4 Consider the system of electrical circuit below Probabilit

4. Consider the system of electrical circuit below. Probabilities on the components are reliabilities and all components are independent. The probability that a single component works is 0.92.
Let S1 be the subsystem consists of 1, 2 and 3. If and only if at least one of components (1, 2 or 3) works the subsystem S1 works. If and only if both S1 and 4 work, the system works.   

a. What is the probability that the subsystem S1 works?

b. What is the probability that the system works? Use p to substitute the answer of part (a) if you cannot solve part (a).

Solution

Let

1, 2, 3, 4 = the events where the respective components are working

a)

P(S1) = P(1 or 2 or 3) = P(1) + P(2) + P(3) - P(1)P(2) - P(2)P(3) - P(1)P(3) + P(1)P(2)P(3) = 0.92 + 0.92 + 0.92 - 0.92*0.92 - 0.92*0.92 - 0.92*0.92 + 0.92*0.92*0.92

P(S1) = 0.999488 [ANSWER]

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B)

P(S1 and 4) = P(S1) P(4) = 0.999488*0.92 = 0.91952896 [ANSWER]