\'1/W} = P( A\'1 n W) / P(W)
P(A\'1) = 1 - P(A) = 0.7
P(A\'1 n W) = P(A1)*P(W) = 0.7*0.12 = 0.084

P(A\'1/W} = P( A\'1 n W) / P(W) = 0.084 / 0.12 = 0.7