1. 1 1 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Chapter 4 __________________________ Introduction to Probability

2. 2 2 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Chapter 4 Introduction to Probability n Experiments, Counting Rules, Events, and Assigning Probabilities n Some Basic Relationships of Probability n Conditional Probability n Bayes’ Theorem

3. 3 3 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Probability ….is a Numerical Measure of the Likelihood of Occurrence of Something

4. 4 4 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Probability as a Numerical Measure of the Likelihood of Occurrence 0 1.5 Increasing Likelihood of Occurrence Probability: The event is very unlikely to occur. The occurrence of the event is just as likely as just as likely as it is unlikely. The event is almost certain to occur.

5. 5 5 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Experiment and Sample Space Experiment: is a process that has well-defined outcomes. Experiment: is a process that has well-defined outcomes. The set of all outcomes of an experiment is known as Sample Space. The set of all outcomes of an experiment is known as Sample Space. An observation in a sample space is called sample An observation in a sample space is called sample point. point. An observation in a sample space is called sample An observation in a sample space is called sample point. point.

6. 6 6 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Counting Experimental Outcomes Single Step Experiment Multiple Step Experiment

7. 7 7 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Single-Step Experiment If an experiment consists of just a single step and n possible results, then the total and n possible results, then the total number of experimental outcomes is n. number of experimental outcomes is n.

8. 8 8 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) A Counting Rule for Multiple-Step Experiments If an experiment consists of a sequence of steps If an experiment consists of a sequence of steps in which there are n 1 possible results for the first step, in which there are n 1 possible results for the first step, n 2 possible results for the second step, and so on, n 2 possible results for the second step, and so on, then, the then, the Total Number of Experimental Outcomes = ( n 1 )( n 2 )... ( n k ).Total Number of Experimental Outcomes = ( n 1 )( n 2 )... ( n k ).

9. 9 9 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Example: Brad’s Investments Brad has invested in two stocks: Oil Industry and Mining Industry. The possible outcomes of his investments three months from now are as follows. Investment Gain or Loss Investment Gain or Loss in 3 Months (in $000) in 3 Months (in $000) Oil Industry Mining Industry 10 10 5 0  20 8  2

10. 10 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Brad’s Investments can be viewed as a Brad’s Investments can be viewed as a two-step experiment. It involves two stocks, each with a set of experimental outcomes. Oil Industry: n 1 = 4 Mining Industry: n 2 = 2 Total Possible Outcomes: n 1 n 2 = (4)(2) = 8 A Counting Rule for Multiple-Step Experiments

11. 11 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Counting Outcomes of Multiple-Step Experiments One method of counting the outcomes of a multiple-step One method of counting the outcomes of a multiple-step experiment involves using a tree diagram. experiment involves using a tree diagram.

12. 12 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Tree Diagram Gain 5 Gain 8 Gain 10 Gain 8 Lose 20 Lose 2 Even Oil Industry (Stage 1) Mining Industry (Stage 2) ExperimentalOutcomes (10, 8) Gain $18,000 (10, -2) Gain $8,000 (5, 8) Gain $13,000 (5, -2) Gain $3,000 (0, 8) Gain $8,000 (0, -2) Lose $2,000 (-20, 8) Lose $12,000 (-20, -2) Lose $22,000

13. 13 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Two other commonly used methods of counting Experimental Outcomes (usually with large sample space) are: 1. Combinations : When Order of Selection is not Important 2. Permutations: When Order of Selection is Important

14. 14 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) A useful method that enables us to count the number of experimental outcomes when n objects are to be selected from a set of N objects. Combinations where: N ! = N ( N  1)( N  2)... (2)(1) n ! = n ( n  1)( n  2)... (2)(1) n ! = n ( n  1)( n  2)... (2)(1) 0! = 1 0! = 1

15. 15 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Example: Survey of Students n In how many ways can a sample consisting two students from a group of five (Adams, Ben, Charlie, David, Edwards) be selected for an interview? n AB, AC, AD, AE, BC, BD, BE, CD, CE, DE Note here that the Order in which we pick each student is not relevant. Picking Adams followed by Ben is the same as Picking Ben followed by Adams

16. 16 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) where: N ! = N ( N  1)( N  2)... (2)(1) n ! = n ( n  1)( n  2)... (2)(1) n ! = n ( n  1)( n  2)... (2)(1) 0! = 1 0! = 1 Permutations A third method of counting the number of experimental outcomes when n objects are to selected from a set of N objects, where the order of selection is important.

17. 17 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Example: Crime Investigation n In how many ways can two individuals from a gang of five suspects (Adams, Ben, Charlie, David, Edwards) be selected for interrogation? n AB, AC, AD, AE, BC, BD, BE, CD, CE, DE n BA, CA, DA, EA, CC, DB, EB, DC, EC, ED Note here that the Order in which we pick the student is Important: Picking A followed by B is considered different from Picking B followed by A by A

18. 18 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) How Do We Assign Probabilities

19. 19 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Assigning Probabilities Classical Method Relative Frequency Method Subjective Method Assign probabilities based on the assumption Assign probabilities based on the assumption of equally likely outcomes of equally likely outcomes Assign probabilities based on experimentation Assign probabilities based on experimentation or historical data or historical data Assign probabilities based on judgment Assign probabilities based on judgment

20. 20 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Classical Method If an experiment has n possible outcomes, this method would assign a probability of 1/ n to each outcome. Experiment: Rolling a die Sample Space: S = {1, 2, 3, 4, 5, 6} Probabilities: Each sample point has a 1/6 chance of occurring 1/6 chance of occurring Example:

21. 21 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Use historical data, identify categories and relative frequency of each category and work out the likelihood of occurrence Relative Frequency Method Probabilities are assigned by dividing class frequency by the total number of observation in the sample space (data).

22. 22 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Relative Frequency Method Number of Cars Rented Number of Days 01234 4 61810 2 Office records of a local car rental company show the following frequencies of daily rentals over the last 40 days. The company wants to know the probabilities of the number of cars it rents out each day. Car Rentals Car Rentals

23. 23 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) In a relative frequency method, probabilities are assigned by dividing class frequency (number of days) by the total observation (total number of days). Relative Frequency Method 4/404/40 Probability Number of Cars Rented Number of Days 01234 4 61810 240.10.10.15.15.45.45.25.25.05.051.00

24. 24 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Subjective Method When economic conditions and company circumstances When economic conditions and company circumstances change rapidly, it might be difficult to assign probabilities based solely on historical data. Thus using available data, experience and intuition, Thus using available data, experience and intuition, We can assign probabilities that express our degree of belief For certain outcome….

25. 25 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Subjective Method Applying the subjective method, a stock analyst made the following probability assignments to Brad’s Investment. Applying the subjective method, a stock analyst made the following probability assignments to Brad’s Investment. Exper. Outcome Net Gain or Loss Probability (10, 8) (10,  2) (5, 8) (5,  2) (0, 8) (0,  2) (  20, 8) (  20,  2) $18,000 Gain $18,000 Gain $8,000 Gain $8,000 Gain $13,000 Gain $13,000 Gain $3,000 Gain $3,000 Gain $8,000 Gain $8,000 Gain $2,000 Loss $2,000 Loss $12,000 Loss $12,000 Loss $22,000 Loss $22,000 Loss.20.08.16.26.10.12.02.06

26. 26 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Which Method is Best? The best probability estimates are often the ones that are The best probability estimates are often the ones that are obtained by combining the estimates from the classical or frequency approach with the subjective estimate. Example: Stock Market Analysts

27. 27 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Events and Their Probabilities

28. 28 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) An event is a collection of sample points. An event is a collection of sample points. The probability of any event is equal to the sum of The probability of any event is equal to the sum of the probabilities of the sample points in the event. the probabilities of the sample points in the event. The probability of any event is equal to the sum of The probability of any event is equal to the sum of the probabilities of the sample points in the event. the probabilities of the sample points in the event. If we can identify all the sample points of an experiment and If we can identify all the sample points of an experiment and assign a probability to each sample point, then we can compute the probability of any event. If we can identify all the sample points of an experiment and If we can identify all the sample points of an experiment and assign a probability to each sample point, then we can compute the probability of any event. Events and Their Probabilities

29. 29 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Given the following information, find the Probability that Brad’s Investment in Oil Industry is Profitable Exper. Outcome Net Gain or Loss Probability (10, 8) (10,  2) (5, 8) (5,  2) (0, 8) (0,  2) (  20, 8) (  20,  2) $18,000 Gain $18,000 Gain $8,000 Gain $8,000 Gain $13,000 Gain $13,000 Gain $3,000 Gain $3,000 Gain $8,000 Gain $8,000 Gain $2,000 Loss $2,000 Loss $12,000 Loss $12,000 Loss $22,000 Loss $22,000 Loss.20.08.16.26.10.12.02.06

30. 30 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Events and Their Probabilities Event O = Oil Industry is Profitable O = {(10, 8), (10,  2), (5, 8), (5,  2)} P (O) = P (10, 8) + P (10,  2) + P (5, 8) + P (5,  2) =.20 +.08 +.16 +.26 =.70

31. 31 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Exper. Outcome Net Gain or Loss Probability (10, 8) (10,  2) (5, 8) (5,  2) (0, 8) (0,  2) (  20, 8) (  20,  2) $18,000 Gain $18,000 Gain $8,000 Gain $8,000 Gain $13,000 Gain $13,000 Gain $3,000 Gain $3,000 Gain $8,000 Gain $8,000 Gain $2,000 Loss $2,000 Loss $12,000 Loss $12,000 Loss $22,000 Loss $22,000 Loss.20.08.16.26.10.12.02.06 Given the following information, find the Probability that Brad’s Investment in the Mining Industry is Profitable

32. 32 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Events and Their Probabilities Event M = Mining Stocks Profitable M = {(10, 8), (5, 8), (0, 8), (  20, 8)} P ( M ) = P (10, 8) + P (5, 8) + P (0, 8) + P (  20, 8) =.20 +.16 +.10 +.02 =.48

33. 33 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Basic Probability Relationships

34. 34 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Basic Probability Relationships The following basic probability relationships can be used to compute the probability of an event without knowledge of The following basic probability relationships can be used to compute the probability of an event without knowledge of probabilities of all the sample points. probabilities of all the sample points. Complement of an Event Complement of an Event Intersection of Two Events Intersection of Two Events Mutually Exclusive Events Mutually Exclusive Events Union of Two Events

35. 35 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Complement of an Event

36. 36 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) The complement of A is denoted by A c. The complement of A is denoted by A c. A complement of an event is another event That consists of all sample points that are NOT in the first event. A complement of an event is another event That consists of all sample points that are NOT in the first event. Complement of an Event Event A AcAcAcAc Sample Space S Sample Space S VennDiagram

37. 37 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Union of Two Events

38. 38 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) The union of events A and B is denoted by A  B  The union of events A and B is denoted by A  B  The union of two events is one that contains all sample The union of two events is one that contains all sample points that are in A or B or both. points that are in A or B or both. The union of two events is one that contains all sample The union of two events is one that contains all sample points that are in A or B or both. points that are in A or B or both. Union of Two Events Sample Space S Sample Space S Event A Event B

39. 39 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Given the following information, find the Probability that Brad’s Investment in the Oil or Mining are Profitable Exper. Outcome Probability (10, 8) (10,  2) (5, 8) (5,  2) (0, 8) (0,  2) (  20, 8) (  20,  2).20.08.16.26.10.12.02.06

40. 40 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Union of Two Events Event O = Oil Socks Profitable Event M = Mining Stocks Profitable O  = Oil Stocks Profitable or Mining Stocks Profitable O  = {(10, 8), (10,  2), (5, 8), (5,  2), (0, 8), (  20, 8)} P ( O  ) = P (10, 8) + P (10,  2) + P (5, 8) + P (5,  2) + P (0, 8) + P (  20, 8) + P (0, 8) + P (  20, 8) =.20 +.08 +.16 +.26 +.10 +.02 =.82

41. 41 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Intersection of Two Events

42. 42 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) The intersection of events A and B is denoted by A  The intersection of events A and B is denoted by A  The intersection of two events is one that contains the set The intersection of two events is one that contains the set of all sample points that are in both events (A and B). The intersection of two events is one that contains the set The intersection of two events is one that contains the set of all sample points that are in both events (A and B). Sample Space S Sample Space S Event A Event B Intersection of Two Events Intersection of A and B

43. 43 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Given the following information, find the probability that Brad’s Investment in both Oil and Mining Industry are Profitable Exper. Outcome Probability (10, 8) (10,  2) (5, 8) (5,  2) (0, 8) (0,  2) (  20, 8) (  20,  2).20.08.16.26.10.12.02.06

44. 44 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Intersection of Two Events Event O = Oil Stocks Profitable Event M = Mining Stocks Profitable O  = Oil Stocks Profitable and Mining Stocks Profitable O  = {(10, 8), (5, 8)} P ( O  ) = P (10, 8) + P (5, 8) =.20 +.16 =.36

45. 45 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Computing Probability of Events is governed by Some Established Rules. These rules are known as Laws of Addition and Multiplication Laws of Addition and Multiplication

46. 46 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Law of Additions The addition law provides a way to compute the The addition law provides a way to compute the probability of event A, or B, or both A and B occurring. probability of event A, or B, or both A and B occurring. The addition law provides a way to compute the The addition law provides a way to compute the probability of event A, or B, or both A and B occurring. probability of event A, or B, or both A and B occurring.

47. 47 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Addition Law The law is written as: The law is written as: P ( A  B ) = P ( A ) + P ( B )  P ( A  B 

48. 48 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Event O = Oil Stocks Profitable Event M = Mining Stocks Profitable P(O  ) Application of Addition Law Given the following information, find the probability that Brad’s Oil OR Mining Stocks are Profitable

49. 49 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) We know: P ( O ) =.70, P ( M ) =.48, P ( O  ) =.36 Thus: P ( O  M ) = P ( O ) + P( M )  P ( O  M) =.70 +.48 .36 =.82 Application of Addition Law (This result is the same as the result obtained earlier using the definition of the probability of an event.) P ( A  B ) = P ( A ) + P ( B )  P ( A  B  P(O  )=

50. 50 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Mutually Exclusive Events Two events are said to be mutually exclusive if the Two events are said to be mutually exclusive if the events have no sample points in common. events have no sample points in common. Two events are said to be mutually exclusive if the Two events are said to be mutually exclusive if the events have no sample points in common. events have no sample points in common. Or when one event occurs, the other cannot occur. Or when one event occurs, the other cannot occur.

51. 51 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Mutually Exclusive Events Sample Space S Sample Space S Event A Event B

52. 52 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Mutually Exclusive Events If events A and B are mutually exclusive, P ( A  B  = 0. If events A and B are mutually exclusive, P ( A  B  = 0. The addition law for mutually exclusive events is: The addition law for mutually exclusive events is: P ( A  B ) = P ( A ) + P ( B ) there’s no need to include “  P ( A  B  ”

53. 53 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Multiplication Law The multiplication law provides a way to compute the probability The multiplication law provides a way to compute the probability of the intersection of two events. of the intersection of two events. The multiplication law provides a way to compute the probability The multiplication law provides a way to compute the probability of the intersection of two events. of the intersection of two events.

54. 54 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Conditional Probability

55. 55 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n The probability of a particular event occurring, given that another event has occurred : P(A|B); P(B|A). n Conditional Probability shows that the two events are related (One depends on the other) Conditional Probability

56. 56 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) A conditional probability is computed as follows : Conditional Probability

57. 57 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) We know: P ( M  C ) =.36, P ( M ) =.70 Thus: Thus: Conditional Probability-Example What is the probability that Brad’s Mining Stocks are Profitable, given that his Oil stocks are Profitable?

58. 58 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Multiplication Law The multiplication law provides a way to compute the probability The multiplication law provides a way to compute the probability of the intersection of two events. of the intersection of two events. The multiplication law provides a way to compute the probability The multiplication law provides a way to compute the probability of the intersection of two events. of the intersection of two events.

59. 59 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Multiplication Law The law is written as: The law is written as: P ( A  B ) = P ( B ) P ( A | B )

60. 60 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Event O = Oil Profitable Event M = Mining Profitable Multiplication Law—Applied Example What is the probability that Brad’s Stocks in Oil and and Mining Industries are Profitable? Mining Industries are Profitable?

61. 61 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Multiplication Law—Applied Example

62. 62 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) We know: P (O) =.70, P ( M | O ) =.5143 Thus: P ( M  C) = P ( O ) P (M |O ) = (.70)(.5143) =.36 (This result is the same as that obtained earlier using the definition of the probability of an event.) Multiplication Law—Applied Example

63. 63 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Independent Events

64. 64 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Independent Events If the probability of event A is not changed by the If the probability of event A is not changed by the existence of event B, we would say that events A existence of event B, we would say that events A and B are independent. and B are independent. If the probability of event A is not changed by the If the probability of event A is not changed by the existence of event B, we would say that events A existence of event B, we would say that events A and B are independent. and B are independent. If two events A and B are independent then : If two events A and B are independent then : P ( A | B ) = P ( A ) P ( B | A ) = P ( B ) or

65. 65 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) The multiplication law also can be used as a test to see The multiplication law also can be used as a test to see if two events are independent. if two events are independent. The multiplication law also can be used as a test to see The multiplication law also can be used as a test to see if two events are independent. if two events are independent. The law is written as: The law is written as: P ( A  B ) = P ( A ) P ( B ) Multiplication Law for Independent Events

66. 66 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Multiplication Law for Independent Events Event M = Markley Oil Profitable Event C = Collins Mining Profitable We know: P ( M  C ) =.36, P ( M ) =.70, P ( C ) =.48 But: P ( M)P(C) = (.70)(.48) =.34, not.36 But: P ( M)P(C) = (.70)(.48) =.34, not.36 Are events M and C independent? Does  P ( M  C ) = P ( M)P(C) ? Hence: M and C are not independent. Hence: M and C are not independent.

67. 67 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Independent Events Events are independent when the occurrence of one event does not change the probability that another event will occur.Events are independent when the occurrence of one event does not change the probability that another event will occur. If A and B are independent, P(A | B) = P(A) because the occurrence of event B does not change the probability that A will occur. If A and B are independent, P(A | B) = P(A) because the occurrence of event B does not change the probability that A will occur. If A and B are independent, then If A and B are independent, then P(A and B) = P(A) P(B)

68. 68 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Bayes’ Theorem

69. 69 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Bayes’ Theorem Often we begin probability analysis with some Often we begin probability analysis with some initial assumptions: Prior probabilities. Then, from a sample, special report, or a product test we Then, from a sample, special report, or a product test we obtain some additional information. Given this information, we calculate revise our initial Given this information, we calculate revise our initial Thought about the vent: Posterior Probabilities. Bayes’ theorem provides the means for revising the Bayes’ theorem provides the means for revising the Prior Probabilities. Prior Probabilities.

70. 70 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Bayes’ Theorem NewInformationNewInformation Application of Bayes’ TheoremApplication TheoremPosteriorProbabilitiesPosteriorProbabilities PriorProbabilitiesPriorProbabilities In the 1700s, Thomas Bayes developed a way to revise the probability that an event occurs from information obtained from a second event.In the 1700s, Thomas Bayes developed a way to revise the probability that an event occurs from information obtained from a second event.

71. 71 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Bayes’ Theorem Bay’s theorem is applicable when the events for which we want Bay’s theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and to compute posterior probabilities are mutually exclusive and their union is the entire sample space.

72. 72 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) A proposed shopping center will provide strong competition for downtown businesses like L. S. Clothiers. If the shopping center is built, the owner of L. S. Clothiers feels it would be best to relocate to the center. Bayes’ Theorem Example: L. S. Clothiers Example: L. S. Clothiers The shopping center cannot be built unless azoning change is approved by the town council. The planning board must first make recommendation,for or against the zoning change, to the council.

73. 73 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Prior Probabilities Let: Bayes’ Theorem A 1 = town council approves the zoning change A 1 = town council approves the zoning change A 2 = town council disapproves the change A 2 = town council disapproves the change P( A 1 ) =.7, P( A 2 ) =.3 Using subjective judgment:

74. 74 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n New Information The planning board has recommended against the zoning change. Let B denote the event of a negative recommendation by the planning board. Given that B has occurred, should L. S. Clothiers revise the probabilities that the town council will approve or disapprove the zoning change? Bayes’ Theorem

75. 75 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Conditional Probabilities Past history with the planning board and the town council indicates the following: Past history with the planning board and the town council indicates the following: Bayes’ Theorem P ( B | A 1 ) =.2 P ( B | A 2 ) =.9 P ( B C | A 1 ) =.8 P ( B C | A 2 ) =.1 Hence:

76. 76 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) P( B c | A 1 ) =.8 P( A 1 ) =.7 P( A 2 ) =.3 P( B | A 2 ) =.9 P( B c | A 2 ) =.1 P( B | A 1 ) =.2  P( A 1  B ) =.14  P( A 2  B ) =.27  P( A 2  B c ) =.03  P( A 1  B c ) =.56 Bayes’ Theorem Tree Diagram Town Council Planning Board ExperimentalOutcomes

77. 77 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Posterior Probabilities Given the planning board’s recommendation not to approve the zoning change, we revise the prior probabilities as follows: Given the planning board’s recommendation not to approve the zoning change, we revise the prior probabilities as follows: Bayes’ Theorem =.34

78. 78 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Conclusion Although, the planning board’s recommendation is good news for L. S. Clothiers, the posterior probability of the town council approving the zoning change is.34 compared to a prior probability of.70. Although, the planning board’s recommendation is good news for L. S. Clothiers, the posterior probability of the town council approving the zoning change is.34 compared to a prior probability of.70. Bayes’ Theorem

79. 79 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Tabular Approach n Step 1 Prepare the following three columns: Column 1  The mutually exclusive events for which Column 1  The mutually exclusive events for which posterior probabilities are desired. posterior probabilities are desired. Column 2  The prior probabilities for the events. Column 2  The prior probabilities for the events. Column 3  The conditional probabilities of the new Column 3  The conditional probabilities of the new information given each event. information given each event.

80. 80 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Tabular Approach (1)(2)(3)(4)(5) Events AiAiAiAi Prior Probabilities P(Ai)P(Ai)P(Ai)P(Ai) Conditional Probabilities P(B|Ai)P(B|Ai)P(B|Ai)P(B|Ai) A1A1A1A1 A2A2A2A2.7.3 1.0.2.9

81. 81 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Tabular Approach n Step 2 Column 4 Compute the joint probabilities for each event and the new information B by using the multiplication law. Compute the joint probabilities for each event and the new information B by using the multiplication law. Multiply the prior probabilities in column 2 by the corresponding conditional probabilities in column 3. That is, P ( A i  B ) = P ( A i ) P ( B | A i ). Multiply the prior probabilities in column 2 by the corresponding conditional probabilities in column 3. That is, P ( A i  B ) = P ( A i ) P ( B | A i ).

82. 82 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Tabular Approach (1)(2)(3)(4)(5) Events A i PriorProbabilities P ( A i ) ConditionalProbabilities P ( B | A i ) A1A1A2A2A1A1A2A2.7.7.3.31.0.2.9.14.27 Joint Probabilities P ( A i P ( A i  B)B)B)B).7 x.2

83. 83 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Tabular Approach n Step 2 (continued) We see that there is a.14 probability of the town We see that there is a.14 probability of the town council approving the zoning change and a negative council approving the zoning change and a negative recommendation by the planning board. recommendation by the planning board. There is a.27 probability of the town council There is a.27 probability of the town council disapproving the zoning change and a negative disapproving the zoning change and a negative recommendation by the planning board. recommendation by the planning board.

84. 84 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Tabular Approach n Step 3 Column 4 Sum the joint probabilities. The sum is the Sum the joint probabilities. The sum is the probability of the new information, P ( B ). The sum.14 +.27 shows an overall probability of.41 of a negative recommendation by the planning board.

85. 85 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Tabular Approach (1)(2)(3)(4)(5) Events A i PriorProbabilities P ( A i ) ConditionalProbabilities P ( B | A i ) A1A1A2A2A1A1A2A2.7.7.3.31.0.2.9.14.27 JointProbabilities P ( A i   B ) P ( B ) P ( B ) =.41

86. 86 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Step 4 Column 5 Compute the posterior probabilities using the basic relationship of conditional probability. Compute the posterior probabilities using the basic relationship of conditional probability. The joint probabilities P ( A i   B ) are in column 4 and the probability P ( B ) is the sum of column 4. The joint probabilities P ( A i   B ) are in column 4 and the probability P ( B ) is the sum of column 4. Tabular Approach

87. 87 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) (1)(2)(3)(4)(5) Events A i PriorProbabilities P ( A i ) ConditionalProbabilities P ( B | A i ) A1A1A2A2A1A1A2A2.7.7.3.31.0.2.9.14.27 JointProbabilities P ( A i   B ) P ( B ) =.41 P ( B ) =.41 Tabular Approach.14/.41.14/.41 Posterior Probabilities P(Ai P(Ai P(Ai P(Ai |B)|B)|B)|B). 3415.6585 1.0000