For the k-out-of-n system described in Problem, assume that each component independently w...

For the k-out-of-n system described in Problem, assume that each component independently works with probability . Find the conditional probability that component 1 is working, given that the system works, when

(a) k = 1, n = 2;

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(b) k = 2,n = 3.

Problem

An engineering system consisting of n components is said to be a k-out-of-n system (k ≤ n) if the system functions if and only if at least k of the n components function. Suppose that all components function independently of one another.

(a) If the ith component functions with probability Pi, i = 1, 2, 3, 4, compute the probability that a 2-out-of-4 system functions.

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(b) Repeat part (a) for a 3-out-of-5 system.

(c) Repeat for a k-out-of-n system when all the Pi equal p (that is, Pi = p, i = 1, 2, ..., n).