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Consider a production system consisting of three Bernoulli machines and a controller, which also obeys the Bernoulli reliability model. This production system is considered up if the controller and at least two machines are up. During each cycle, the controller is up with probability 0.8 and each machine is up with probability 0.9. The controller and the machines fail independently.
(a) Calculate the probability that the production system is up.
(b) Assume that a second controller, identical to the first one, is added to the system, and the re-designed system is up if at least one controller and two machines are up. Calculate the probability that the re-designed production system is up.
(c) Explain the reason for the difference between the two numbers you calculated.
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