1. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-1 Lesson 3: A Survey of Probability Concepts

2. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-2 Outline Learning Exercises Definitions Basic rules of Probability Independence Tree Diagram Bayes’ Theorem

3. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-3 Learning exercise 1: University Demographics Current enrollments by college and by sex appear in the following table. CollegeAg-ForArts-SciBus-EconEducEngrLawUndeclTotals Female500150040010002001008004500 Male9001200500 13002009005500 Totals140027009001500 300170010000 If we select a student at random, what is the probability that the student is : A female or male, i.e., P(Female or Male). Not from Agricultural and Forestry, i.e., P(not-Ag-For) A female given that the student is known to be from BusEcon, i.e., P(Female |BusEcon). A female and from BusEcon, i.e., P(Female and BusEcon). From BusEcon, i.e., P(BusEcon).

4. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-4 Learning exercise 1: University Demographics CollegeAg-ForArts-SciBus-EconEducEngrLawUndeclTotals Female500150040010002001008004500 Male9001200500 13002009005500 Totals140027009001500 300170010000 P(Female or Male)=(4500 + 5500)/10000 = 1 P(not-Ag-For)=(10000 – 1400) /10000 = 0.86 P(Female | BusEcon)= 400 /900 = 0.44 P(Female and BusEcon)= 400 /10000 = 0.04 P(BusEcon)= 900 /10000 = 0.09 P(Female and BusEcon) = P(BusEcon) P(Female | BusEcon)

5. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-5 Learning exercise 2: Predicting Sex of Babies Many couples take advantage of ultrasound exams to determine the sex of their baby before it is born. Some couples prefer not to know beforehand. In any case, ultrasound examination is not always accurate. About 1 in 5 predictions are wrong. In one medical group, the proportion of girls correctly identified is 9 out of 10, i.e., applying the test to 100 baby girls, 90 of the tests will indicate girls. and the number of boys correctly identified is 3 out of 4. i.e., applying the test to 100 baby boys, 75 of the tests will indicate boys. The proportion of girls born is 48 out of 100. What is the probability that a baby predicted to be a girl actually turns out to be a girl? Formally, find P(girl | test says girl).

6. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-6 Learning exercise 2: Predicting Sex of Babies P(girl | test says girl) In one medical group, the proportion of girls correctly identified is 9 out of 10 and the number of boys correctly identified is 3 out of 4. The proportion of girls born is 48 out of 100. Think about the next 1000 births handled by this medical group. 480 = 1000*0.48 are girls 520 = 1000*0.52 are boys Of the girls, 432 (=480*0.9) tests indicate that they are girls. Of the boys, 130 (=520*0.25) tests indicate that they are girls. In total, 562 (=432+130) tests indicate girls. Out of these 562 babies, 432 are girls. Thus P(girl | test says girl ) = 432/562 = 0.769

7. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-7 Learning exercise 2: Predicting Sex of Babies Test says girlTest says boyTotals Girl(4) 432(7) 48(2) 480 Boy(5) 130(8) 390(3) 520 Totals(6) 562(9) 438(1) 1000 With the information given, we can fill in the following table in sequence from (1) to (9), with the initial assumption of 1000 babies in total. For the question at hand, i.e., P(girl | test says girl ), we only need to fill in the cells from (1) to (6). P(girl | test says girl ) = 432/562 = 0.769

8. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-8 Learning exercise 2: Predicting Sex of Babies 480 = 1000*0.48 are girls 520 = 1000*0.52 are boys Of the girls, 432 (=480*0.9) tests indicate that they are girls. Of the boys, 130 (=520*0.25) tests indicate that they are girls. In total, 562 tests indicate girls. Out of these 562 babies, 432 are girls. Thus P(girls | test syas girls ) = 432/562 = 0.769 1000*P(girls) 1000*P(boys) 1000*[P(girls)*P(test says girls|girls) + P(boys)*P(test says girls|boys)] 1000*P(boys)*P(test says girls | boys) 1000*P(girls)*P(test says girls|girls) 1000*[P(girls)*P(test says girls|girls) + P(boys)*P(test says girls|boys)]

9. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data A fair die is rolled once. Peter is concerned with whether the resulted number is even, i.e., 2, 4, 6. Paul is concerned with whether the resulted number is less than or equal to 3, i.e., 1, 2, 3. Mary is concerned with whether the resulted number is 6. Sonia is concerned with whether the resulted number is odd, i.e., 1, 3, 5. A fair die is rolled twice. John is concerned with whether the resulted number of first roll is even, i.e., 2, 4, 6. Sarah is concerned with whether the resulted number of second roll is even, i.e., 2, 4, 6. Example 1 (to be used to illustrate the definitions)

10. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-10 Definitions: Experiment and outcome A random experiment is the observation of some activity or the act of taking some measurement. The experiment is rolling the one die in the first example, and rolling one die twice in the second example. An outcome is the particular result of an experiment. The possible outcomes are the numbers 1, 2, 3, 4, 5, and 6 in the first example. The possible outcomes are number pairs (1,1), (1,2), …, (6,6), in the second example. Sample Space – the collection of all possible outcomes of a random experiment.

11. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-11 Definition: Event An event is the collection of one or more outcomes of an experiment. For Peter: the occurrence of an even number, i.e., 2, 4, 6. For Paul: the occurrence of a number less than or equal to 3, i.e., 1, 2, 3. For Mary: the occurrence of a number 6. For Sonia: the occurrence of an odd number, i.e., 1, 3, 5. For John: the occurrence of (2,1), (2,2), (2,3), …, (2,6), (4,1), …,(4,6), (6,1), …,(6,6) [John does not care about the result of the second roll].

12. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-12 Definition: Intersection of Events Intersection of Events – If A and B are two events in a sample space S, then the intersection, A ∩ B, is the set of all outcomes in S that belong to both A and B AB ABAB S The intersection of Peter’s event and Paul’s event contains 2. The intersection of Peter’s event and Mary’s event contains 6. The intersection of Paul’s event and Mary’s event contains nothing. The intersection of Peter’s event and Sonia’s event contains nothing.

13. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-13 Definition: Mutually Exclusive events A and B are Mutually Exclusive Events if they have no basic outcomes in common i.e., the set A ∩ B is empty A B S

14. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-14 Example 2: Intersections and Mutually Exclusive events Peter’s event and Paul’s event are not mutually exclusive – both contains 2. The intersection of Peter’s event and Paul’s event contains 2. Peter’s event and Mary’s event are not mutually exclusive – both contains 6. The intersection of Peter’s event and Mary’s event contains 6. Paul’s event and Mary’s event are mutually exclusive – no common numbers. The intersection of Paul’s event and Mary’s event contains nothing. Peter’s event and Sonia’s event are mutually exclusive – no common numbers. The intersection of Peter’s event and Sonia’s event contains nothing.

15. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-15 Definition: Union Union of Events – If A and B are two events in a sample space S, then the union, A U B, is the set of all outcomes in S that belong to either A or B AB The entire shaded area represents A U B S

16. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-16 Definition: Exhaustive events Events are collectively exhaustive if at least one of the events must occur when an experiment is conducted. Peter ’ s event (even numbers) and Sonia ’ s event (odd numbers) are collectively exhaustive. Peter ’ s event (even numbers) and Mary ’ s event (number 6) are not collectively exhaustive. Events E 1, E 2, … E k are Collectively Exhaustive events if E 1 U E 2 U... U E k = S i.e., the events completely cover the sample space

17. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-17 Complement The Complement of an event A is the set of all basic outcomes in the sample space that do not belong to A. The complement is denoted A S

18. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-18 Example 3 Let the Sample Space be the collection of all possible outcomes of rolling one die: S = [1, 2, 3, 4, 5, 6] Let A be the event “Number rolled is even” Let B be the event “Number rolled is at least 4” Then the two events contains A = [2, 4, 6] B = [4, 5, 6]

19. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-19 Example 3 S = [1, 2, 3, 4, 5, 6] A = [2, 4, 6] B = [4, 5, 6] Complements: Intersections: Unions:

20. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-20 Mutually exclusive: Are A and B mutually exclusive? No. The outcomes 4 and 6 are common to both Collectively exhaustive: Are A and B collectively exhaustive? No. A U B does not contain 1 and 3 Example 3 S = [1, 2, 3, 4, 5, 6] A = [2, 4, 6] B = [4, 5, 6]

21. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-21 Probability Probability – the chance that an uncertain event will occur (always between 0 and 1) 0 ≤ P(A) ≤ 1 For any event A Certain Impossible.5 1 0

22. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-22 Assessing Probability There are three approaches to assessing the probability of an uncertain event: 1. classical probability (Assumes all outcomes in the sample space are equally likely to occur.)

23. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-23 Counting the Possible Outcomes Use the Combinations formula to determine the number of combinations of n things taken k at a time where n! = n(n-1)(n-2)…(1) 0! = 1 by definition

24. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-24 Example 4: Combination Suppose there are 2 stocks in our portfolio. We would like to select 1 stocks and sell them. What are all the possible combinations. [1], [2] C(2,1) = 2!/(1!(2-1)!)=2

25. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-25 Example 4: Combination Suppose there are 3 stocks in our portfolio. We would like to select 1 stocks and sell them. What are all the possible combinations. [1], [2], [3] C(3,1) = 3!/(1!(3-1)!)=3*2*1/[1*(2*1)]=3 Suppose there are 3 stocks in our portfolio. We would like to select 2 stocks and sell them. What are all the possible combinations. [1,2], [1,3], [2,3] C(3,2) = 3!/(2!(3-2)!)=3*2*1/[(2*1)*1]=3

26. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-26 Example 5: Combination Suppose there are 4 stocks in our portfolio. We would like to select 1 stocks and sell them. What are all the possible combinations. [1], [2], [3], [4] C(4,1) = 4!/(1!(4-1)!)=4*3*2*1/[1*(3*2*1)]=4 Suppose there are 4 stocks in our portfolio. We would like to select 2 stocks and sell them. What are all the possible combinations. [1,2], [1,3], [1,4], [2,3], [2,4], [3,4] C(4,2) = 4!/(2!(4-2)!)=4*3*2*1/[(2*1)*(2*1)]=6

27. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-27 Assessing Probability Three approaches (continued) 2. relative frequency probability the limit of the proportion of times that an event A occurs in a large number of trials, n 3. subjective probability an individual opinion or belief about the probability of occurrence

28. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-28 Probability Postulates 1. If A is any event in the sample space S, then 2. Let A be an event in S, and let O i denote the basic outcomes (mutually exclusive). Then (the notation means that the summation is over all the basic outcomes in A) 3.P(S) = 1

29. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-29 Probability Rules The Complement rule: The Addition rule: The probability of the union of two events is

30. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-30 A Probability Table B A Probabilities and joint probabilities for two events A and B are summarized in this table:

31. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-31 Example 5: Addition Rule Consider a standard deck of 52 cards, with four suits: ♥ ♣ ♦ ♠ Let event A = card is an Ace Let event B = card is from a red suit

32. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-32 P(Red U Ace) = P(Red) + P(Ace) - P(Red ∩ Ace) = 26/52 + 4/52 - 2/52 = 28/52 Don’t count the two red aces twice! Black Color Type Red Total Ace 224 Non-Ace 24 48 Total 26 52 Example 5: Addition Rule

33. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-33 A conditional probability is the probability of one event, given that another event has occurred: The conditional probability of A given that B has occurred The conditional probability of B given that A has occurred Conditional Probability

34. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-34 Example 6: Conditional Probability What is the probability that a car has a CD player, given that it has AC ? i.e., we want to find P(CD | AC) Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both.

35. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-35 Example 6: Conditional Probability No CDCDTotal AC.2.5.7 No AC.2.1.3 Total.4.61.0 Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both.

36. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-36 Example 6: Conditional Probability No CDCDTotal AC.2.5.7 No AC.2.1.3 Total.4.6 1.0 Given AC, we only consider the top row (70% of the cars). Of these, 20% have a CD player. 20% of 70% is 28.57%.

37. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-37 Multiplication Rule Multiplication rule for two events A and B: also Examples: 1.P(test says girl and girl) = P(girls) * P(test says girls | girls) 2.P(test says boy and boy) = P(boys) * P(test says boys | boys)

38. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-38 Example 7: Multiplication Rule P(Red ∩ Ace) = P(Red| Ace)P(Ace) Black Color Type Red Total Ace 224 Non-Ace 24 48 Total 26 52

39. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-39 Example 8: Independence Should I go to a party without my girlfriend? The probability of the my going to party is 0.7 (i.e., I go to 70 out of 100 parties on average). If I tend to go to whichever party my girlfriend (Venus) goes, my party behavior depends on Venus’s. That is, my probability of going to a party conditional on Venus’s presence is larger than 0.7 (extreme case being 1.0). If I tends to avoid going to whichever party Venus goes, my party behavior also depends on Venus. That is, my probability of going to a party conditional on Venus’s presence is less than 0.7 (extreme case being 0.0). If in making the party decision, I never consider whether Venus is going to a party, my party behavior does not depends on Venus’s. That means, the probability of going to a party conditional on Venus’s presence is 0.7.

40. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-40 Example 8: Independence Should I go to a party without my girlfriend? Define events: A: a young man goes to a party B: his girlfriend goes to the same party. Assume P(A) =0.7 His party behavior does not depend on his girlfriend’s only if P(A|B) =P(A) = 0.7. And, event A is said to be independent of event B. P(the young man and his girlfriends shows up in a party) = P(A & B) = P(B)*P(A|B). If he always goes to whichever party his girlfriend goes, P(A|B) = 1. Hence, P(A & B) = P(B)*P(A|B) = P(B). If he always avoid to whichever party his girlfriend goes, P(A|B) = 0. Hence, P(A & B) = P(B)*P(A|B) = 0. If in making the party decision, he never considers whether his girlfriend is going to a party, P(A|B) = 0.7. Hence, P(A & B) = P(B)*P(A|B) = P(B)*P(A) = P(B)*0.7.

41. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-41 Statistical Independence Two events are statistically independent if and only if: Events A and B are independent when the probability of one event is not affected by the other event If A and B are independent, then if P(B)>0 if P(A)>0

42. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-42 Example 9: Statistical Independence No CDCDTotal AC.2.5.7 No AC.2.1.3 Total.4.61.0 Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both. Are the events AC and CD statistically independent?

43. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-43 Example 9: Statistical Independence No CDCDTotal AC.2.5.7 No AC.2.1.3 Total.4.61.0 P(AC ∩ CD) = 0.2 P(AC) = 0.7 P(CD) = 0.4 P(AC)P(CD) = (0.7)(0.4) = 0.28 P(AC ∩ CD) = 0.2≠ P(AC)P(CD) = 0.28 So the two events are not statistically independent

44. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-44 Bivariate Probabilities B1B1 B2B2...BkBk A1A1 P(A 1  B 1 )P(A 1  B 2 )... P(A 1  B k ) A2A2 P(A 2  B 1 )P(A 2  B 2 )... P(A 2  B k ).............................. AhAh P(A h  B 1 )P(A h  B 2 )... P(A h  B k ) Outcomes for bivariate events:

45. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-45 Joint and Marginal Probabilities The probability of a joint event, A ∩ B: Computing a marginal probability: Where B 1, B 2, …, B k are k mutually exclusive and collectively exhaustive events

46. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-46 Example 10: Marginal Probability P(Ace) Black Color Type Red Total Ace 224 Non-Ace 24 48 Total 26 52

47. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-47 Tree Diagrams A tree diagram is useful for portraying conditional and joint probabilities. It is particularly useful for analyzing business decisions involving several stages.

48. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-48 EXAMPLE 11: Tree Diagram R1 B1 R2 B2 R2 B2 7/12 5/12 6/11 5/11 7/11 4/11 In a bag containing 7 red chips and 5 blue chips you select 2 chips one after the other without replacement. Construct a tree diagram showing this information.

49. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-49 EXAMPLE 11: Tree Diagram R1 B1 R2 B2 R2 B2 7/12 5/12 6/11 5/11 7/11 4/11 The tree diagram is very illustrative about the relation between joint probability and conditional probability Let A (B) be the event of a red chip in the first (second) draw. P(B|A) = 6/11 P(A) = 7/12 P(A and B) = P(A)*P(B|A) = 6/11 * 7/12 7/12 6/11

50. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-50 Example 12: Using a Tree Diagram Has AC Does not have AC Has CD (.2/.7) Does not have CD (.5/.7) Has CD (.2/.3) Does not have CD (.1/.3) P(AC)=.7 P(AC)=.3 P(AC ∩ CD) =.2 P(AC ∩ CD) =.5 P(AC ∩ CD) =.1 P(AC ∩ CD) =.2 All Cars Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both.

51. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-51 Odds The odds in favor of a particular event are given by the ratio of the probability of the event divided by the probability of its complement The odds in favor of A are

52. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-52 Example 13: Odds Calculate the probability of winning if the odds of winning are 3 to 1: Now multiply both sides by 1 – P(A) and solve for P(A): 3 x (1- P(A)) = P(A) 3 – 3P(A) = P(A) 3 = 4P(A) P(A) = 0.75

53. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-53 Overinvolvement Ratio Does A 1 depends on B 1 and B 2 differently? The probability of event A 1 conditional on event B 1 divided by the probability of A 1 conditional on activity B 2 is defined as the overinvolvement ratio: An overinvolvement ratio greater than 1 implies that event A 1 increases the conditional odds ratio in favor of B 1 :

54. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-54 Example 14: Overinvolvement ratio If the probability to get lung cancer is 0.5% for smokers and 0.1% for non-smokers, what is the overinvolvement ratio? Prob(lung cancer | smokers) = 0.005 Prob(lung cancer | non-smokers) = 0.001 Overinvolvement ratio = P(L|S)/P(L|N) = 5

55. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-55 Bayes’ Theorem where: E i = i th event of k mutually exclusive and collectively exhaustive events A = new event that might impact P(E i )

56. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-56 Bayes’ Theorem Bayes’ Theorem can be derived based on simple manipulation of the general multiplication rule. P(A 1 |B) = P(A 1 & B) /P(B) = [P(A 1 ) P(B|A 1 )] / P(B) = [P(A 1 ) P(B|A 1 )] / [P(A 1 & B) + P(A 2 & B)] = [P(A1) P(B|A 1 ) ]/ [P(A 1 ) P(B|A 1 ) + P(A 2 ) P(B|A 2 )]

57. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-57 Example 15: Bayes’ Theorem A drilling company has estimated a 40% chance of striking oil for their new well. A detailed test has been scheduled for more information. Historically, 60% of successful wells have had detailed tests, and 20% of unsuccessful wells have had detailed tests. Given that this well has been scheduled for a detailed test, what is the probability that the well will be successful?

58. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-58 Example 15: Bayes’ Theorem Let S = successful well U = unsuccessful well P(S) =.4, P(U) =.6 (prior probabilities) Define the detailed test event as D Conditional probabilities: P(D|S) =.6 P(D|U) =.2 Goal is to find P(S|D)

59. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-59 Example 15: Bayes’ Theorem Apply Bayes’ Theorem: So the revised probability of success (from the original estimate of.4), given that this well has been scheduled for a detailed test, is.667

60. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-60 Example 16: Bayes’ Theorem Duff Cola Company recently received several complaints that their bottles are under-filled. A complaint was received today but the production manager is unable to identify which of the two Springfield plants (A or B) filled this bottle. The following table summarizes the Duff production experience. What is the probability that the under-filled bottle came from plant A?

61. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-61 Example 16: Bayes’ Theorem The likelihood the bottle was filled in Plant A is reduced from.55 to.4783. Without the information about U, the manager will say the under-filled bottle is likely from plant A. With the additional information about U, the manager will say the under-filled bottle is likely from plant B. What is the probability that the under-filled bottle came from plant A?

62. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson3-62 - END - Lesson 3: A Survey of Probability Concepts