Copyright © 2011 Pearson Education, Inc. Probability: Living with the Odds Discussion Paragraph 7B 1 web 59. Lottery Chances 60. HIV Probabilities 1 world 61. Combined Probability in the News 62. Combined Probability in Your Life

Copyright © 2011 Pearson Education, Inc. Slide 7-3 Unit 7C The Law of Large Numbers

7-C Copyright © 2011 Pearson Education, Inc. Slide 7-4 The Law of Large Numbers The law of large numbers applies to a process for which the probability of an event A is P(A) and the results of repeated trials are independent. If the process is repeated over many trials, the proportion of the trials in which event A occurs will be close to the probability P(A). The larger the number of trials, the closer the proportion should be to P(A).

7-C Copyright © 2011 Pearson Education, Inc. Slide 7-5 The Law of Large Numbers Illustrate the law of large numbers with a die rolling experiment. The following figure shows the results of a computer simulation of rolling a die.

7-C Roulette CN (1a-b) A roulette wheel has 38 numbers: 18 are black, 18 are red, and the numbers 0 and 00 are green. a. What is the probability of getting a red number on any spin? b. If patrons in a casino spin the wheel 100,000 times, how many times would you expect a red number? Copyright © 2011 Pearson Education, Inc. Slide 7-6

7-C Copyright © 2011 Pearson Education, Inc. Slide 7-7 Consider two events, each with its own value and probability. Expected Value = (event 1 value) (event 1 probability) + (event 2 value) (event 2 probability) The formula can be extended to any number of events by including more terms in the sum. Expected Value

7-C Copyright © 2011 Pearson Education, Inc. Slide 7-8 Expected Value Example: Suppose an automobile insurance company sells an insurance policy with an annual premium of $200. Based on data from past claims, the company has calculated the following probabilities: An average of 1 in 50 policyholders will file a claim of $2,000. An average of 1 in 20 policyholders will file a claim of $1,000. An average of 1 in 10 policyholders will file a claim of $500. Assuming that the policyholder could file any of the claims above, what is the expected value to the company for each policy sold?

7-C Copyright © 2011 Pearson Education, Inc. Slide 7-9 Expected Value Let the $200 premium be positive (income) with a probability of 1 since there will be no policy without receipt of the premium. The insurance claims will be negative (expenses). The expected value is (This suggests that if the company sells many policies, then the return per policy, on average, is $60.)

7-C Lottery Expectations CN (2) Suppose that $1 lottery tickets have the following probabilities: 1 in 5 to win a free ticket (worth $1), 1 in 100 to win $5, 1 in 100,000 to win $1,000, and 1 in 10 million to win $1 million. 2. What is the expected value of a lottery ticket? Discuss the implications. (Note: Winners do not get back the $1 they spend on the ticket.) Copyright © 2011 Pearson Education, Inc. Slide 7-10

7-C Art Auction CN (3) You are at an art auction, trying to decide whether to bid $50,000 for a particular painting. You believe that you have a ½ probability of reselling the painting to a client in New York for $70K and ¼ probability of reselling the painting to a client in San Francisco for $80K. Otherwise you will have to keep the painting—an option that has benefits, but not monetary ones. 3. What is the expected value of a $50,000 bid? Copyright © 2011 Pearson Education, Inc. Slide 7-11

7-C Copyright © 2011 Pearson Education, Inc. Slide 7-12 The gambler's fallacy is the mistaken belief that a streak of bad luck makes a person "due" for a streak of good luck. The Gambler’s Fallacy

7-C Continued Losses CN (4) Suppose you are playing the coin toss game, in which you win $1 for heads and lose $1 for tails. After 100 tosses you are $10 in the hole because you flip perhaps 45 heads and 55 tails. The empirical probability is 0.45 for heads. So you keep playing the game. With 1,000 tosses, perhaps you get 490 heads and 510 tails. The empirical ratio is now 0.49, but you are $20 in the hole at this point. Because you know the law of averages helps us to know that the eventual theoretical probability is 0.50, you decide to play the game just a little more. 4.a. Does the result agree with what we expect from the law of large numbers? b. Have you gained back any of your losses? Explain Slide 7-13

7-C Copyright © 2011 Pearson Education, Inc. Slide 7-14 Another common misunderstanding that contributes to the gambler’s fallacy involved expectations about streaks. Streaks For tossing a coin 6 times, the outcomes HTTHTH and HHHHHH (all heads) are equally likely. The Gambler’s Fallacy Streaks

7-C Hot hand at the Craps Table CN (5a-b) 5. The popular casino game of craps involves rolling dice. Suppose you are playing craps and suddenly find yourself with a “hot hand”: You roll a winner on ten consecutive bets. a. Is your hand really “Hot”? b. Should you increase your bet because you are on a hot streak? Assume you are making bets with a.486 probability of winning on a single play (the best odds available in craps). Copyright © 2011 Pearson Education, Inc. Slide 7-15

7-C The Gambler’s Fallacy The House Edge A casino makes money because games are set up so that the expected earnings of patrons are negative (that is losses). Because the casino earns whatever patrons lose, the casino’s earnings are positive. The House Edge The expected value to the casino of a particular bet Copyright © 2011 Pearson Education, Inc. Slide 7-16

7-C The House Edge in Roulette CN (6a-b) 6. The game of roulette is usually set up so that betting on red is a 1 to 1 bet. That is, you win the same amount of money as you bet if red comes up. Betting on a single number is a 35 to 1 bet. That is, you win 35 times as much as you bet if your number comes up. a. What is the house edge in each of these two cases? b. If patrons wager $1 million on such bets, how much should the casino expect to earn? Copyright © 2011 Pearson Education, Inc. Slide 7-17

7-C Quick Quiz CN (7) 7. Please answer the 10 quick quiz multiple choice questions on p Copyright © 2011 Pearson Education, Inc. Slide 7-18

7-C Homework 7C Discussion Paragraph 7B Class Notes 1-7 p. 445: web 39. Lotteries 40. The Morality of Gambling 1 world 41. Law of Large Numbers 42. Personal Law of Large Numbers 43. The Gambler’s Fallacy Copyright © 2011 Pearson Education, Inc. Slide 7-19