1. Please turn off cell phones, pagers, etc. The lecture will begin shortly.

2. Exam 3 preliminary results None of the 33 items was clearly ineffective. The exam was more difficult than anticipated. 4 items were rated “difficult” 28 were “moderately difficult” 1 was “easy” An adjustment will be made to the scores.

3. Lecture 29 Today’s lecture will cover material related to Sections 16.6 and 18.5 1.Review expected value (Section 16.6) 2.Expectation and personal decisions (Section 18.5)

4. 1. Review of Expected Value A probability distribution is a list of all possible outcomes, and the probabilities of those outcomes Probability distribution Example A couple has two children. Find the probability distribution for the number of boys. # of boys = 0 with probability ¼ (GG) # of boys = 1 with probability ½ (GB or BG) # of boys = 2 with probability ¼ (BB)

5. If you know the probability distribution, you can find the expected value (or “expectation”): Expected value multiply each outcome by its probability, and add them up Example A couple has two children. Find the expected number of boys. 0 × ¼ = 0 1 × ½ = ½ 2 × ¼ = ½ The expected value is 0 + ½ + ½ = 1.

6. Interpretation of Expected Value The expected value is the average value of the outcome over the long-run, if the experiment were repeated many times. For the last example, suppose that took a very large, random sample of families with two children. And suppose that we recorded the number of boys in each of these families: 0, 2, 1, 0, 1, 1, 2, 1, 1, 0, 0, 2, 1, … The average of all of these numbers would be the expected value, which in this case is 1.

7. Another example Suppose you roll a single die. What is the expected value? 1 2 outcome 3 4 5 6 1/6 prob. 1/6 2/6 3/6 4/6 5/6 6/6 outcome × prob. Expected value = 1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 6/6 = 21/6 = 3.5 Note that this is not one of the possible outcomes. It’s the average value of the outcomes if the experiment is repeated many times.

8. Weird example How much would you pay to play this game? “St. Petersburg paradox” “I will flip this coin until it comes up ‘heads.’ If I get heads on the first try, I will give you $2. If it takes two tries, I will give you $4. If it takes three tries, I will give you $8, and so on.” (For each additional try, the prize is doubled.) $5 $10 $25 $50 A gentleman from Russia approaches you and says:

9. List the possible outcomes and their probabilities: 1/2 1/4 Outcome ($)prob. 1/8 1/16 1/32 1 1 1 1 1 …and so on outcome × prob. 2H 4HH 8HHH 16HHHH 32HHHHH Expected value = 1 + 1 + 1 + 1 + 1 + 1 … Your expected winnings are infinite. So why were you unwilling to pay lots of money to play this game? = ∞

10. 2. Expectation and personal decisions It makes sense to base financial decisions purely on expectation or expected value if the experiment is truly going to be repeated many times. For example, your average gains (or losses) will be approximately equal to the expected value if you play the lottery every week sell insurance to many customers are a professional gambler operate a gambling casino

11. Individuals who are making one-time decisions typically do not make those decisions purely on the basis of expected values. Personal decisions When making personal decisions, people look at both the outcome and the probability, not just the outcome × probability. For example, suppose you could buy a raffle ticket for $10 that would give you a 1 in 3,000 chance to win a new car worth $25,000. Would you buy it?

12. Your expected winnings in this raffle are $ 25,000 × 1/3000 = $ 8.33 From a standpoint of expectation, it does not make sense to buy the raffle ticket, because $8.33 is less than $10. But many people would buy the raffle ticket. Many are willing to trade a small amount of money for a small chance to win a large amount of money. It’s because losing a small amount of money makes little difference in their lives, whereas winning a large amount would make a substantial difference.

13. Another example Suppose I offer you two choices: a check for $25,000 or a 1 in 40 chance to win $1,000,000 Unless you are already quite wealthy, you would probably take the check for $25,000. In each case, however, the expected value of the prize is the same: $ 25,000 × 1 = $ 25,000 $ 1,000,000 × 1/40 = $ 25,000 If individuals made personal decisions based on expected values, they would be indifferent to these two choices. But they are not indifferent; most would take the check.

14. Health insurance Suppose there is a 1 in 50 or 2% chance that you will have a serious illness or injury this year requiring a hospital stay and medical care that costs $50,000 or more. Would you pay $100 per month for a medical plan that would cover the cost of medical care in those extreme circumstances? Your expected cost for the year if you buy insurance is $ 1,200 × 1 = $ 1,200 Your expected cost for the year if you don’t buy insurance is $ 50,000 × 1/50 = $ 1,000 Health insurance is always more expensive than the expected medical bill. But many people would buy it anyway, because it gives them peace of mind.

15. When expected values matter Who would be willing to accept $1,200 and then agree to pay all your hospital bills next year, if you should have a serious illness? An insurance company. The company has a large number of customers. These customers function as repeated experiments. The average hospital bill that they will have to pay per customer will be close to the expected value. So it is profitable for them to assume the risk for large numbers of people, if the expected value of their customers’ claims is less than the selling price of their coverage.

16. Why do people gamble? Gambling is a losing proposition for the gambler. In pure games of chance (slot machines, lotteries, roulette, etc.) the expected values always favor the house. So why do people gamble? A decision to gamble can be rational if the gambler derives some non-monetary benefit (e.g. fun or excitement), and the gambler does not gamble too often, given his or her financial situation

17. Problem gambling If a person gambles too often, the law of expected values inevitably takes over, causing the gambler to steadily lose money at a greater rate than he can afford. The problem gambler makes a small decision to gamble “just this once.” But he does it over and over. Each decision could be rational if it were an isolated event. But taken together, these individual decisions can bring disastrous loss. A problem gambler fails to place these momentary decisions into the broader context of life.