Bayes Theorem

The managers of Red Valley Auto Products are considering purchasing a new piece of customized automation equipment for their main production facility, from ABC Automation.  The operating efficiency of the machine will either be high, medium or low and estimated return on investment for the equipment under each of these operating conditions is estimated to be $80 million, $15 million and -$40 million, respectively.  The company’s automation engineering manager estimates that there is a 0.3 probability that the machine operate with high efficiency, a 0.4 probability that it will have medium efficiency and a 0.3 probability that it will be low.

a.       On the basis of the manager’s probabilities, determine whether the machine should be purchased.

b.      Rather than making an immediate decision, the company has the opportunity to visit one of ABC’s existing customers who has a similar piece of equipment.  Red Valley sends a team of engineers to this facility to do some benchmarking.  This delays the project, and this delay, together with other costs from the investigation costs the company $1 million.  However, the benchmarking gives Red Valley some additional information.  It gives them an indication that their machine will operate with high efficiency.  However, this indication is not perfect, as the benchmarked piece of equipment is not exactly the same as the one Red Valley plans to purchase.  The reliability of this indication is as follows:

                                                               i.      If Red Valley’s new machine is such that high efficiency can be achieved, then there is a probability of 0.80 that the benchmarking would in fact indicate high efficiency.

                                                             ii.      If Red Valley’s new machine is such that medium efficiency can be achieved, then there is a probability of 0.25 that the benchmarking would incorrectly indicate high efficiency.

                                                           iii.      If Red Valley’s new machine is such that low efficiency can be achieved, then there is a probability of 0.10 that the benchmarking would incorrectly indicate high efficiency.

In light of this new information, should the company change their decision?  Discuss whether you think it was worth spending $1 million to collect this new information.

Please show all work and decision and use bayes theorem

please explain on the steps