A system consists of five components is connected in series as shown below. As soon as one component fails, the entire system will fail. Assume that the components fail independently of one another. (a) Suppose that each of the first two components have lifetimes that are exponentially distributed with mean 99 weeks, and that each of the last three components have lifetimes that are exponentially distributed with mean 129 weeks. Find the probability that the system lasts at least 47 weeks. (b) Now suppose that each component has a lifetime that is exponentially distributed with the same mean. What must that mean be (in years) so that 83% of all such systems lasts at least one year?

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