Kate is a buyer for a men's fashion retail store. She will order a new cloth overcoat from Paris for the fall fashion season. Based on her experience, she expects to sell at least 100 coats, and at most 400, but she feels that any number of sales in between is equally likely. Therefore, she estimates that her sales are uniformly distributed between 100 and 400.The total cost to the store is $100 per coat, and the retail price is set at $180. Any coats left over at the end of season would be sold at $60 each. a) How many coats should Kate buy if she wants to maximize profits?

b) Assume Kate buys the number of coats suggested in part a), what is the probability that the coats sell out? What is the probability that they do not sell out?