All Boots is a retailer of boots. It sources a kind of waterproof hunting boots from an Asian supplier for $35 each and sells them to customers for $60 each. Leftover boots at the end of season will be sold to an outlet mall at $20 each. Given the $60 retail price, All Boots forecasts the demand distribution as follows:

Now suppose All Boots found a reliable vendor in the United States that can produce boots very quickly but at a higher price than All Boots' Asian supplier. Hence, in addition to boots from Asia, All Boots can buy an unlimited quantity of additional boots from this American vendor at $45 each after demand is know.

a) Suppose All Boots orders 500 boots from the Asian supplier (Note that the first order quantity of 500 units is given, which may not be the optimal order quantity). What is the probability that All Boots will order from the American supplier once demand is known, i.e., the probability of placing a second order? (Hint: given the 1st order quantity of 500 units, under what situation in terms of demand will All Boots need to place a second order?)

b) Again assume that All Boots orders 500 boots from the Asian supplier. On average, how many boots should the American supplier expect that All Boots will order, i.e., the expected second order quantity? Parts c) and d) are separate from parts a) and b)

c) Given the opportunity to order from the American supplier at $45 per boot, what order quantity from its Asian supplier now maximizes All Boots' expected profit, i.e., optimal first order quantity? d) Given the order quantity in part c) (not the quantity in parts a and b), what is All Boots' expected profit? [Hint: expected profit = maximum profit - mismatch cost, where maximum profit = (p-c)* ]