The following system will work provided there is at least one functional path ac

Question

The following system will work provided there is at least one functional path across (from left to right) the entire system. Let P(A) = 80 and P(C) = .99 represent the probability that components A and C work properly. If the three components are independent and the probability that the entire system works must be at lest .95, what is the required probability that component B works? Let A, B, and C be events and that all three have positive probabilities (i.e. none have probability 0). Construct a Venn diagram that contains three events A, B, and C such that P(A|C) = 1 and P(B|C) = 0.

Explanation / Answer

In the given system A and B are parellel and C is in series.

HEre P(A) = 0.80 and P(C) = 0.99 and P(B) = b

so Here P(System) >= 0..95

P (system) = P(C) * [P(A) P(B)]

so P(A) P(B) = 1 - ( 1- 0.80) * (1-x) = 1- 0.2(1-x) = 1- 0.2 + 0.2x = 0.8 + 0.2x

and P(C) * [P(A) P(B)] = 0.99 * (0.8 + 0.2x) = 0.792 + 0.198x

P (system) =  0.792 + 0.198x

P(System) >= 0..95

0.792 + 0.198x >= 0..95

0.198x >= 0.158

x >= 0.798

P(B) >= 0.798 is the required probability for system to work.

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