POSC 202A: Lecture 5

1. POSC 202A: Lecture 5 Today: Expected Value

2. Expected Value- Is the mean outcome of a probability distribution. It is our long run expectation of the expected return of some (social) process. Expected Value

3. The Law of Large Numbers- If a random phenomenon with numerical outcomes is repeated many times independently, the mean of the actually observed outcomes approaches the expected value. Expected Value

4. To calculate, we need to know: • The benefit from something occurring (B). • The probability the benefit occurs (P) • The cost (benefit) of something not happening (Bc). • The probability this cost does not occur (1-P) • Expected Value= (B*P)+ Bc*(1-P) Expected Value

5. Expected Value= (B*P)+ Bc*(1-P) In the overwhelming majority of cases Bc=0. So, EV reduces to B*P Expected Value

6. Expected Value- A random phenomenon that has multiple outcomes is found by multiplying each outcome by its probability and adding all of the products. Expected Value

7. Expected Value= (B*P)+ Bc*(1-P) Here we use B to be the net benefit. In the overwhelming majority of cases Bc=0. Think of this as the return (profit+investment). So, EV reduces to B*P Expected Value

8. A roulette wheel has 38 slots, numbered 0,00, and 1-36. 18 are red, 18 are black, and 2 are green. The wheel is balanced so that the ball is equally likely to land on any slot. Expected Value: Roulette

9. Three main bets: One number: win if the number comes up. One column (or dozen): win if any in the column comes up. One color: win if the color comes up. Expected Value: Roulette

10. The key probabilities are: One number: 1/38 One column (or dozen): 12/38 Black or Red: 18/38 Expected Value: Roulette

11. The key bets are: One number: returns $36 (win $35) One column (or dozen): returns $3 (Win $2) Black or Red: returns $2 (win $1) Expected Value: Roulette

12. What are the expected values? (Recall, B*P) One number: One column: One color: Expected Value: Roulette

13. What are the expected values? Recall, = (B*P)+ Bc*(1-P) One number: (1/38 * $35)+(37/38*-$1)= (35/38)-(37/38) = -.052 One column: (12/38* $2)+(26/38*$-1)= (24/38)-(26/38) = -.052 One color: (18/38* $1)+(20/38*$-1)= (18/38)-(20/38) = -.052 What does this mean? Which gives us the best chance of winning money? Expected Value: Roulette

14. Shortcut method: return for each $1 bet. Recall, B*P One number: 1/38 * $36=.947 One column: 12/38* $3= .947 One color: 18/38* $2= .947 What does this mean? Which gives us the best chance of winning money? Expected Value: Roulette

15. Which gives us the best chance of winning money? To answer this question we can use what we learned about the normal curve to solve for the areas. How would we do this? Expected Value: Roulette

16. How would we do this? Convert each bet type to standard units and solve for the area that corresponds to a profit. Expected Value: Roulette

17. How would we do this? Conceptually, draw and label our curve: Expected Value: Roulette $0 -.052

18. How would we do this? Next put into Standard Units. Recall Or 1-.947 S.D. Clearly, we need to find the SD. Expected Value: Roulette

19. Clearly, we need to find the SD. We can use the SD formula from last week. Expected Value: Roulette But, how do we find observations on which to calculate it?

20. The areas (probabilities) from the Z table, differ on each bet: Expected Value: Roulette

21. Recall that underlying distributions converge around the sample mean as the number of trials increase.

22. REMEMBER Expected Value- Is the AVERAGE outcome of a probability distribution. It is our long run expectation of the expected return of some (social) process. The outcome in any particular trial, instance, or case, will vary. Expected Value