1. 4 - 1 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Statistics for Managers Using Microsoft Excel Basic Probability & Discrete Probability Distributions Chapter 4 n Define experiment, outcome, event, sample space, & probability n Use a contingency table to find probabilities n Describe 4 discrete probability distributions n Find the probability of discrete random variables

2. 4 - 2 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Thinking Challenge What’s the probability of getting a head on the toss of a single fair coin? Use a scale from 0 (no way) to 1 (sure thing). So toss a coin twice. Do it! Did you get one head & one tail? What’s it all mean?

3. 4 - 3 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Many Repetitions!* Number of Tosses Total Heads Number of Tosses

4. 4 - 4 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Introduction to Probability n Experiment l Process of obtaining an observation, outcome or simple event n Outcome l Result of an experiment

5. 4 - 5 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Sample space depends on experimenter! n Experiment l Process of obtaining an observation, outcome or simple event n Outcome l Result of an experiment n Sample space (S) l Collection of all possible outcomes l Defined by experimenter Experiments & Outcomes

6. 4 - 6 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Outcome Examples Toss a coin, note faceHead, tail Toss 2 coins, note facesHH, HT, TH, TT Select 1 card, note kind 2, 2 ,..., A  (52) Select 1 card, note colorRed, black Play a football gameWin, lose, tie Inspect a part, note qualityDefective, good Observe genderMale, female ExperimentSample Space

7. 4 - 7 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Outcome Properties n Mutually exclusive l 2 outcomes can not occur at the same time s Example: Both male & female in same person n Collectively exhaustive l 1 outcome in sample space must occur s Example: Male or female Experiment: Observe gender © 1984-1994 T/Maker Co.

8. 4 - 8 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Events n Any collection of outcomes n Simple event l Outcome with 1 characteristic n Compound event l Collection of outcomes or simple events l 2 or more characteristics l Joint event: special case s 2 events occurring simultaneously

9. 4 - 9 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Event Examples Sample spaceHH, HT, TH, TT 1 head & 1 tailHT, TH Heads on 1st coinHH, HT At least 1 headHH, HT, TH Heads on bothHH Experiment: Toss 2 coins. Note faces. EventOutcomes in Event

10. 4 - 10 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Visualizing Sample Space n Listing l S = {Head, Tail} n Contingency Table

11. 4 - 11 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Contingency Table Experiment: Toss 2 coins. Note faces. S = {HH, HT, TH, TT}Sample space Outcome (% or count) Simple event (head on 1st coin)

12. 4 - 12 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Joint Events (Event A and Event B) Sample space (S): 2 R, 2 R , 2 B ,..., A B  Experiment: Draw 1 card. Note kind, color & suit. Joint event Ace AND Black: A B , A B  Simple event Ace: A R, A R , A B , A B  Simple event black: 2 B ,..., A B 

13. 4 - 13 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Either-Or Events (Event A or Event B) Sample space (S): 2 R, 2 R , 2 B ,..., A B  Experiment: Draw 1 card. Note kind, color & suit. Joint event Ace OR Black: A R,..., A B ,  2 B ,..., K B  Simple event Ace: A R, A R , A B , A B  Simple event Black: 2 B ,..., A B 

14. 4 - 14 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e   Special Events n Null event l Club & Diamond on 1 card draw n Complement of event For event A, all events not in A: A n Mutually exclusive event l Events do not occur simultaneously Null Event Q Q

15. 4 - 15 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e What is Probability? n Numerical measure of likelihood that event will occur l P(Event) l P(A) l Prob(A)

16. 4 - 16 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e What is Probability? n Numerical measure of likelihood that event will occur l P(Event) l P(A) l Prob(A) n Lies between 0 & 1 n Sum of events is 1 1.5 0 Certain Impossible © 1984-1994 T/Maker Co.

17. 4 - 17 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Assigning Event Probabilities n a priori classical method n Empirical classical method n Subjective method

18. 4 - 18 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e a priori Classical Method n Prior knowledge of process n Before experiment n P(Event) = X / T l X = No. of event outcomes l T = Total outcomes in sample space l Each of T outcomes is equally likely s P(Outcome) = 1/T © 1984-1994 T/Maker Co.

19. 4 - 19 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Empirical Classical Method n Actual data collected n After experiment n P(Event) = X / T l Repeat experiment T times l Event observed X times n Also called relative frequency method Of 100 parts inspected, only 2 defects!

20. 4 - 20 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Subjective Method n Individual knowledge of situation n Before experiment n Unique process l Not repeatable n Different probabilities from different people © 1984-1994 T/Maker Co.

21. 4 - 21 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Thinking Challenge n that a box of 24 bolts will be defective? n that a toss of a coin will be a tail? n that Tom will default on his PLUS loan? n that a student will earn an ‘A’ in this class? n that a new store on RTE. 1 will succeed? Which method should be used to find the probability...

22. 4 - 22 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Joint Event Probability n Numerical measure of likelihood that joint event will occur n Can often use contingency table l 2 variables only n Formula methods l Addition rule l Conditional probability formula l Multiplication rule

23. 4 - 23 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Event Probability Using Contingency Table Joint Probability Marginal (Simple) Probability

24. 4 - 24 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Contingency Table Example Experiment: Draw 1 card. Note kind, color & suit. P(Ace) P(Ace AND Red)P(Red)

25. 4 - 25 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Thinking Challenge What’s the probability? P(A) = P(D) = P(C and B) = P(A or D) = P(B and D) =

26. 4 - 26 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Solution* The probabilities are: P(A) = 6/10 P(D) = 5/10 P(C and B) = 1/10 P(A or D) = 9/10 P(B and D) = 3/10

27. 4 - 27 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Addition Rule n P(A or B)= P(A) + P(B) - P(A and B) n For mutually exclusive events: P(A or B) = P(A) + P(B)

28. 4 - 28 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Addition Rule Example Experiment: Draw 1 card. Note kind, color & suit. P(Ace OR Black) = P(Ace)+P(Black) - P(Ace AND Black)

29. 4 - 29 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Thinking Challenge Using the Addition Rule, what’s the probability? P(A or D) = P(B or C) =

30. 4 - 30 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Solution* Using the Addition Rule, the probabilities are:

31. 4 - 31 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Conditional Probability n Event probability given that another event occurred n Revise original sample space to account for new information l Eliminates certain outcomes n P(A | B) = P(A and B) P(B)

32. 4 - 32 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Conditional Probability Using Contingency Table Experiment: Draw 1 card. Note kind, color & suit. Revised sample space

33. 4 - 33 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e n Event occurrence does not affect probability of another event l e.g., toss 1 coin twice n Causality not implied n Tests for independence l P(A | B) = P(A) l P(A and B) = P(A)*P(B) Statistical Independence

34. 4 - 34 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Thinking Challenge Using the table then the formula, what’s the probability? P(A|D) = P(C|B) = Are C & B independent?

35. 4 - 35 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Solution* Using the formula, the probabilities are: Dependent

36. 4 - 36 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Multiplication Rule n P(A and B)= P(A)*P(B|A) = P(B)*P(A| B) n For independent events: P(A and B) = P(A)*P(B)

37. 4 - 37 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Multiplication Rule Example Experiment: Draw 1 card. Note kind, color & suit.

38. 4 - 38 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Thinking Challenge Using the Multiplication Rule, what’s the probability? P(C and B) = P(B and D) = P(A and B) =

39. 4 - 39 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Solution* Using the Multiplication Rule, the probabilities are: P(C and B) = P(C)  P(B|C) = P(B and D) = P(B)  P(D|B) =

40. 4 - 40 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Discrete Random Variables

41. 4 - 41 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Thinking Challenge You’re taking a 33 question multiple choice test. Each question has 4 choices. Clueless on 1 question, you decide to guess. What’s the chance you’ll get it right? If you guessed on all 33 questions, what would be your grade? Pass?

42. 4 - 42 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Data Types

43. 4 - 43 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Random Variable n A numerical outcome of an experiment l Number of tails in 2 coin tosses s Observe 0, 1, or 2 tails n Discrete random variable l Whole number (0, 1, 2, 3 etc.) l Obtained by counting l Usually finite number of values  Poisson random variable is exception (  )

44. 4 - 44 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Discrete Random Variable Examples

45. 4 - 45 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Discrete Probability Distribution n List of all possible [ X i, P(X i ) ] pairs l X i = Value of random variable (outcome) l P(X i ) = Probability associated with value n Mutually exclusive (no overlap) n Collectively exhaustive (nothing left out) 0  P(X i )  1  P(X i ) = 1

46. 4 - 46 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Discrete Probability Distribution Example Probability Distribution Values, X i Probabilities, P(X i ) 01/4 =.25 12/4 =.50 21/4 =.25 Experiment: Toss 2 coins. Count # tails. © 1984-1994 T/Maker Co.

47. 4 - 47 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Visualizing Discrete Probability Distributions { (0,.25), (1,.50), (2,.25) } Listing Table GraphEquation

48. 4 - 48 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Summary Measures Notation for a Population n Expected value l Mean of probability distribution l Weighted average of all possible values  = E(X) =  X i P(X i ) n Variance l Weighted average squared deviation about mean  2 = E[ (X i    (X i    P(X i )

49. 4 - 49 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Summary Measures Calculation Table

50. 4 - 50 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Thinking Challenge You toss 2 coins. You’re interested in the number of tails. What are the expected value & standard deviation of this random variable, number of tails? © 1984-1994 T/Maker Co.

51. 4 - 51 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Expected Value & Variance Solution*

52. 4 - 52 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Discrete Probability Distribution Function n Type of model l Representation of some underlying phenomenon n Mathematical formula n Represents discrete random variable n Used to get exact probabilities PXx x x () !  e -

53. 4 - 53 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Discrete Probability Distribution Models

54. 4 - 54 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Discrete Probability Distribution Models

55. 4 - 55 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Binomial Distribution n Number of ‘successes’ in a sample of n observations (trials) n # reds in 15 spins of roulette wheel n # defective items in a batch of 5 items n # correct on a 33 question exam n # customers who purchase out of 100 customers who enter store

56. 4 - 56 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Binomial Distribution Properties n Two different sampling methods l Infinite population without replacement l Finite population with replacement n Sequence of n identical trials n Each trial has 2 outcomes l ‘Success’ (desired outcome) or ‘failure’ n Constant trial probability n Trials are independent

57. 4 - 57 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Binomial Probability Distribution Function P(X) = Probability of X ‘successes’ n=Sample size p=Probability of ‘success’ x=Number of ‘successes’ in sample (X = 0, 1, 2,..., n)

58. 4 - 58 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Binomial Probability Distribution Example Experiment: Toss 1 coin 4 times in a row. Note # tails. What’s the probability of 3 tails?

59. 4 - 59 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Binomial Distribution Characteristics n = 5 p = 0.1 n = 5 p = 0.5 Mean Standard Deviation

60. 4 - 60 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Binomial Distribution Thinking Challenge You’re a telemarketer selling service contracts for Macy’s. You’ve sold 20 in your last 100 calls (p =.20). If you call 12 people tonight, what’s the probability of A. No sales? B. Exactly 2 sales? C. At most 2 sales? D. At least 2 sales?

61. 4 - 61 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Binomial Distribution Solution* A. P(0) =.0687 B. P(2) =.2835 C. P(at most 2)= P(0) + P(1) + P(2) =.0687 +.2062 +.2835 =.5584 D. P(at least 2)= P(2) + P(3)...+ P(12) = 1 - [P(0) + P(1)] = 1 -.0687 -.2062 =.7251

62. 4 - 62 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Discrete Probability Distribution Models

63. 4 - 63 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Poisson Distribution n Number of events that occur in an area of opportunity l Events per unit s Example: Time, length, area, space n Examples l # customers arriving in 20 minutes l # strikes per year in the U.S. l # defects per lot (group) of VCR's

64. 4 - 64 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Poisson Process n Constant event probability l Average of 60/hr. is 1/min. for 60 1-minute intervals n One event per interval l Don’t arrive together n Independent events l Arrival of 1 person does not affect another’s arrival © 1984-1994 T/Maker Co.

65. 4 - 65 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Poisson Probability Distribution Function P(X) = Probability of X ‘successes’ =Expected (mean) number of ‘successes’ e=2.71828 (base of natural logs) x=Number of ‘successes’ per unit

66. 4 - 66 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Poisson Distribution Characteristics  = 0.5  = 6 Mean Standard Deviation

67. 4 - 67 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Poisson Distribution Example Customers arrive at a rate of 72 per hour. What is the probability of 4 customers arriving in 3 minutes? © 1995 Corel Corp.

68. 4 - 68 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Poisson Distribution Solution 72 per hr. = 1.2 per min. = 3.6 per 3 min. interval

69. 4 - 69 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Thinking Challenge You work in Quality Assurance for an investment firm. A clerk enters 75 words per minute with 6 errors per hour. What is the probability of 0 errors in a 255-word bond transaction? © 1984-1994 T/Maker Co.

70. 4 - 70 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Poisson Distribution Solution: Finding * 75 words/min = (75 words/min)(60 min/hr) = 4500 words/hr 6 errors/hr= 6 errors/4500 words =.00133 errors/word In a 255-word transaction (interval): = (.00133 errors/word )(255 words) =.34 errors/255-word transaction

71. 4 - 71 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Poisson Distribution Solution: Finding P(0)*

72. 4 - 72 © 1998 Prentice-Hall, Inc. Statistics for Managers Using Microsoft Excel, 1/e Conclusion n Defined experiment, outcome, event, sample space, & probability n Used a contingency table to find probabilities n Described 4 discrete probability distributions n Found the probability of discrete random variables