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The mean time between failures (MTBF) is a common metric used to measure the performance of manufacturing systems. MTBF is the elapsed time between failures of a system during normal operations. The failures could be caused by broken machines or computer errors, among other failures. Suppose that the MTBF for a new automated manufacturing system follows an exponential distribution with a mean of 12.7 hours

a. What is the probability that the automated manufacturing system runs for more than 15 hours without failure?

b. What is the probability that the automated manufacturing system runs for 8 or fewer hours without failure?

c. What is the probability that the automated manufacturing system runs for more than 6 hours but less than 10 hours before a failure?