Ks OBJECTIVES a O EE оо М Me И НН НН - 1 Compute expected value. | Use expected value to solve Ao pues jue mG eR БЫ ББК EM EIN BN ES EE И Wm wd WE applied problems. : Use expected value to determine the average payoff or loss in a game of chance. | Compute expected value. "очен он к он но EEN EN EN WE ES ESE EOF MS DON SECTION 11.8 Expected Value 659 EI we mew mee eee ee A RC GN ME ES BEN SEN HE BEN НН БЫ Бонн НЫ Бы НЫ EGE BSN НЫ БН BOO НЮ ОМК DN WD DOR НМ МЫ BM ee ME Кан GN НН НН НН Em ee MN EC EE но Ня ED ER НН НЫ НН Ee Re MM Но вая mad ses me me Expected Value . Would you be willing to spend $50 a year for an insurance . policy that pays $200,000 if you become too ill to .¥ continue your education? It is unlikely that this will Insurance companies make money by compensating us for events that have a low © probability. If one in every 5000 … students needs to quit college due to “ serious illness, the probability of this event is =000- Multiplying the amount of the claim, $200,000, by its probability, в i, tells the insurance company what to expect to pay out on average for each policy: 1 Ea $200,000 X 5000 7 $40. Amount of Probability of the claim paying the claim Over the long run, the insurance company can expect to pay $40 for each policy it sells. By selling the policy for $50, the expected profit 1s $10 per policy. If 400,000 stu- dents choose to take out this insurance, the company. can expect to make 400,000 Xx $10, or $4,000,000. Expected value is a mathematical way to use probabilities to determine what to expect in various situations over the long run. Expected value is used to determine premiums on insurance policies, weigh the risks versus the benefits of alternatives in business ventures, and indicate to a player of any game of chance what will happen if the game is played repeatedly. The standard way to find expected value is to multiply each possible outcome by its probability, and then add these products. We use the letter £ to represent expected value. Computing Expected Value Find the expected value for the outcome of the roll of a fair die. Solution The outcomes are 1, 2, 3, 4, 5, and 6, each with a probability of Е The expected value, E, is computed by multiplying each outcome by its probability and then adding these products. 1 1 1 1 1 1 .— + .— *— + . — *— + . — des ar EE 6 1594344455 | - 6 Te 699 E The expected value of the roll of a fair die is 3.5. This means that if the die is rolled repeatedly, there is an average of 3.5 dots per roll over the long run. This expected value cannot occur on a single roll of the die. However, it is a long-run average of the various outcomes that can occur when a fair die 1s rolled. It is equally probable that a pointer will land on any one of four regions, numbered 1 through 4. Find the expected value for where the pointer will stop. 660 CHAPTER 11 Counting Methods and Probability Theory TABLE 11.8 Outcomes and Probabilities for the Number of Girls in a Three-Child Family Outcome: Number of Girls Probability 0 ol 1 o 2 cop cols 3 TABLE 11.9 Number of Heads | Probability fe a >) — E S| 5s [Sle [5s | 5 Use expected value to solve applied problems. TABLE 11.10 Probabilities for Auto Claims | Amouni of Claim (to the nearest $2000) Probability $0 0.70 $2000 0.15 $4000 0.08 $6000 0.05 $8000 0.01 $10,000 0.01 Computing Expected Value Find the expected value for the number of girls for a family with three children. Solution A family with three children can have 0, 1,2, or 3 girls. There are eight ways these outcomes can occur. One way No girls : Boy Boy Boy One girl: Girl Boy Boy, Boy Girl Boy, Boy Boy Girl ee Two girls : Girl Girl Boy, Girl Boy Girl, Boy Girl Girl — "tree ways Three girls : Girl Girl Girl Three ways One way Table 11.8 shows the probabilities for 0,1, 2, and 3 girls. The expected value, E, is computed by multiplying each outcome by its probability and then adding these products. 1 3 3 Ps 3 +643 12 3 E == 0: сев Те НЕ же Ве = — = — = 1.5 3 8 8 8 8 8 2 The expected value 1s 1.5. This means that if we record the number of girls in many different three-child families, the average number of girls for all these families will be 1.5. In a three-child family, half the children are expected to be girls, so the expected value of 1.5 1s consistent with this observation. A fair coin 1s tossed four times in succession. Table 11.9 shows the probabilities for the different number of heads that can arise. Find the expected value for the number of heads. Applications of Expected Value Empirical probabilities can be determined in many situations by examining what has occurred in the past. For example, an insurance company can tally various claim amounts over many years. If 15% of these amounts are for a $2000 claim, then the probability of this claim amount 1s 0.15. By studying sales of similar houses in a particular area, a realtor can determine the probability that he or she will sell a listed house, another agent will sell the house, or the listed house will remain unsold. Once probabilities have been assigned to all possible outcomes, expected value can indicate what 1s expected to happen in the long run. These ideas are illustrated in Examples 3 and 4. Determining an Insurance Premium An automobile msurance company has determined the probabilities for various claim amounts for drivers ages 16 through 21, shown in Table 11.10. a. Calculate the expected value and describe what this means in practical terms. b. How much should the company charge as an average premium so that it does not lose or gain money on its claim costs? Solution a. The expected value, FE, is computed by multiplying each outcome by its probability and then adding these products. E = $0(0.70) + $2000(0.15) + $4000(0.08) + $6000(0.05) + $8000(0.01) + $10,000(0.01) = $0 + $300 + $320 + $300 + $80 + $100 = $1100 The expected value is $1100. This means that in the long run the average cost of a claim is $1100. The insurance company should expect to pay $1100 per car insured to people in the 16-21 age group. THE REALTOR'S SUMMARY HEET My Cost: $5000 My Possible Income: | | I sell house: $30,000 Another agent sells house: $15,000 House unsold after 4 months: $0 The Probabilities: I sell house: 0.3 Another agent sells house: 0.2 House unsold after 4 months: | 0.5 My Bottom Line: I take the listing only 1f 1 anticipate earning at least $6000. SECTION 11.8 Expected Value 661 b. At the very least, the amount that the company should charge as an average premium for each person in the 16-21 age group is $1100. In this way, it will not lose or gain money on its claims costs. It’s quite probable that the company will charge more, moving from break-even to profit. Work Example 3 again if the probabilities for claims of $0 and $10,000 are “ reversed. Thus, the probability of a $0 claim is 0.01 and the probability of a $10,000 claim is 0.70. E SA A A WO AN A m БАШ Gr kl | PEN MAE ME Mm De tb! Ме Мед DE pe dar die a ma Em A TE We AE ee De WD MI Dm A ee SW Te A A DO RR nd EA meh ni ii Business decisions are interpreted in terms of dollars and cents. In these situations, expected value is calculated by multiplying the gain or loss for each possible outcome by its probability. The sum of these products is the expected value. Expectation in a Business Decision You are a realtor considering listing a $500,000 house. The cost of advertising and providing food for other realtors during open showings is anticipated to cost you $5000. The house is quite unusual and you are given a four-month listing. It the house is unsold after four months, you lose the listing and receive nothing. You anticipate that the probability you sell your own listed house is 0.3, the probability that another agent sells your listing is 0.2, and the probability that the house is unsold after 4 months is 0.5. If you sell your own listed house, the commission 15 а hefty $30,000. If another realtor sells your listing, the commission is $15,000. The bottom line: You will not take the listing unless you anticipate earning at least $6000. Should you list the house? Solution Shown in the margin is a summary of the amounts of money and probabilities that will determine your decision. The expected value in this situation is the sum of each income possibility times its probability. The expected value represents the amount you can anticipate earning if you take the listing. If the expected value is not at least $6000, you should not list the house. The possible incomes listed in the margin, $30,000, $15,000, and $0, do not take into account your $5000 costs. Because of these costs, each amount needs to be reduced by $5000. Thus, you can gain $30,000 — $5000, or $25,000, or you can gain $15,000 — $5000, or $10,000. Because $0 — $5000 = —$5000, you can also lose $5000. Table 11.11 summarizes possible outcomes if you take the listing, and their respective probabilities. TABLE 11.11 Gains, Losses, and Probabilities for Listing a $500,000 House Outcome | Gain or Loss Probability Sells house $25,000 0.3 Another agent sells house $10,000 0.2 House doesn't sell —$5000 0.5 The expected value, E, is computed by multiplying each gain or loss in Table 11.11 by its probability and then adding these results. Е = $25,000(0.3) + $10,000(0.2) + (—$5000)(0.5) = $7500 + $2000 + (—$2500) = $7000 You can expect to earn $7000 by listing the house. Because the expected value exceeds $6000, you should list the house. 662 CHAPTER 11 Counting Methods and Probability Theory Use expected value to determine the average payoff or loss in a game of chance. AWAY probabilities and payoffs that result in negative expected values. This means that players will lose money in the long run. With no clocks on the walls or windows to look out of, casinos bet that players will forget about time and play longer. This brings casino owners closer to what they can expect to earn, while negative expected values sneak up on losing patrons. (No wonder they provide free beverages!) A Potpourri of Gambling- Related Data * Population of Las Vegas: 535,000 * Slot Machines in Las Vegas: 150,000 * Average Earnings per Slot Machine per Year: $100,000 * Average Amount Lost per Hour in Las Vegas Casinos: $696,000 * Average Amount an American Adult Loses Gambling per Year: $350 * Average Amount a Nevada Resident Loses Gambling per Year: $1000 Source: Paul Grobman, Vital Statistics, Plume, 2005. get ONE OND NE NR ONE ONS MR oS RR osm em em O O вн Sm O он он нон но но он EE EE EN EE EN OO = me Re me кн mm mm ME EE ME о EE ME MEN ME ME EE RE Re fe es es ee Ee в новы о EN MN MN ME ME но кн оно ны кн ны вн вы вы ок pl O we O ши нож ни ME вн ош me кн ош к о оно оно о Rom wm mm em mE RE RE WE EE MS EE ee ен we mm mm Rm mm БЫ жи к WN NN и пы RE ни вн EN NR SS RR Bm Mm ши WN MN ME EN EN EN но кн вы кн на оны кн кв mm em ож ож мн жи mm mm mm mm mm жа Em mm Em ны Me Me EE Sm а шв ны кн кн жи вн mm mm mm mm mm mm mm Em Em Mm Em Em em A me em ee ое == == == = The SAT 1s a multiple-choice test. Each question has five possible answers. The test taker must select one answer for each question or not answer the question. One point is awarded for each correct response and + point is subtracted for each wrong answer. No points are added or subtracted for answers left blank. Table 11.12 summarizes the information for the outcomes of a random guess on an SAT question. Find the expected point value of a random guess. Is there anything to gain or lose on average by guessing? Explain your answer. TABLE 11.12 Gains and Losses for Guessing on the SAT Outcome Gain or Loss Probability Guess correctly 1 Е T 1 4 Guess incorrectly 4 = Expected Value and Games of Chance Expected value can be interpreted as the average payoff in a contest or game when either 1s played a large number of times. To find the expected value of a game, multiply the gain or loss for each possible outcome by its probability. Then add the products. A game is played using one die. If the die is rolled and shows 1, 2, or 3, the player wins nothing. If the die shows 4 or 5, the player wins $3. If the die shows 6, the player wins $9. If there is a charge of $1 to play the game, what is the game’s expected value? Describe what this means in practical terms. Solution Because there is a charge of $1 to play the game, a player who wins $9 gains $9 — $1, or $8. A player who wins $3 gains $3 — $1, or $2. If the player gets $0, there is a loss of $1 because $0 — $1 = —$1. The outcomes for the die, with their respective gains, losses, and probabilities, are summarized in Table 11.13. TABLE 11.13 Gains, Losses, and Probabilities in a Game of Chance Outcome Gain or Loss Probability 1,2,0r 3 —$1 я 4 or 5 $2 $ 6 $8 : Expected value, E, is computed by multiplying each gain or loss in Table 11.13 by its probability and then adding these results. Е = с50)(2) + (5) #65) — —$3 + $4 + $8 _ 59 _ = 6 = 5 = $1.50 The expected value is $1.50. This means that in the long run, a player can expect to win an average of $1.50 for each game played. However, this does not mean that the player will win $1.50 on any single game. It does mean that if the game is played repeatedly, then, in the long run, the player should expect to win about $1.50 per play on the average. If 1000 games are played, one could expect to win $1500. However, if only three games are played, one’s net winnings can range between —$3 and $24, even though the expected winnings are $1.50(3), or $4.50. I E i E | i i i I i i i i i 1 I I i I i 1 ! i I I 1 i 1 1 I I i i I E i E i i i i 1 1 I i i i i i 1 i 1 i 1 i i i I I i i i % FIGURE 11.14 A U.S. roulette wheel BLITZER BONUS HOW TO WIN AT ROULETTE A player may win or lose at roulette, but in the long run the casino always wins. Casinos make an average of three cents on every dollar spent by gamblers. There is a way to win at roulette in the long run: Own the casino. "a Ey e ME O O O e e e O O БЫ БЫ БЫ ББ EN O E O e e me ee OO O AS O EN O и нон НН НЫ Ee SO O Em SE EE DE O O mE = но = == == SECTION 11.8 Expected Value 663 A charity is holding a raffle and sells 1000 raffle tickets for $2 each. One of the tickets will be selected to win a grand prize of $1000. Two other tickets will be selected to win consolation prizes of $50 each. Fill in the missing column in Table 11.14. Then find the expected value if you buy one raffle ticket. Describe what this means in practical terms. What can you expect to happen if you purchase five tickets? TABLE 11.14 Gains, Losses, and Probabilities in a Raffle Outcome Gain or Loss Probability Win Grand Prize тот Win Consolation Prize 100 Win Nothing т а бы le А TE ни цы ей ны Ц AR RE ET EN ЦН СВР САБЫ ЧЕМ el De LL i ЕП НА Unlike the game in Example 5, games in gambling casinos are set up so that players will lose in the long run. These games have negative expected values. Such a game is roulette, French for “little wheel.” We first saw the roulette wheel in Section 11.7. It is shown again in Figure 11.14. Recall that the wheel has 38 numbered slots (1 through 36, 0, and 00). In each play of the game, the dealer spins the wheel and a small ball in opposite directions. The ball is equally likely to come to rest in any one of the slots, which are colored black, red, or green. Gamblers can place a number of different bets in roulette. Example 6 illustrates one gambling option. Expected Value and Roulette One way to bet in roulette is to place $1 on a single number. If the ball lands on that number, you are awarded $35 and get to keep the $1 that you paid to play the game. If the ball lands on any one of the other 37 slots, you are awarded nothing and the $1 that you bet is collected. Find the expected value for playing roulette if you bet $1 on number 20. Describe what this means. Solution Table 11.15 contains the two outcomes of interest: the ball landing on your number, 20, and the ball landing elsewhere (in any one of the other 37 slots). The outcomes, their respective gains, losses, and probabilities, are summarized in the table. TABLE 11.15 Playing One Number with a 35 to 1 Payoff in Roulette Outcome Gain or Loss Probability Ball lands on 20 $35 + Ball does not land on 20 —$1 E E Expected value, E, is computed by multiplying each gain or loss in Table 11.15 by its probability and then adding these results. — ea 1 6/37) $35 $37 _ 2 в = $5(55) + < (5) a AAA The expected value is approximately — $0.05. This means that in the long run, a player can expect to lose about 5¢ for each game played. If 2000 games are played, one could expect to lose $100. In the game of one-spot keno, a card is purchased for $1. It allows a player to choose one number from 1 to 80. A dealer then chooses twenty numbers at random. If the player’s number is among those chosen, the player is paid $3.20, but does not get to keep the $1 paid to play the game. Find the expected value of a $1 bet. Describe what this means. 664 CHAPTER 11 Counting Methods and Probability Theory Practice and Application Exercises 5. An architect is considering bidding for the design of a new | : museum. The cost of drawing plans and submitting a model In Exercises 1-2, the numbers that each pointer can land on and is $10,000. The probability of being awarded the bid is 0.1, their respective probabilities are shown. Compute the expected and anticipated profits are $100,000, resulting in a possible value for the number on which each pointer lands. gain of this amount minus the $10,000 cost for plans and a 1. Outcome | Probability model. What is the expected value 1 this situation? Describe ES ZA what this value means. 1 3 6. A construction company is planning to bid on a building 9 + contract. The bid costs the company $1500. The probability : _ that the bid is accepted is =. If the bid is accepted, the 3 4 company will make $40,000 minus the cost of the bid. Find the expected value in this situation. Describe what this value means. 7. It is estimated that there are 27 deaths for every 10 million people who use airplanes. A company that sells flight insurance provides $100,000 in case of death in a plane crash. у Outcome | Probability A policy can be purchased for $1. Calculate the expected value nn bt and thereby determine how much the insurance company can 1 = make over the long run for each policy that it sells. > ia + tes 8. A 25-year-old can purchase a one-year life insurance policy | — for $10,000 at a cost of $100. Past history indicates that the 3 ; probability of a person dying at age 25 is 0.002. Determine 4 1 TEA the company's expected gain per policy. Exercises 9-10 are related to the SAT, described in Check Point 4 on page 602. 9. Suppose that you can eliminate one of the possible five answers. Modify the two probabilities shown in the final The tables in Exercises 3—4 show claims and their probabilities for column in Table 11.12 on page 662 by finding the an insurance company. probabilities of guessing correctly and guessing incorrectly a. Calculate the expected value and describe what this means under these circumstances. What is the expected point value in practical terms. of a random guess? Is it advantageous to guess under these b. How much should the company charge as an average ~~ circumstances? premium so that it breaks even on its claim costs? 10. Suppose that you can eliminate two of the possible five c. How much should the company charge to make a profit of answers. Modify the two probabilities shown in the final $50 per policy? column in Table 11.12 on page 662 by finding the 3. PROBABILITIES FOR HOMEOWNERS' probabilities of guessing correctly and guessing incorrectly INSURANCE CLAIMS under these circumstances. What is the expected point value Amount of Claim (to the of a random guess? Is it advantageous to guess under these nearest $50,000) Probability circumstances? E 7 0.65 11. A store specializing in mountain bikes is to open in one of . two malls. If the first mall is selected, the store anticipates a $50,000 0.20 | : . ————— yearly profit of $300,000 if successful and a yearly loss of _ $100,000 010° $100,000 otherwise. The probability of success is +. If the $150,000 0.03 second mall 1s selected, it is estimated that the yearly profit $200,000 | … 001 | will be $200,000 if successful; otherwise, the annual loss will $250,000 EA 5 be $60,000. The probability of success at the second mall is 3. Which mall should be chosen in order to maximize the 4. PROBABILITIES FOR MEDICAL INSURANCE expected profit? CLAIMS 12. An oil company is considering two sites on which to drill, Amount of Claim (to the described as follows: teca | aah Site A: Profit if oil is found: $80 million $0 0.70 | Loss if no oil is found: $10 million $20,000 0.20 Probability of finding oil: 0.2 PO à PE Site B: Profit if oil is found: $120 million anne a $60,000 0.02 Loss if no oil is found: $18 million $80,000 0.01 Probability of finding oil: 0.1 pL00.000 Sle) Which site has the larger expected profit? By how much? 13. 14. In a product liability case, a company can settle out of court for a loss of $350,000, or go to trial, losing $700,000 if found guilty and nothing if found not guilty. Lawyers for the company estimate the probability of a not-guilty verdict to be 0.8. a. Find the expected value of the amount the company can lose by taking the case to court. b. Should the company settle out of court? A service that repairs air conditioners sells maintenance agreements for $80 a year. The average cost for repairing an air conditioner is $350 and 1 in every 100 people who purchase maintenance agreements have air conditioners that require repair. Find the service’s expected profit per maintenance agreement. Exercises 15-19 involve computing expected values in games of chance. 15. 16. 17. 18. 19. Yr vy 20. 21. 22. a a WF PS BE в ое п еб Fo ER РО Ee ¡Line in Mar ET - E 3 A game is played using one die. If the die is rolled and shows 1, the player wins $5. If the die shows any number other than 1, the player wins nothing. If there is a charge of $1 to play the game, what is the game’s expected value? What does this value mean? A game is played using one die. If the die is rolled and shows 1, the player wins $1; if 2, the player wins $2; if 3, the player wins $3. If the die shows 4, 5, or 6, the player wins nothing. If there is a charge of $1.25 to play the game, what is the game’s expected value? What does this value mean? Another option in a roulette game (see Example 6 on page 663) is to bet $1 on red. (There are 18 red compartments, 18 black compartments, and 2 compartments that are neither red nor black.) If the ball lands on red, you get to keep the $1 that you paid to play the game and you are awarded $1. If the ball lands elsewhere, you are awarded nothing and the $1 that you bet is collected. Find the expected value for playing roulette if you bet $1 on red. Describe what this number means. The spinner on a wheel of fortune can land with an equal chance on any one of ten regions. Three regions are red, four are blue, two are yellow, and one is green. A player wins $4 if the spinner stops on red and $2 if it stops on green. The player loses $2 if it stops on blue and $3 if it stops on yellow. What is the expected value? What does this mean if the game is played ten times? For many years, organized crime ran a numbers game that is now run legally by many state governments. The player selects a three-digit number from 000 to 999. There are 1000 such numbers. A bet of $1 is placed on a number, say number 115. If the number is selected, the player wins $500. If any other number is selected, the player wins nothing. Find the expected value for this game and describe what this means. i E a : д 2 BT A, Y “e y e pe вела 5 вы в i What does the expected value for the outcome of the roll of a fair die represent? Explain how to find the expected value for the number of girls for a family with two children. What is the expected value? How do insurance companies use expected value to determine what to charge for a policy? SECTION 11.8 Expected Value 665 23. Describe a situation in which a business can use expected value. 24. If the expected value of a game is negative, what does this mean? Also describe the meaning of a positive and a zero expected value. 25. The expected value for purchasing a ticket in a raffle is —$0.75. Describe what this means. Will a person who purchases a ticket lose $0.75? Critical Thinking Exercises Make Sense? In Exercises 26-29, determine whether each statement makes sense or does not make sense, and explain your reasoning. 26. Ifound the expected value for the number of boys for a family with five children to be 2.5.1 must have made an error because a family with 2.5 boys cannot occur. 27. Here’s my dilemma: I can accept a $1000 bill or play a dice game ten times. For each roll of the single die, ® I win $500 for rolling 1 or 2. * J win $200 for rolling 3. e lose $300 for rolling 4, 5, or 6. Based on expected value, I should accept the $1000 bill. 28. Гуе 1051 а fortune playing roulette, so I'm bound to reduce my losses if I play the game a little longer. 29. My expected value in a state lottery game is $7.50. 30. A popular state lottery is the 5/35 lottery, played in Arizona, Connecticut, Illinois, Iowa, Kentucky, Maine, Massachusetts, New Hampshire, South Dakota, and Vermont. In Arizona's version of the game, prizes are set: First prize is $50,000, second prize is $500, and third prize is $5. To win first prize, you must select all five of the winning numbers, numbered from 1 to 35. Second prize is awarded to players who select any four of the five winning numbers, and third prize is — awarded to players who select any three of the winning numbers. The cost to purchase a lottery ticket is $1. Find the expected value of Arizona's “Fantasy Five” game, and describe what this means in terms of buying a lottery ticket over the long run. 31. Refer to the probabilities of dying at any given age on page 630 to solve this exercise. A 20-year-old woman wants to purchase a $200,000 one-year life insurance policy. What should the insurance company charge the woman for the policy if it wants an expected profit of $60? Group Exercise 32. This activity is a group research project intended for people interested in games of chance at casinos. The research should culminate in a seminar on games of chance and their expected values. The seminar is intended to last about 30 minutes and should result in an interesting and informative presentation made to the entire class. Each member of the group should research a game avail- able at a typical casino. Describe the game to the class and compute its expected value. After each member has done this, so that class members now have an idea of those games with the greatest and smallest house advantages, a final group member might want to research and present ways for currently treating people whose addiction to these games has caused their lives to swirl out of control.