Part A Keystone can only properly market a limited number of properties. They are trying to figure out which property they should select to add to their portfolio among four candidates (and in so doing, gain some insight into how to make such decisions in the future). There is some uncertainty as the selling price (and therefore commission/profit) depends on the state of the real estate market 6 months in the future. Based on their experience, Keystone has provided an estimate of selling prices for a "good" real estate market and a "bad" real estate market for the four candidate properties (see Table 1). Table 1: Sales price (in millions of $) Bad Good Alternatives Market Market Property 1 Property 2 Property 3 Property 4 2.1 3.7 1.7 3.8 2.7 2.6 2.2 2.4 Even with these estimates, though, Keystone managers are unsure which property to choose. Keystone managers consider themselves to be optimistic about the future, but would like to consider a variety of ways to make this decision. When asked how likely they think it is that the market will be good, Keystone said "about a 70% chance". Keystone also mentioned that they are curious about "opportunity loss", but they have no idea what this means or how to incorporate it into their decision making process. Keystone noted that they could pay for market forecasts that will help predict good and bad markets, but they have not done so in the past. They would like some help deciding whether to purchase the forecasts and what they should pay for a forecast. (Guiding questions: What are the expected values of each alternative? What would be the expected value of the total return if you have no forecast? What would the expected value be if you have a perfect forecast? What would be the value of such a forecast?) Part B Keystone puts a lot of work into showcasing properties to potential buyers (something like a personal open house). Each showcase results in either a 'success' (the buyer purchases the property) or a 'failure' (the buyer refuses to buy the property). Historically, Keystone figures that on any given showcase there is a 15% chance they will be successful and make a sale. Keystone has 6 showcases planned for this month, and they would like to explore some probabilities surrounding these showcases to help them manage this process and forecast revenues. In particular, Keystone wants to see at least two of these showcases turn into sales. What are the chances that exactly 2 of the showcases will result in a sale? What are the chances that they will make at least 2 sales? What are the chances that they will make fewer than 2 sales? Sometimes buyers will request a second viewing of the property. Based on historical evidence, buyers ask for a second viewing on 40% of successful showcases and on only 10% of unsuccessful showcases. What are the chances that a showcase will be successful, given that a potential buyer has asked for a second viewing? (Guiding questions: What are the chances that exactly 2 of the showcases will result in a sale? What are the chances that they will make at least 2 sales? What are the chances that they will make fewer than 2 sales? What are the chances that a showcase will be successful, given that a potential buyer has asked for a second viewing?) Part C Keystone is curious how likely they are to reach their annual revenue target. Past experience suggests that annual revenue follows a typical bell-shaped distribution, with a mean of $32 million and a standard deviation of $5 million. Keystone has set a revenue goal of $38 million for this year. How likely are they to reach the target based on the historical data? In order to remain in good standing with investors, Keystone needs to make at least $26 million. What are the chances they fail fall into poor standing with their investors? (Guiding questions: How likely are they to reach the target based on the historical data? What are the chances they fail fall into poor standing with their investors?)