2. (20 points) Consider a small bakery that makes excellent fruitcakes. All cakes are baked fresh in the morning and sold during the day and unsold cakes are discarded when the store is closed in the evening. Assume that the baker bakes 10 fruitcakes each morning. Most cus- tomers arrive during the day before 18.00. The number of customers that arrive before 18.00 is a uniform discrete random variable between 5 and 10 (P(Dai) = 1/6 for i = 5,6,..., 10). There are also a 1 few long-term customers who arrive after 18.00. Let De denote the number of these customers. The probability mass function of De is given by: P(De = 0) = 0.2, P(De = 1) = 0.2, P(De = 2) = 0.3, P(De = 3) = 0.3. The baker knows that the evening customers are valuable because they sometimes buy additional products other than the fruitcake. Assume that the price of the fruitcake is 10 TL and that the early (before 18.00) customers buy only the fruitcake but that each evening customer buys additional products worth an additional 40 TL with probability of additional purchase 3 = 1/4 (and buys only the cake with probability 3/4). Assume that the baker reserves two cakes for the evening customers. (a) (4 points) What is the probability that the baker's revenue is less than 65 TL on a given day? (b) (6 points) What is the change in the baker's expected revenue if he were to reserve one cake instead of two for late customers (reason using a decision tree)? (c) (4 points) What is optimal number of fruitcakes to reserve to late customers? (d) (6 points) If the probability of additional purchase were small, it might not be necessary to reserve any fruitcakes for evening customers. What is the largest value of 3 for which it is optimal not to reserve any fruitcakes? 1