A k out of n system is one in which there is a group of n components, and the system will function if at least k of the components function. Assume the components function independently of one another.
a. In a 3 out of 5 system, each component has probability 0.9 of functioning. What is the probability that the system will function?
b. In a 3 out of n system, in which each component has probability 0.9 of functioning, what is the smallest value of n needed so that the probability that the system functions is at least 0.90?
(a) Let X be the number of components that function. Then X ~ Bin(5,0.9).
(b) We need to find the smallest value of n so that P(X ? 2) < 0.10 when X ~ Bin(n, 0.9). Consulting Table A.1, we find that if n = 3, P(X ? 2)=0.271, and if n=4, P(X ? 2)=0.052. The smallest value of n is therefore n = 4.
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