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Chapter 06

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Operations Management and Information System

OMIS 2010

Alan Marshall

Fall

Description

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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