11.3 Expected Value (Expectation) 681 = ⋅ + ⋅ = + = + = E P P (does not rain) (number of people) (rain) (number of people) 0.7(20,000) 0.3(12,000) 14,000 3600 17,600 Thus, the average, or expected, number of people who will attend the concert is 17,600. ■ Now try Exercise 31 Example 2 A New Flight Route Ateimpo Airlines is considering adding a route from El Paso, TX to Tucson, AZ. Before making a decision, the company needs to consider many factors, including potential profit or loss. After considerable research, Ateimpo estimates that if it adds the route, there is a 60% chance of making an annual $800,000 profit, a 10% chance of breaking even, and a 30% chance of losing $1,000,000. How much can Ateimpo expect to make annually on this new route? Solution Three amounts are to be considered: a gain of $800,000, breaking even at $0, and a loss of $1,000,000. The probability of gaining $800,000 is 0.6, the probability of breaking even is 0.1, and the probability of losing $1,000,000 is 0.3. = + + = + + − = PA P A P A Ateimpo’s expectation (0.6)($800, 000) (0.1)($0) (0.3)( $1, 000, 000) $180,000 1 1 2 2 3 3 Ateimpo Airlines has an expectation, or expected average annual gain, of $180,000 for adding this particular route. Thus, if the company opened routes like this one, with these particular probabilities and amounts, in the long run it would have an average annual gain of $180,000 per route. However, you must remember that there is a 30% chance that Ateimpo Airlines will lose $1,000,000 on this particular route or any particular route with these probabilities and amounts. ■ Now try Exercise 7 Learning Catalytics Keyword: Angel-SOM-11.3 (See Preface for additional details.) Example 3 Test-Taking Strategy Maria is taking a multiple-choice test in which there are five possible choices for each question. Of these choices, only one is the correct answer. The instructions indicate that she will be awarded 2 points for each correct response, that she will lose 1 2 point for each incorrect response, and that no points will be added or subtracted for questions left blank. a) If Maria does not know the correct answer to a question and she guesses at the answer, determine her expectation for that question. b) If Maria does not know the correct answer to a question, is it to her advantage or disadvantage to guess at an answer? c) If Maria can eliminate one of the possible choices and she guesses at the answer, determine her expectation. d) If she can eliminate one of the possible choices, is it to her advantage or disadvantage to guess at an answer? Solution a) To determine the expected value if Maria guesses at an answer, recall only one of five possible choices is the correct answer. Bychykhin Olexandr/ Shutterstock
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