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Chapter 06

# OMIS 2010 Chapter Notes - Chapter 06: Ford Tempo, Mutual Exclusivity, Probability Distribution

by OC9945

School

York UniversityDepartment

Operations Management and Information SystemCourse Code

OMIS 2010Professor

Alan MarshallChapter

06Chapter 6: Probability

6.1 Introduction

This chapter introduced the basic concepts of probability. It outlined rules and techniques for

assigning probabilities to events. At the completion of this chapter, you are 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 the 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 then 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 reword the statement of a given event so

that it conforms with one of the two expressions given above. For example,

suppose your friend Karen is 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 she’ll pass the accounting exam,

or she’ll 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 doesn’t receive the lowest rating.

M: Stock B doesn’t 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 .M

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 pair of mutually 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 registration form is selected at random from a file of the registration forms for the

200,000 cars, and the type of car appearing on the form is observed.

53

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###### Document Summary

This chapter introduced the basic concepts of probability. It outlined rules and techniques for assigning probabilities to events. How to recognize when it is appropriate to use a poisson distribution, and how to use the table of. 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.

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