A foundry has received an order for 20 custom-designed castings. The casting process costs $700 per unit scheduled. If a casting is good, then it is machined to specifications at an added cost of $500 per unit. If a casting is not sold, it has a recycle value of $300. The customer has indicated a willingness to pay $2000 per casting for 20 acceptable castings; the customer has also agreed to pay $1500 each for one or two additional castings. However, the customer is unwilling to purchase fewer than 20 or more than 22 castings. Based on historical records, the following probability distributions have been estimated. How many castings should be scheduled for production to maximize expected profit? What is the probability of losing money?

# Good Number of Castings Scheduled Castings 20 21 22 23 24 25 26 27 28 29 30 12 0.10 0.05 0.00 0.00 0.00 0.00 0.00 0.00 0.00 D.00 0.00 13 0.10 010 0. 05 0.05 0.05 0.05 0.00 0.00 0.00 0.00 0.00 14 0.10 0.10 0. 10 0.05 0.05 0.05 0.05 0.00 0.00 0.00 15 0.00 0.10 0.10 0. 10 0.10 0.05 0.05 0.05 0.00 0.00 0.00 0.00 16 0.10 0.10 0. 10 0.10 0.05 0.05 0.05 0.05 0.00 D.00 17 0.00 0.10 0.10 0. 10 0.10 0.10 0.05 0.05 0.05 0.05 0.00 0.00 18 0.20 0.10 0. 10 0.10 0.10 0.05 0.05 0.10 0.05 D.05 19 0.05 0.10 0.15 0. 10 0.10 0.10 0.10 0. 10 0.10 0.10 0.05 0.05 20 0.10 0.10 0. 15 0.10 0.10 0.10 0. 10 0.10 0.10 0.10 0.05 21 0.DO 0.10 0. 10 0.10 0.10 0.10 0. 10 0.10 0.10 D. LO 0.10 22 0.00 0.0D 0. 10 0.10 0.10 0.10 0. 10 0.10 0.10 D.LO 0.10 23 0.00 0.00 0.00 0.10 0.10 0.10 0. 10 0.10 0.10 0.10 24 0.10 0.00 0.00 0.00 0.00 0.10 0.10 0. 10 0.10 0.10 D. 10 0.10 25 0.D0 0.00 0.00 0.00 0.0D 0.10 0. 10 0.10 0.10 D.10 0.10 26 0.00 000 0.00 0:00 0.00 0.00 0 05 0.05 0.10 0.10 0.10 27 0.DO 0.00 0.00 0.00 0.00 0.00 0.00 0.05 0.05 0.10 0.10 28 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.05 D.05 0.05 29 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 D.05 0.05 30 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.0D 0.00 0.00 0.05 Shown below is the matrix of net income for each combination of Q and x