Suppose that n components are connected in series. For each unit, there is a backup unit, and the system fails if and only if both a unit and its backup fail. Assuming that all the units are independent and fail with probability p, what is the probability that the system works? For n = 10 and p = .05, compare these results with those of Example F in Section 1.6.
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Suppose that a system consists of components connected in a series, so the system fails if any one component fails. If there are n mutually independent components and each fails with probability p, what is the probability that the system will fail? It is easier to find the probability of the complement of this event; the system works if and only if all the components work, and this situation has probability (1 − p)n. The probability that the system fails is then 1− (1 − p)n. For example, if n = 10 and p = .05, the probability that the system works is only .9510 = .60, and the probability that the system fails is .40. Suppose, instead, that the components are connected in parallel, so the system fails only when all components fail. In this case, the probability that the system fails is only .0510 = 9.8 × 10−14.
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