Assignment:
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.
Provide complete and step by step solution for the question and show calculations and use formulas.