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Probability, Profit / Loss and Expected Values

A coats manufacturer knows that there is a probability of 1/3 of their coats
having defects that will prevent a retailer from selling the coat. The cost of
manufacturing the coats is $60 per coat. To persuade retailers to buy from them,
they make the following offer:
? A sample of five coats is to be taken, at random, from each consignment
of 30 coats.
? If there are no defective coats in the sample then the retailer will buy the
whole consignment at full cost of $100 per coat.
? A reduction of $20 per coat is made in the price for every defective coat
found in the sample. So, for example, if 3 defects were found then the
retail would buy all 30 coats for $40 each, ($1200 in total); if all five are
defective then the retailer would get all 30 coats for free.
Once a retailer has bought the consignment they will go through them and
throw out the defective coats, and put the good coats on sale for $150.
1. Calculate the expected profit/loss for the manufacturer if this arrangement
is continued over the long term.
2. Calculate the expected profit/loss for the retailer.
3. If the manufacturer wanted the mean profit per coat (i.e., the expected
profit) to be more than $30, what is the maximum reduction that can be
offered per defect in the sample.

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Probability, Profit / Loss and Expected Values are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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