A fast food outlet has a soft serve machine. At the end of each day the machine runs an automatic cleaning cycle. There is a probability of 0.4 that the machine will fail during the cleaning cycle. If the machine fails then there is a probability of 0.5 that the following day will be spent waiting for parts, otherwise repairs can commence. If the technician is waiting for parts on any day then there is a probability of 0.9 that they will arrive during that day, and repairs can commence the following day. Once repairs commence, there is a probability of 0.7 that repairs are successful and the machine can return to service the following day, and a probability of 0.2 that it must wait for more parts. The possible states of the machine each day are operating, waiting for parts, and being repaired. Write down a transition matrix that represents this 3-state Markov chain. If the machine is operating today, what is the probability that it will be oper- ating in two days time? The probability matrix P has a fixed point n' = [0.52 0.18 0.30]. You know that the machine is not operating today. What is the probability that it is waiting for parts? What is the probability that is being repaired? What is the probability that it will be operating tomorrow?

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