Donut Lovers is a small bakery in Duluth that makes delicious pastries and in particular, donuts. On Halloween day, the owner, Jimmy, must decide how many Halloween donuts to prepare (these are special donuts shaped like ghosts which are only sold on Halloween day) before opening the bakery at 5 am. The selling price of a Halloween donut is $1.5. The total cost of making a Halloween donut is 25 cents. Any unsold Halloween donut at the end of Halloween day is first wrapped in plastic then donated to Minnie's food pantry; it costs Jimmy 10 cents per donut to wrap it in plastic. If the bakery runs out of Halloween donuts before the end of the day, Jimmy is always able to convince customers who wanted a Halloween donut to buy a "regular" donut instead. The regular donuts are sold at $1.25 and cost 20 cents to make. Assume there is an unlimited supply of regular donuts. Based on his experience selling donuts on Halloween day over the last few years, Jimmy estimates that the number of Halloween donuts he can sell on Halloween day this year can take the following values with the following probabilities:

Demand for donuts Probability 100 0.05 120 0.05 140 0.1 160 0.1 180 0.1 200 0.2 220 0.1 240 0.1 260 0.1 280 0.05 300 0.05 (a) What is the expected demand for Halloween donuts? (b) Suppose that Jimmy decides to prepare 160 Halloween donuts. How much profit can he expect to make from selling Halloween donuts (and regular donuts as a substitute)? (c) How many Halloween donuts should Jimmy prepare before opening the bakery on Halloween day in order to maximize her expected profit? Comment on how this number compares to the expected demand for Halloween donuts (your answer to (a)) and explain why it is higher or lower.