1. Lecture 5. Basic Probability David R. Merrell 90-786 Intermediate Empirical Methods For Public Policy and Management

2. I think that the team that wins game five will win the series...Unless we lose game five. -- Charles Barkley

3. Regularity: Empirical Rule contains 68% of data contains 95% of data contains 99.9% of data

4. How to Verify? Try Monte Carlo simulations Easy to use Minitab Let’s do that!

5. Terminology Probability trial: a process giving observations with uncertain values Repeated probability trials: independently repeated under the same conditions Outcome: a most basic happening Event: set of outcomes

6. Assignment of Probabilities 1. Symmetry--Classical 2. Relative Frequency 3. Betting Odds--Subjective

7. Classical Approach Elementary outcomes are equally likely Probability is defined to be the proportion of times that an event can theoretically be expected to occur Used in standard games of chance We can determine the probability of an event occurring without any experiments or trials ever taking place

8. Example 1 - Rolling a die Experiment: Roll a die Sample space: S = {1, 2, 3, 4, 5, 6} Number of possible outcomes: 6 P(4) = 1/6 P(even) = 3/6 P(number < 3) = 2/6

9. Example 2 - Flipping a coin Experiment: Flip 2 coins Sample space: S = {HH, TH, HT, TT} Number of possible outcomes: 4 P(both heads) = 1/4 P(at least one tail) = 3/4

10. Example 3 - Drawing a card Experiment: Draw a card from a deck of 52 Number of possible outcomes: 52 P(ace) = 4/52 P(diamond) = 13/52 P(red and ace) = 2/52

11. Relative Frequency Approach Used when classical approach is not applicable and repeated probability trials are possible Probability is the proportion of times an event is observed to occur in a large number of trials

12. Example 4--Relative Frequencies In 1985, 22.9% of whites were below the poverty level In 1977, the percent urban in Iraq was 64. In 1984, the divorce rate in Maine was 3.6 per 1000 population. (Problems here!)

13. Law of Large Numbers

14. “Law of Small Numbers” Toss a coin 1000 times and it will show up heads 500 times???

15. “Law of Averages” “I’ve lost money every time I bought a stock...I’m due!”

16. Subjective Approach Used when repeated probability trials are not feasible. Probability is subjective--an educated guess, a personal assessment

17. Well-Calibrated Probability Forecaster Link subjective probability to repeated probability trials P(MSFT goes up tomorrow) =.55 Does it go up 55% of the time?

18. Example 5--Subjective Probability What is the probability that the Pittsburgh Steelers will win next week? What is the probability that Al Gore will be elected president in the year 2000?

19. Odds vs. Probabilities Odds are a restatement of probability If the probability that an event will occur is 3/5, then the odds in favor of the event occurring are 3:2 Odds against an event occurring are the reverse of odds in favor of occurring. In this case 2:3. To calculate the probability, given the odds 1:3 1 1 + 3 4 probability is 1/4

20. Odds Odds of a:b in favor of an event A Bet in Favor Bet Against A Occurs A Does Not b-b -aa

21. Probability Notation P(A) - probability that event A occurs P(A’) - probability that event A will not occur (A’ is the complement of A) P(A  B) - probability that A will occur or B will occur or both (Union of A and B) P(A  B) - probability that A and B will occur simultaneously (Joint probability of A and B) P(A | B) - probability of A, given that B is known to have occurred. (Conditional probability)

22. Probability Axioms 1. P(A) > 0 2. P(S) = 1 3. A i mutually exclusive,

23. Addition Law for Probability P(A or B) = P(A) + P(B) - P(A and B) Example: A left engine functions B right engine functions

24. “Proof by Paint” A B 110 “paint and scrape” A B 1 2 1 2 1

25. If Mutually Exclusive... P(A or B) = P(A) + P(B) Note simplification of Addition Rule

26. If Independent... P(A and B) = P(A)P(B) Note simplification of Multiplication Rule

27. Some Connections... LogicSet Arithmetic Simplification and x independence or + mutually exclusive Note: independence is NOT mutual exclusivity

28. Multiplication Law for Probability P(A and B) = P(A B) = P(A)P(B|A) = P(A|B)P(B) Example Sell cocaine and go to jail A B

29. Example 6--Probability Calculations P(adult male is a Democrat) = 0.6, P(belongs to a labor union) = 0.5 P(Democrat and labor union) = 0.35, Find the probability that an adult male chosen at random: is a Democrat or belongs to a labor union does not belong to a labor union is a Democrat given that he belongs to a labor union

30. Conditional Probability Events A, B P(A and B) = P(B |A)P(A) = P(A|B)P(B) Definition:

31. Example 7--Conditional Probability

32. Contingency Table Help determine probabilities when we have two variables Joint and conditional probabilities are in the cells Marginal probabilities are on the “margins” of the table

33. Educational Achievement: Coding of Ordinal Variable 1 if grade 4 or less 2 if grades 5-7 3 if grade 8 4 if high school incomplete (9-11) 5 if high school graduate (12) 6 if technical, trade, or business after high school 7 if college/ university incomplete 8 if college/university graduate or more

34. Educational Achievement Table

35. Count--Absolute Frequency

36. Joint Probability

37. Marginal Probability

38. P(Ed =4|F) P(F|Ed=4) Conditional Probabilities:

39. Marginal Probability Conditional Probabilities Joint Probability Absolute Frequencies

40. Example 8--More Probability Calculations Find the probability that the individual: is a high school graduate is female is male or has incomplete high school is female and did not complete college graduated from college given that he is a male is male given that he graduated from college

41. Next Time... Bayes Rule Total Probability Rule Applications