1. © 2012 McGraw-Hill Ryerson Limited1 © 2009 McGraw-Hill Ryerson Limited

2. © 2012 McGraw-Hill Ryerson Limited2 Lind Marchal Wathen Waite

3. © 2012 McGraw-Hill Ryerson Limited3 Define the terms probability distribution and random variable. Distinguish between discrete and continuous probability distributions. Calculate the mean, the variance, and the standard deviation of a discrete probability distribution. Describe the characteristics of, and compute, probabilities using the binomial probability distribution. Learning Objectives LO 2 3 4 1

4. © 2012 McGraw-Hill Ryerson Limited4 Describe the characteristics of, and compute, probabilities using the hypergeometric probability distribution. Describe the characteristics of, and compute, probabilities using the Poisson probability distribution. Learning Objectives LO 6 5

5. © 2012 McGraw-Hill Ryerson Limited5 What Is Probability Distribution? LO 1

6. © 2012 McGraw-Hill Ryerson Limited6 It gives the entire range of values that can occur based on an experiment. It is similar to a relative frequency distribution. However, instead of describing the past, it describes a possible future event and how likely that event is. Probability Distribution LO 1

7. © 2012 McGraw-Hill Ryerson Limited7 Suppose we are now interested in getting a suit of cards from a well-shuffled deck. What is the probability distribution for the suit diamonds? The possible results are: diamond, club, spade, heart LO 1 Example – Probability Distributions Suit Probability of Outcome, P(x) diamond club heart spade Total

8. © 2012 McGraw-Hill Ryerson Limited8 LO 1 Solution – Probability Distributions

9. © 2012 McGraw-Hill Ryerson Limited9 There are 52 possible outcomes. LO 1 Solution – Probability Distributions

10. © 2012 McGraw-Hill Ryerson Limited10 LO 1 Solution – Probability Distributions Let E be the event “drawing a diamond”. An examination of the sample space shows that there are 52 cards, of which 13 are diamonds, so n(E) = 13 and n(S) = 52. Hence the probability of event E occurring is P(E) = 13 / 52 = 0.25.

11. © 2012 McGraw-Hill Ryerson Limited11 1.The probability of a particularoutcome is between 0 and 1 inclusive. 2.The outcomes are mutually exclusive events. 3.The list is exhaustive. The sum of the probabilities of the various events is equal to 1. Characteristics of a Probability Distribution LO 1

12. © 2012 McGraw-Hill Ryerson Limited12 You Try It Out! The possible outcomes of an experiment involving a card drawn from a well-shuffled deck of 52 cards: a)Develop a probability distribution for drawing a king. b)Portray the probability distribution graphically. c)What is the sum of the probabilities? LO 1

13. © 2012 McGraw-Hill Ryerson Limited13 Random Variables LO 2

14. © 2012 McGraw-Hill Ryerson Limited14 In any experiment of chance, the outcomes occur randomly. So the numerical value associated with an outcome is often called a random variable. Random Variables LO 2

15. © 2012 McGraw-Hill Ryerson Limited15 The following diagram illustrates the terms: experiment outcome event random variable First, for the experiment of drawing a card from a well- shuffled deck of 52 cards, there are 52 possible outcomes. In this experiment, we are interested in the event of drawing a diamond. LO 2 Example – Probability Distributions

16. © 2012 McGraw-Hill Ryerson Limited16 The random variable is a card drawn from a well-shuffled deck of 52 cards. We want to know the probability of the event given that the random variable equals 1. The result is P (suit of diamond) = 0.25 LO 2 Example – Probability Distributions

17. © 2012 McGraw-Hill Ryerson Limited17 Discrete: A variable that can assume only certain, clearly separated values. It is usually the result of counting something. Examples: 1. The number of heads appearing when a coin is tossed three times. 2. The number of students earning an A in this class. LO 2 Example – Probability Distributions

18. © 2012 McGraw-Hill Ryerson Limited18 Continuous: Can assume an infinite number of values within a given range. It is usually the result of some type of measurement. Examples: 1.The length of each song on a CD. 2.The height of each student in the class. LO 2 Example – Probability Distributions

19. © 2012 McGraw-Hill Ryerson Limited19 The Mean, Variance, and Standard Deviation of a Probability Distribution LO 3

20. © 2012 McGraw-Hill Ryerson Limited20 The mean is a typical value used to represent the central location within a probability distribution. It also is the long-run average value of the random variable The mean of a probability distribution is also referred to as its expected value. It is a weighted average where the possible values of a random variable are weighted by their corresponding probabilities of occurrence. LO 3 The Mean of a Probability Distribution

21. © 2012 McGraw-Hill Ryerson Limited21 Measures the amount of spread in a distribution The computational steps are: 1.Subtract the mean from each value, and square this difference. 2.Multiply each squared difference by its probability. 3.Sum the resulting products to arrive at the variance. Find the standard deviation by taking the positive square root of the variance. LO 3 The Variance and Standard Deviation of a Probability Distribution

22. © 2012 McGraw-Hill Ryerson Limited22 John Ragsdale sells new cars for Pelican Ford. John usually sells the largest number of cars on a Saturday. He has the following probability distribution for the number of cars he expects to sell on a particular Saturday. LO 3 Example – Mean, Variance, and Standard Deviation Number of Cars Sold, x Probability, P(x) 00.05 10.15 20.35 30.30 40.15 Total1.0

23. © 2012 McGraw-Hill Ryerson Limited23 1.What type of distribution is this? 2.On a typical Saturday, how many cars does John expect to sell? 3.What is the variance of the distribution? LO 3 Example – Probability Distributions Continued

24. © 2012 McGraw-Hill Ryerson Limited24 1.This is a discrete probability distribution for the random variable “number of cars sold”. John expects to sell only within a certain range of cars. Also, the outcomes are mutually exclusive. He cannot sell a total of both three and four cars on the same Saturday. The sum of all possible outcomes totals 1. Hence, these circumstances qualify as a probability distribution. LO 3 Solution – Probability Distributions

25. © 2012 McGraw-Hill Ryerson Limited25 2.The mean number of cars sold is computed by weighting the number of cars sold by the probability of selling that number and adding the product. LO 3 Solution – Probability Distributions Continued

26. © 2012 McGraw-Hill Ryerson Limited26 The calculations are summarized in the following table. LO 3 Solution – Probability Distributions Continued Number of Cars Sold, x Probability, P(x) xP(x) 00.050.00 10.15 20.350.70 30.300.90 40.150.60 Total1.0

27. © 2012 McGraw-Hill Ryerson Limited27 Again, a table is useful for organizing the calculations for the variance, LO 3 Solution – Probability Distributions Continued Number of Cars Sold, x Probability, P(x) 00.050–2.355.52250.2761 10.151–2.351.82250.2734 20.352–2.350.12250.0429 30.303–2.350.42250.1268 40.154–2.352.72250.4084

28. © 2012 McGraw-Hill Ryerson Limited28 You Try It Out! A soft drink seller offers three sizes of cola—small, medium, and large—for wholesale. The colas are sold for $0.70, $1, and $1.30, respectively. Twenty percent of the orders are for small, 55 percent are for medium, and 25 percent are for the large sizes. Organize the size of the colas and the probability of a sale into a probability distribution. a)Is this a discrete probability distribution? Indicate why or why not. b)Compute the mean amount charged for a cola. c)What is the variance in the amount charged for a cola? The standard deviation? LO 3

29. © 2012 McGraw-Hill Ryerson Limited29 The Binomial Probability Distribution LO 4

30. © 2012 McGraw-Hill Ryerson Limited30 Characteristics of a Binomial Probability Distribution 1.An outcome on each trial of an experiment is classified into one of two mutually exclusive categories—a success or a failure. 2.The random variable counts the number of successes in a fixed number of trials. 3.The probability of success and failure stays the same for each trial. 4.The trials are independent, meaning that the outcome of one trial does not affect the outcome of any other trial. LO 4 The Binomial Probability Distribution

31. © 2012 McGraw-Hill Ryerson Limited31 Binomial Probability Distribution given as: Where: C denotes a combination. n is the number of trials. x is a number of successes. p is the probability of a success on each trial. LO 4 The Binomial Probability Distribution

32. © 2012 McGraw-Hill Ryerson Limited32 Suppose a die is tossed six times. a)What is the probability of getting exactly two 4s? b)What is the probability of getting exactly three 6s? LO 4 EXAMPLE – The Binomial Probability Distribution

33. © 2012 McGraw-Hill Ryerson Limited33 a) b) LO 4 Solution – The Binomial Probability Distribution

34. © 2012 McGraw-Hill Ryerson Limited34 LO 4 The Mean and Variance of a Binomial Distribution

35. © 2012 McGraw-Hill Ryerson Limited35 Suppose a die is tossed five times. a)What is the average number of getting exactly two 4s? b)What is the variance of getting exactly two 4s? LO 4 Example: The Binomial Probability Distribution

36. © 2012 McGraw-Hill Ryerson Limited36 Example – The Binomial Probability Distribution SOLUTION – The Binomial Probability Distribution.) a) b) LO 4

37. © 2012 McGraw-Hill Ryerson Limited37 Solution – Mean and Variance: Another Solution Number of 4s, x P(x)P(x)xP(x) 00.40190.0000-0.83330.69440.2791 10.4019 0.16670.02780.0112 20.16070.32151.16671.36120.2188 30.03210.09642.16674.69460.1509 40.00320.01293.166710.02800.0322 50.00010.00064.166717.36140.0022 LO 4

38. © 2012 McGraw-Hill Ryerson Limited38 Example – Binomial Distributions: Using Tables Ten percent of the worm gears produced by an automatic, high-speed milling machine are defective. What is the probability that out of five gears selected at random none will be defective? Exactly one? Exactly two? Exactly three? Exactly four? Exactly five out of five? LO 4

39. © 2012 McGraw-Hill Ryerson Limited39 Probability of 0, 1, 2, … Successes for a p of 0.05, 0.10, 0.20, 0.50, and 0.70 and n of 5 LO 4 EXAMPLE – Binomial Distributions: Using Tables

40. © 2012 McGraw-Hill Ryerson Limited40 Binomial Distributions In Excel – Megastat

41. © 2012 McGraw-Hill Ryerson Limited41 Binomial Distributions In Excel

42. © 2012 McGraw-Hill Ryerson Limited42 You Try It Out! Hospital records show that 75% of the patients suffering from a certain disease, will die of it. Suppose we select a random sample of six patients. a)Does this situation fit the assumptions of the binomial distribution? b)What is the probability that the six patients will recover? c)Use formula (5-3) to determine the exact probability that four of the six sampled patients will recover. d)Use Appendix A to verify your answers to parts (b) and (c). LO 4

43. © 2012 McGraw-Hill Ryerson Limited43 LO 4 Example – Probability Distributions x0.050.10.20.30.40.50.60.70.80.90.95 00.5990.3490.1070.0280.0060.0010.000 10.3150.3870.2680.1210.040.0100.0020.000 20.0750.1940.3020.2330.1210.0440.0110.0010.000 30.010.0570.2010.2670.2150.1170.0420.0090.0010.000 40.0010.0110.0880.2000.2510.2050.1110.0370.0060.000 5 0.0010.0260.1030.2010.2460.2010.1030.0260.0010.000 6 0.0060.0370.1110.2050.2510.2000.0880.0110.001 70.000 0.0010.0090.0420.1170.2150.2670.2010.0570.010 80.000 0.0010.0110.0440.1210.2330.3020.1940.075 90.000 0.0020.0100.0400.1210.2680.3870.315 100.000 0.0010.0060.0280.1070.3490.599

44. © 2012 McGraw-Hill Ryerson Limited44 Binomial – Shapes for Varying P (n constant) P = 0.05 n = 10 P = 0.1 n = 10 P = 0.2 n = 10 LO 4

45. © 2012 McGraw-Hill Ryerson Limited45 Binomial – Shapes for Varying P (n constant) P = 0.5 n = 10 P = 0.7 n = 10 LO 4

46. © 2012 McGraw-Hill Ryerson Limited46 Binomial – Shapes for Varying n P = 0.1, n = 7P = 0.7, n = 12 LO 4 P = 0.1, n = 40 P = 0.1, n = 20

47. © 2012 McGraw-Hill Ryerson Limited47 We might be interested in the cumulative binomial probability of obtaining 45 or fewer heads in 100 tosses of a coin. Or we may be interested in the probability of selecting at most two defectives at random from production during the previous hour. In these cases we need cumulative frequency distributions similar to the ones developed in Chapter 2. Cumulative Binomial Distributions LO 4

48. © 2012 McGraw-Hill Ryerson Limited48 The probability that a student is accepted to a prestigious college is 0.6. If 10 students from the same school apply: a)What is the probability that exactly 7 are accepted? b)What is the probability that at most 2 are accepted? LO 4 Example – Probability Distributions

49. © 2012 McGraw-Hill Ryerson Limited49 a) b) LO 4 Solution – Probability Distributions

50. © 2012 McGraw-Hill Ryerson Limited50 Cumulative Binomial Distributions in Excel LO 4

51. © 2012 McGraw-Hill Ryerson Limited51 You Try It Out! For a case in which n = 10 and p = 0.45, determine the probability that: a) x = 6 b) x ≤ 6 c) x > 6 LO 4

52. © 2012 McGraw-Hill Ryerson Limited52 Hypergeometric Probability Distribution LO 5

53. © 2012 McGraw-Hill Ryerson Limited53 Hypergeometric Probability Distribution A probability function f(x) that gives the probability of obtaining exactly x elements of one kind and n − x elements of another if n elements are chosen at random without replacement from a finite population containing N elements of which S are of the first kind and N − S are of the second kind and that has the form. LO 5

54. © 2012 McGraw-Hill Ryerson Limited54 Hypergeometric Probability Distribution Hypergeometric Distribution is: Where: N is the size of the population. S is the number of successes in the population. x is the number of successes in the sample. It may be 0, 1, 2, 3,.... n is the size of the sample or the number of trials. C is the symbol for a combination. LO 5

55. © 2012 McGraw-Hill Ryerson Limited55 Assume that a jar contains 5 white marbles and 45 black marbles. You close your eyes and draw 10 marbles without replacement. What is the probability that exactly 4 of the 10 marbles are white? LO 5 Example – Probability Distributions

56. © 2012 McGraw-Hill Ryerson Limited56 LO 5 Solution – Probability Distributions

57. © 2012 McGraw-Hill Ryerson Limited57 When the binomial requirement of a constant probability of success cannot be met, then use the hypergeometric distribution. A rule of thumb: If the selected items are not returned to the population and the sample size is less than 5 percent of the population, the binomial distribution can be used to approximate the hypergeometric distribution. LO 5 Example – Probability Distributions

58. © 2012 McGraw-Hill Ryerson Limited58 PlayTime Toys Inc. employs 50 people in the assembly department. Forty of the employees belong to a union and 10 do not. Five employees are selected at random to form a committee to meet with management regarding shift starting times. LO 5 Hypergeometric vs. Binomial Number of Union Members on Committee Hypergeometric Probability, P(x) Binomial Probability (n = 5 and p = 0.80) 00.000 10.0040.006 20.0440.051 30.2100.205 40.4310.410 50.3110.328 1.000

59. © 2012 McGraw-Hill Ryerson Limited59 Hypergeometric Distribution in Excel LO 5

60. © 2012 McGraw-Hill Ryerson Limited60 You Try It Out! Kolzak Appliance Outlet just received a shipment of 15 TV sets. Shortly after they were received, the manufacturer called to report that he had inadvertently shipped five defective sets. Ms. Kolzak, the owner of the outlet, decided to test four of the 10 sets she received. What is the probability that none of the four sets tested is defective? LO 5

61. © 2012 McGraw-Hill Ryerson Limited61 POISSON PROBABILITY DISTRIBUTION LO 6

62. © 2012 McGraw-Hill Ryerson Limited62 The Poisson probability distribution describes the number of times some event occurs during a specified interval. The interval may be time, distance, area, or volume. Characteristics: 1.The random variable is the number of times some event occurs during a defined period. 2.The probability of the event is proportional to the size of the interval. 3.The intervals do not overlap and are independent. LO 6 The Poisson Distribution

63. © 2012 McGraw-Hill Ryerson Limited63 Poisson Distribution is: Where: µ is the mean number of occurrences (successes) in a particular interval. e is the constant 2.718 28… (base of the Napierian logarithmic system). x is the number of occurrences (successes). P(x) is the probability for a specified value of x. LO 6 Example – Probability Distributions

64. © 2012 McGraw-Hill Ryerson Limited64 The mean number of successes,, can be determined in binomial situations by np, where n is the number of trials and p the probability of a success. The variance of the Poisson distribution is also equal to np. The Mean and Variance of the Poisson Probability Distribution LO 6

65. © 2012 McGraw-Hill Ryerson Limited65 Example – Poisson Probability Distribution Suppose a random sample of 1000 salespeople sold a total of 3000 life insurance policies per week. Thus, the average number of policies sold per week by a salesperson is 3(3000/1000). Use Poisson’s law to calculate the probability that a salesperson will not sell a policy in a given week. LO 6

66. © 2012 McGraw-Hill Ryerson Limited66 EXAMPLE – Poisson Probability Distribution Table x0.10.20.30.40.50.60.70.80.9 00.90480.81870.74080.67030.60650.54880.49660.44930.4066 10.09050.16370.22220.26810.30330.32930.34760.35950.3659 20.00450.01640.03330.05360.07580.09880.12170.14380.1647 30.00020.00110.00330.00720.01260.01980.02840.03830.0494 40.00000.00010.00030.00070.00160.00300.00500.00770.0111 50.0000 0.00010.00020.00040.00070.00120.0020 60.0000 0.00010.00020.0003 70.0000 LO 6

67. © 2012 McGraw-Hill Ryerson Limited67 Poisson Probability Distribution in Excel LO 6

68. © 2012 McGraw-Hill Ryerson Limited68 Poisson Probability Distribution Shape Always positively skewed. Has no specific upper limit. As μ becomes larger, the Poisson distribution becomes more symmetrical. LO 6

69. © 2012 McGraw-Hill Ryerson Limited69 LO 6 Shape of Poisson Probability Distribution

70. © 2012 McGraw-Hill Ryerson Limited70 You Try It Out! If electricity power failures occur according to a Poisson distribution with an average of three failures every twenty weeks, calculate the probability that there will not be more than one failure during a particular week. LO 6

71. © 2012 McGraw-Hill Ryerson Limited71 I.A random variable is a numerical value determined by the outcome of a random experiment. II.A probability distribution is a listing of all possible outcomes of an experiment and the probability associated with each outcome. A.A discrete probability distribution can assume only certain values. The main features are: 1.The sum of the probabilities is 1. 2.The probability of a particular outcome is between 0 and 1. 3.The outcomes are mutually exclusive. B.A continuous distribution can assume an infinite number of values within a specific range. Chapter Summary

72. © 2012 McGraw-Hill Ryerson Limited72 III.The mean and variance of a discrete probability distribution are computed as follows: A.The mean is equal to: [5–1] B.The variance is equal to: [5–2] Chapter Summary

73. © 2012 McGraw-Hill Ryerson Limited73 IV.The binomial distribution has the following characteristics: A.Each outcome is classified into one of two mutually exclusive categories. B.The distribution results from a count of the number of successes in a fixed number of trails. C.The probability of a success remains the same from trial to trial. D.Each trial is independent. E.A binomial probability is determined as follows: [5–3] Chapter Summary

74. © 2012 McGraw-Hill Ryerson Limited74 F.The mean is computed as [5–4] G.The variance is: [5–5] Chapter Summary

75. © 2012 McGraw-Hill Ryerson Limited75 V.The hypergeometric distribution has the following characteristics: A.There are only two possible outcomes. B.The probability of a success is not the same on each trial. C.The distribution results from a count of the number of successes in a fixed number of trials. D.A hypergeometric probability is computed from the following equation. [5–6] Chapter Summary

76. © 2012 McGraw-Hill Ryerson Limited76 VI.The Poisson distribution has the following characteristics: A.It describes the number of times an event occurs during a specified interval. B.The probability of a “success” is proportional to the length of the interval. C.Non-overlapping intervals are independent. D.It is a limiting form of the binomial distribution when n is large and p is small. Chapter Summary

77. © 2012 McGraw-Hill Ryerson Limited77 E.A Poisson probability is determined from the following equation: [5–7] F.The mean and the variance are: Chapter Summary