Three switches, namely S, A & B control a single indicator lamp as shown in Fig.

Question

Three switches, namely S, A & B control a single indicator lamp as shown in Fig. 0. At any given time, at least two of the switches need to be on for the lamp to be on; any single switch on does not allow sufficient current through to turn the lamp on due to the internal resistance of the switch. Switches A and B are suspected to intermittently fail and get stuck at the OFF ('0') position; both switches are outfitted with a fuse so they do not get stuck to the ON ('1') position. The following observation data have been reported over a period of one day (24hrs): Switch S is inaccessible and therefore the setting and setting changes have not been recorded during observation day. Determine the probabilities of the following events: Switch A fails alone Switch B fails alone Switch A and B fail jointly Failure is defined as the condition at which A or B is stuck at 0 regardless of its apparent setting; the switch can move into or recover from the fail condition at any time without warning and independently of setting or other context.

Explanation / Answer

(a) P(A) = Inteval in which only A fails/total interval observed = 173.25/(173.25+53.19+39.60)

=0.65

(b)P(B) =  Inteval in which only B fails/total interval observed = 53.19/(173.25+53.19+39.60)

=0.199932341

(c) Both A and B fail = P(both have 0 and 0 values) = 0/(173.25+53.19+39.60) = 0

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