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An engineering system consisting of n components is said
Chapter 3, Problem 67P(choose chapter or problem)
Problem 67P
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.
(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).
Questions & Answers
QUESTION:
Problem 67P
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.
(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).
ANSWER:
Step 1 of 3
(a)
Suppose that all components function independently of one another.
If the component functions with probability
we are asked to compute the probability that a
system functions.
Let be the event that we have a functioning system.
We must have two or more of the four components functioning so we can have that
\(P(E)=p_{1} p_{2} p_{3} p_{4}+\left(1-p_{1}\right) p_{2} p_{3} p_{4}+p_{1}\left(1-p_{2}\right) p_{3} p_{4}+p_{1} p_{2}\left(1-p_{3}\right) p_{4}+p_{1} p_{2} p_{3}\left(1-p_{4}\right)+\left(1-p_{1}\right)\left(1-p_{2}\right) p_{3} p_{4}\)