# 14.4. Expected Value Objectives. Expected Value

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1 . Expected Value Objectives. Understand the meaning of expected value. 2. Calculate the expected value of lotteries and games of chance.. Use expected value to solve applied problems. Life and Health Insurers Profits Skyrocket 2%...* How do insurance companies make so much money? When you buy car insurance, you are playing a sort of mathematical game with the insurance company. You are betting that you are going to have an accident the insurance company is betting that you won t. Similarly, with health insurance, you are betting that you will be sick the insurance company is betting that you will stay well. With life insurance, you are betting that,... well,... you get the idea. Expected Value Casinos also amass their vast profits by relying on this same mathematical theory called expected value, which we will introduce to you in this section. Expected value uses probability to compare alternatives to help us make decisions. Because of an increase in theft on campus, your school now offers personal property insurance that covers items such as laptops, ipods, cell phones, and even books. Although *According to a report by Weiss Ratings, Inc., a provider of independent ratings of financial institutions. Copyright 200 Pearson Education, Inc.

4 698 CHAPTER y Probability This amount means that, on the average, the casino expects you to lose slightly more than 5 cents for every dollar you bet. Now try Exercises to 8. ] The roulette wheel in Example is an example of an unfair game. DEFINITIONS If a game has an expected value of 0, then the game is called fair. A game in which the expected value is not 0 is called an unfair game. Outcome Value Probability You win \$99,000 You lose -\$ 999,000 TABLE.0 Values and probabilities associated with playing the daily number. Although it would seem that you would not want to play an unfair game, in order for a casino or a state lottery to make a profit, the game has to be favored against the player. EXAMPLE Determining the Fair Price of a Lottery Ticket Assume that it costs \$ to play a state s daily number. The player chooses a three-digit number between 000 and 999, inclusive, and if the number is selected that day, then the player wins \$500 (this means the player s profit is \$500 - \$ = \$99.) a) What is the expected value of this game? b) What should the price of a ticket be in order to make this game fair? SOLUTION: a) There are,000 possible numbers that can be selected. One of these numbers is in your favor and the other 999 are against your winning. So, the probability of 999 you winning is and the probability of you losing is. We summarize, the values for this game with their associated probabilities in Table.0. The expected value of this game is therefore probability of winning probability of losing This means that the player, on average, can expect to lose 50 cents per game. Notice that playing this lottery is 0 times as bad as playing a single number in roulette. b) Let x be the price of a ticket for the lottery to be fair. Then if you win, your profit will be x and if you lose, your loss will be x. With this in mind, we will recalculate the expected value to get a # (500 - x)b + a 999 # x2-999x 500 -,000x (-x)b = =.,000,000,000,000 We want the game to be fair, so we will set this expected value equal to zero and solve for x, as follows: 500 -,000x = 0,000,000 # 500 -,000x =,000 # 0 = 0,000 Multiply both sides of the equation by, ,000x = 0 Cancel,000 and simplify. 500 =,000x Add,000x to both sides. 500,000 =,000x,000 = x x = 500,000 = 0.50, ( ),000 amount won Divide both sides by,000. Simplify. amount lost , , Copyright 200 Pearson Education, Inc.

6 700 CHAPTER y Probability Quiz Yourself Calculate the expected value as in Example 5(b), but now assume that the student can eliminate two of the choices. Interpret this result. b) If you eliminate one of the choices and choose randomly from the remaining four choices, the probability of being correct is with a value of + point; the probability of being incorrect is with a value of -. The expected value is now a # b + a # a - bb = + - = 0. You now neither benefit nor are penalized by guessing. Now try Exercises 9 to 22. ] Businesses have to be careful when ordering inventory. If they order too much, they will be stuck with a surplus and might take a loss. On the other hand, if they do not order enough, then they will have to turn customers away, losing profits. EXAMPLE 6 Using Expected Value in Business Cher, the manager of the U2 Coffee Shoppe, is deciding on how many of Bono s Bagels to order for tomorrow. According to her records, for the past 0 days the demand has been as follows: Demand for Bagels 0 0 Number of Days with These Sales 6 She buys bagels for \$.5 each and sells them for \$.85. Unsold bagels are discarded. Find her expected value for her profit or loss if she orders 0 bagels for tomorrow morning. SOLUTION: We can describe the expected value in words as P(demand is 0) * (the profit or loss if demand is 0) + P(demand is 0) * (the profit or loss if demand is 0). For of the last 0 days the demand has been for 0 bagels, so P(demand is 0) = Similarly, P(demand is 0) = 6 0 = = 0.6 We will now consider her potential profit or loss. If the demand is for 0 bagels, she will sell all of the bagels and make 0(\$.85 - \$.5) = 0(\$0.0) = \$6.00 profit. If the demand is for 0 bagels, she will make 0(\$0.0) = \$2.00 profit on the sold bagels, but will lose 0(\$.5) = \$.50 on the 0 unsold bagels. So, she will lose \$2.50. The following table summarizes our discussion so far if she orders 0 bagels. Demand Probability Profit or Loss 0 0. \$ \$2.50 Quiz Yourself 5 Redo Example 6 assuming that Cher orders 0 bagels. Thus, the expected value in profit or loss for ordering 0 bagels is (0.0)(6) + (0.60)(-2.50) = (-.50) =.90. So she can expect a profit of \$.90 if she orders 0 bagels. ] 5 Copyright 200 Pearson Education, Inc.

9 .5 y Looking Deeper: Binomial Experiments 70. Mike sells the Town Crier, a local paper, at his newsstand. Over the past 2 weeks, he has sold the following number of copies. Number of Copies Sold Number of Days with These Sales 2 Each copy of the paper costs him 0 cents, and he sells it for 60 cents. Assume that these data will be consistent in the future. a. What is Mike s expected daily profit if he orders 80 copies per day? (Hint: First compute the profit Mike will earn if he can sell 90, 85, 80, and 75 papers.) b. What is Mike s expected daily profit if he orders 85 copies per day? 5. a. Calculate the expected total if you roll a pair of standard dice. b. Unusual dice, called Sicherman dice, are numbered as follows. Red:, 2, 2,,, Green:,,, 5, 6, 8 Calculate the expected total if you roll a pair of these dice. 6. Suppose we have two pairs of dice (these are called Efron s dice) numbered as follows. Pair One: Red: 2, 2, 2, 2, 6, 6 Green: 5, 5, 6, 6, 6, 6 Pair Two: Red:,,, 5, 5, 5 Green:,,,, 2, 2 If you were to play a game in which the highest total wins when you roll, what is the better pair to play with? 7. In playing a lottery, a person might buy several chances in order to improve the likelihood (probability) of winning. Discuss whether this is the case. Does buying several chances change your expected value of the game? Explain. 8. A lottery in which you must choose 6 numbers correctly from 0 m 0 possible numbers is called a lottery. In general, an 6 n lottery is one in which you must correctly choose n numbers from m possible numbers. Investigate what kind of lotteries there are in your state. What is the probability of winning such a lottery? Copyright 200 Pearson Education, Inc.

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