$(20$ points$)$ A job shop has received an order for high$-$precision formed parts. The cost of producing each part is estimated to be $$5,000.$ The customer requires that either $10,$ or $11$ parts be supplied. Each good part sold will produce revenue of $$12,000.$ However, if fewer than $10$ good parts are produced, none will be purchased; if more than $11$ good parts are produced, the excess will not be purchased. The company can recycle the unpurchased parts and get $$1000$ for each part. The probability of an individual part being acceptable equals $0\mathrm{.}8.$ Determine the expected profits for batch production quantities of $10,11,$ and $12.$ For each, determine the expected value on the transaction. Of the three choices, which is most preferred? Why?