Chapter 06 Textbook Study Guide

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Operations Management and Information System
OMIS 2010
Alan Marshall

Chapter 6: Probability 6.1 Introduction This ch apter in troduced th e b asic co ncepts o f p robability. It o utlined ru les an d tech niques for assigning probabilities to events. At the completion of this chapter, you ar e expected to know the following: 1. The meaning of the many new terms introduced. 2. The three general approaches for assigning probabilities. 3. How to define a sample space for a random experiment. 4. The meaning of conditional probability and independent events. 5. How to employ the three rules of probability. 6. How to construct and use a probability tree. 7. The concept of a random variable and its probability distribution. 8. How to compute the mean and standard deviation of a discrete probability distribution. 9. How to recognize when it is appropriate to use a binomial distribution, and how to use the table of binomial probabilities. 10. How to recognize when it is appropriate to use a Poisson distribution, and how to use the table of Poisson probabilities. 6.2 Assigning Probabilities to Events This section introduced the notion of a random experiment and described the outcomes, or events, that may result from such an experiment. When attempting to solve any problem involving probabilities, you should begin by defining the random experiment and the sample space. You are expected to know the meaning of the many new terms introduced in this section, such as simple event, mutually exclusive, exhaustive, union, intersection, and complement. This section also described procedures for assigning probabilities to events and outlined the basic requirements that must be satisfied by probabilities assigned to simple events. Probabilities can be assigned to the simple events (or, for that matter, to any events) using the classical approach, the relative frequency approach, or the subjective approach. Whatever method is used to assign probabilities to th e simple events that form a sample space, two basic requirements must be satisfied: 1. Each simple event probability must lie between 0 and 1, inclusive. 2. The probabilities assigned to the simple events in a sample space must sum to 1. The probability of any event A is th en obtained by summing the probabilities assigned to the simple events contained in A. 51 Question: How do I know whether I should combine two events A and B using and or or? Answer: The key here is to fully understand the meaning of the combined statement. P(A and B) = P(A and B both occur) P(A or B) = P(A or B or both occur) Sometimes it will be necessary to rewo rd the statement of a given event so that it conforms with one of t he two expressions given above. For example, suppose your friend Karen i s about to write two exams and you define the events as follows: A: Karen will pass the statistics exam. B: Karen will pass the accounting exam. The event Karen will pass at least one of the two exams can be reworded as Karen will either pass the statistics exam or shell pass the accounting exam, or shell pass both exams. This new event can therefore be denoted (A or B). On the other hand, the event Karen will not fail either exam is the same as Karen will pass both her statistics exam and her accounting exam. This event can therefore be denoted (A and B). Example 6.1 An investor has asked his stockbroker to rate three stocks (A, B, and C) and list them in the order in which she would recommend them. Consider the following events: L: Stock A doesnt receive the lowest rating. M: Stock B doesnt receive the lowest rating. N: Stock C receives the highest rating. a) Define the random experiment and list the simple events in the sample space. b) List the simple events in each of the events L, M, and N. c) List the simple events belonging to each of the following events: (L or N), (L and M), and d) Is there a pair of mutually exclusive events among L, M, and N? e) Is there a pair of exhaustive events among L, M, and N? 52 Solution a) The random experiment consists of observing the order in which the stockbroker recommends the three stocks. The sample space consists of the set of all possible orderings: S = {ABC, ACB, BAC, BCA, CAB, CBA} b) L = {ABC, ACB, BAC, CAB} M = {ABC, BAC, BCA, CBA} N = {CAB, CBA} c) The event (L or N) consists of all simple events in L or N or both: ( L or N) = {ABC, ACB, BAC, CAB, CBA} The event (L and M) consists of all simple events in both L and M: (L and M) = {ABC, BAC} The complement of M consists of all simple events that do not belong to M: M = {ACB, CAB} d) No, there is not a pai r of m utually exclusive events among L, M, and N, since each pair of events has at least one simple event in common. (L and M) = {ABC, BAC} (L and N) = {CAB} (M and N) = {CBA} e) Yes, L and M are an exhaustive pair of events, since every simple event in the sample space is contained either in L or M, or both. That is, (L or M) = S. Example 6.2 The five top-selling cars in Canada in the 1986 model year are shown below, together with assumed sales levels. One regi stration form is selected at random from a fi le of t he registration forms for the 200,000 cars, and the type of car appearing on the form is observed. 53 Car Sales Level Ford Tempo 50,000 Hy undai Pony 44,000 Pont iac 6000 40,000 C hevrolet Cavalier 34,000 Ch evrolet Celebrity 32,000 200,000 a) List the simple events in the sample space for this random experiment. b) Assign a probability to each simple event. c) Find the probability that each of the following events will occur: L: The form selected is for a car made by Ford. M: The form selected is for a North American car. N: The form selected is not for a car made by General Motors. Solution a) There are five possible simple events that could be observed, distinguished from one another by the five different types of cars that could appear on the registration form selected. Let Tempo represent the simple event that the registration fo rm selected is for a Ford Tem po, and define the other four simple events in a similar manner. Then the sample space is S = {Tempo, Pony, 6000, Cavalier, Celebrity} b) Since each of the 200,000 registration form s has the same chance of being selected and there 50,000 are 50,000 forms for a Tempo, P(Tempo) = = .25 200,000 44,000 Similarly, P(Pony) = 200,000 = .22 40,000 P(6000) = = .20 200,000 34,000 P(Cavalier) = 200,000 = .17 P(Celebrity) = 32,000 = .16 200,000 54
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