Transcript 7C PPT
Casinos don’t gamble. Chapter 7C The Law of Large Numbers Expected Value Gambler’s Fallacy 7-A When tossing a fair coin, which of the following events is most likely? A. Getting 8 tails in 10 tosses B. Getting 80 tails in 100 tosses C. Getting 800 tails in 1000 tosses The Law of Large Numbers 7-A 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 through 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). The Law of Large Numbers This figure shows the results of a computer simulation of rolling a die. 7-A The Law of Large Numbers A roulette wheel has 37 numbers: 18 are black, 18 are red, and one green. 1. What is the probability of getting a red number on any spin? 2. If patrons in a casino spin the wheel 100,000 times, how many times should you expect a red number? The Law of Large Numbers 7-A In a large casino, the house wins on its blackjack games with a probability of 50.7%. Which of the following events is the most likely? Most unlikely? A. You win at a single game. B. You come out ahead after playing forty times. C. You win at a single game given that you have just won ten games in a row. D. You win at a single game given that you have just lost ten games in a row. 7-A Expected Value Expected Value 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. 7-A Expected Value You have been asked to play a dice game. You will roll one die and: If you roll a 1, 2, 3 or 4, you will lose $30. If you roll a 5, you will win $48 If you roll a 6, you will win $60 What is the expected value (to you) of this game? 7-A Expected Value Consider the following game. The game costs $1 to play and the payoffs are $5 for red, $3 for blue, $2 for yellow, and nothing for white. The following probabilities apply. What are your expected winnings? Does the game favor the player or the owner? Outcome Red Blue Yellow White Probability 0.2 0.4 0.16 0.78 7-A Expected Value 7-A You are given 5 to 1 odds against tossing three heads with three coins. This means that you win $5 if you succeed and you lose $1 if you fail. Find the expected value (to you) of the game. NOTE: A game is said to be "fair" if the expected value for winnings is 0, that is, in the long run, the player can expect to win 0. Expected Value 7-A 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? 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 negatives (expenses). The expected value is (This suggests that if the company sold many, many policies, on average, the return per policy is a positive $60.) 7-A Expected Value 7-A An insurance company knows that the average cost to build a home in a new subdivision is $100,000, and that in any particular year, there is a 1 in 50 chance of a wildfire destroying all homes in the subdivision. Based on these data, and assuming the insurance company wants a positive expected value when it sells policies, what is the minimum that the company must charge for fire insurance policies in this subdivision? Expected Value Textbook problems Orange: pg448 #35, 38 Green: pg489 #45, 49 7-A Homework Chapter 7C Orange: Pg 445 Quick Quiz: 1, 2, 4, 5, 6 Exercises: 1, 2, 9, 15, 19, 31, 35, 38 Green: Pg 485 1, 2, 4, 5, 6, 11, 12, 19, 25, 29, 41, 45, 49 In a family of six children, which sequence do you think is more likely? a. MFFMFM b. MMMMMM c. Both are equally likely