1. Probability Theory

2. Topics Basic Probability Concepts: Sample Spaces and Events, Simple Probability, and Joint Probability, Conditional Probability Bayes’ Theorem Probability Distribution

3. Sample Spaces Collection of all Possible Outcomes e.g. All 6 faces of a die: e.g. All 52 cards of a bridge deck:

4. Events Simple Event: Outcome from a Sample Space with 1 Characteristic Red Card e.g. A Red Card from a deck of cards. Joint Event: Involves 2 Outcomes Simultaneously AceRed Card e.g. An Ace which is also a Red Card from a deck of cards.

5. Visualizing Events Contingency Tables Ace Not Ace Total Red 2 24 26 Black 2 24 26 Total 4 48 52

6. Simple Events The Event of a Happy Face 5 There are 5 happy faces in this collection of 18 objects

7. Joint Events AND The Event of a Happy Face AND Light Colored 3 Happy Faces which are light in color

8.   Special Events Null event Club & diamond on 1 card draw Complement of event For event A, All events not In A: Null Event

9. 3 Items: 3 Happy Faces Given they are Light Colored Dependent or Independent Events The Event of a Happy Face GIVEN it is Light Colored E = Happy Face  Light Color

10. Contingency Table A Deck of 52 Cards Ace Not an Ace Total Red Black Total 224 2 26 44852 Sample Space Red Ace

11. 2500 Contingency Table 2500 Employees of Company ABC Agree Neutral Opposed | Total MALE FEMALE Total 900 200 300 100 400 | 1500 600 | 1000 1200300 1000 | Sample Space

12. Tree Diagram Event Possibilities Red Cards Black Cards Ace Not an Ace Ace Not an Ace Full Deck of Cards

13. Probability Probability is the numerical measure of the likelihood that the event will occur. Value is between 0 and 1. Sum of the probabilities of all mutually exclusive and collective exhaustive events is 1. Certain Impossible.5 1 0

14. Computing Probability The Probability of an Event, E: Each of the Outcome in the Sample Space equally likely to occur. e.g. P ( ) = 2/36 (There are 2 ways to get one 6 and the other 4) P(E) = Number of Event Outcomes Total Number of Possible Outcomes in the Sample Space = X T

15. Computing Joint Probability The Probability of a Joint Event, A and B: e.g. P(Red Card and Ace) P(A and B) Number of Event Outcomes from both A and B Total Number of Possible Outcomes in Sample Space = =

16. P(A 2 and B 1 ) P(A 1 and B 1 ) Event Total 1 Joint Probability Using Contingency Table Joint Probability Marginal (Simple) Probability P(A 1 )A1A1 A2A2 B1B1 B2B2 P(B 1 ) P(B 2 ) P(A 1 and B 2 ) P(A 2 and B 2 ) P(A 2 )

17. Computing Compound Probability The Probability of a Compound Event, A or B: e.g. P(Red Card or Ace)

18. 2500 Contingency Table 2500 Employees of Company ABC Agree Neutral Opposed | Total MALE FEMALE Total 900 200 300 100 400 | 1500 600 | 1000 1200300 1000 | Sample Space

19. The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion Calculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposal

20. The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion Calculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposal 600/2500 = 0.24

21. The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion Calculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposal 600/2500 = 0.24 2. Neutral

22. The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion Calculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposal 600/2500 = 0.24 2. Neutral 300/2500 = 0.12

23. The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion Calculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposal 600/2500 = 0.24 2. Neutral 300/2500 = 0.12 3. Opposed to the proposal, GIVEN that the employee selected is a female

24. The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion Calculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposal 600/2500 = 0.24 2. Neutral 300/2500 = 0.12 3. Opposed to the proposal, GIVEN that the employee selected is a female 600/1000 = 0.60

25. The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion Calculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposal 600/2500 = 0.24 2. Neutral 300/2500 = 0.12 3. Opposed to the proposal, GIVEN that the employee selected is a female 600/1000 = 0.60 4. Either a female or opposed to the proposal

26. The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion Calculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposal 600/2500 = 0.24 2. Neutral 300/2500 = 0.12 3. Opposed to the proposal, GIVEN that the employee selected is a female 600/1000 = 0.60 4. Either a female or opposed to the proposal ……….. 1000/2500 + 1000/2500 - 600/2500 = 1400/2500 = 0.56

27. The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion Calculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposal 600/2500 = 0.24 2. Neutral 300/2500 = 0.12 3. Opposed to the proposal, GIVEN that the employee selected is a female 600/1000 = 0.60 4. Either a female or opposed to the proposal ……….. 1000/2500 + 1000/2500 - 600/2500 = 1400/2500 = 0.56 5. Are Gender and Opinion (statistically) independent?

28. The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion Calculate the probability that an employee selected (at random) from this group will be: 1. A female opposed to the proposal 600/2500 = 0.24 2. Neutral 300/2500 = 0.12 3. Opposed to the proposal, GIVEN that the employee selected is a female 600/1000 = 0.60 4. Either a female or opposed to the proposal ……….. 1000/2500 + 1000/2500 - 600/2500 = 1400/2500 = 0.56 5. Are Gender and Opinion (statistically) independent? For Opinion and Gender to be independent, the joint probability of each pair of A events (GENDER) and B events (OPINION) should equal the product of the respective unconditional probabilities….clearly this does not hold…..check, e.g., the prob. Of MALE and IN FAVOR against the prob. of MALE times the prob. of IN FAVOR …they are not equal….900/2500 does not equal 1500/2500 * 1200/2500

29. P(A 1 and B 1 ) P(B 2 )P(B1)P(B1) P(A 2 and B 2 ) P(A 2 and B 1 ) Event Total 1 Compound Probability Addition Rule P(A 1 and B 2 )P(A 1 )A1A1 A2A2 B1B1 B2B2 P(A 2 ) P(A 1 or B 1 ) = P(A 1 ) +P(B 1 ) - P(A 1 and B 1 ) For Mutually Exclusive Events: P(A or B) = P(A) + P(B)

30. Computing Conditional Probability The Probability of Event A given that Event B has occurred: P(A  B) = e.g. P(Red Card given that it is an Ace) =

31. Black Color Type Red Total Ace 224 Non-Ace 24 48 Total 26 52 Conditional Probability Using Contingency Table Conditional Event: Draw 1 Card. Note Kind & Color Revised Sample Space

32. Conditional Probability and Statistical Independence Conditional Probability: P(A  B) = P(A and B) = P(A  B) P(B) Multiplication Rule: = P(B  ) P(A)

33. Conditional Probability and Statistical Independence (continued) Events are Independent: P(A  B) = P(A) Or, P(A and B) = P(A) P(B) Events A and B are Independent when the probability of one event, A is not affected by another event, B. Or, P(B  A) = P(B)

34. Bayes’ Theorem P(B i  A) = Adding up the parts of A in all the B’s Same Event

35. Bayes’ Theorem Given a hypothesis H and an event E P(H|E) = P(E|H) x P(H) P(E) Where P(E) = P(E|H) x P(H) + P(E|  H) x P(  H)

36. Example You have a physical complaint which is one of the symptoms of a rare and unpleasant disease. 1 in 500 people have the disease. The probability of having the physical symptom if you have the disease is 0.7. The probability of having the symptom without also having the disease is 0.01. How likely is it that you are suffering from the disease?

37. Example P(E) = P(E|H) x P(H) + P(E|  H) x P(  H) P(E) = 0.7 x 0.002 + 0.01 x 0.998 = 0.01138 P(H|E) = P(E|H) x P(H) P(E) P(H|E) = 0.7 x 0.002 = 0.123 0.01138

38. Probability Distribution Models Probability Distributions ContinuousDiscrete