1. 5- 1
Chapter Five
A Survey of Probability Concepts
GOALS
When you have completed this chapter, you will be able to:
ONE
Define probability.
TWO
Describe the classical, empirical, and subjective approaches to
probability.
THREE
Understand the terms: experiment, event, outcome, permutations, and
combinations.
FOUR
Define the terms: conditional probability and joint probability.
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2. 5- 2
Chapter Five continued
A Survey of Probability Concepts
GOALS
When you have completed this chapter, you will be able to:
FIVE
Calculate probabilities applying the rules of addition and the
rules of multiplication.
SIX
Use a tree diagram to organize and compute probabilities.
SEVEN
Calculate a probability using Bayes’ theorem.
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3. 5- 3
Definitions
A probability is a measure of the likelihood that an event
in the future will happen.
It it can only assume a value between 0 and 1.
A value near zero means the event is not likely to
happen. A value near one means it is likely.
There are three definitions of probability: classical,
empirical, and subjective.
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4. 5- 4
Definitions continued
 The classical definition applies when there are n equally
likely outcomes.
 The empirical definition applies when the number of
times the event happens is divided by the number of
observations.
 Subjective probability is based on whatever information
is available.
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5. 5- 5
Definitions continued
An experiment is the observation of some activity
or the act of taking some measurement.
An outcome is the particular result of an
experiment.
An event is the collection of one or more
outcomes of an experiment.
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6. 5- 6
Mutually Exclusive Events
Events are mutually exclusive if the occurrence
of any one event means that none of the others
can occur at the same time.
Events are independent if the occurrence of one
event does not affect the occurrence of another.
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7. 5- 7
Collectively Exhaustive Events
Events are collectively exhaustive if at least one
of the events must occur when an experiment is
conducted.
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8. 5- 8
Example 1
A fair die is rolled once.
The experiment is rolling the die.
The possible outcomes are the numbers 1, 2,
3, 4, 5, and 6.
An event is the occurrence of an even
number. That is, we collect the outcomes 2,
4, and 6.
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9. 5- 9
EXAMPLE 2
Throughout her teaching career Professor Jones has
awarded 186 A’s out of 1,200 students. What is the
probability that a student in her section this semester
will receive an A?
 This is an example of the empirical definition of
probability.
 To find the probability a selected student earned an A:
186
P( A) = = 0.155
1200
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10. 5- 10
Subjective Probability
Examples of subjective probability are:
 estimating the probability the Washington
Redskins will win the Super Bowl this year.
 estimating the probability mortgage rates for home
loans will top 8 percent.
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11. 5- 11
Basic Rules of Probability
If two events A and B are mutually exclusive,
the special rule of addition states that the
probability of A or B occurring equals the sum
of their respective probabilities:
P(A or B) = P(A) + P(B)
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12. 5- 12
EXAMPLE 3
 New England Commuter Airways recently
supplied the following information on their
commuter flights from Boston to New York:
Arrival Frequency
Early 100
On Time 800
Late 75
Canceled 25
Total 1000
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13. 5- 13
EXAMPLE 3 continued
 If A is the event that a flight arrives early, then
P(A) = 100/1000 = .10.
IfB is the event that a flight arrives late, then
P(B) = 75/1000 = .075.
The probability that a flight is either early or late
is:
P(A or B) = P(A) + P(B) = .10 + .075 =.175.
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14. 5- 14
The Complement Rule
The complement rule is used to determine the
probability of an event occurring by subtracting
the probability of the event not occurring from 1.
If P(A) is the probability of event A and P(~A) is
the complement of A,
P(A) + P(~A) = 1 or P(A) = 1 - P(~A).
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15. 5- 15
The Complement Rule continued
A Venn diagram illustrating the complement rule
would appear as:
A ~A
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16. 5- 16
EXAMPLE 4
Recall EXAMPLE 3. Use the complement rule to find
the probability of an early (A) or a late (B) flight
IfC is the event that a flight arrives on time, then P(C) =
800/1000 = .8.
If D is the event that a flight is canceled, then P(D) =
25/1000 = .025.
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17. 5- 17
EXAMPLE 4 continued
P(A or B) = 1 - P(C or D)
= 1 - [.8 +.025] =.175
D
C
.025
.8
~(C or D) = (A or B)
.175
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18. 5- 18
The General Rule of Addition
If A and B are two events that are not mutually
exclusive, then P(A or B) is given by the
following formula:
P(A or B) = P(A) + P(B) - P(A and B)
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19. 5- 19
The General Rule of Addition
The Venn Diagram illustrates this rule:
B
A and B
A
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20. 5- 20
EXAMPLE 5
In a sample of 500 students, 320 said they had
a stereo, 175 said they had a TV, and 100 said
they had both:
TV
175
Both
Stereo 100
320
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21. 5- 21
EXAMPLE 5 continued
 If a student is selected at random, what is the
probability that the student has only a stereo, only a
TV, and both a stereo and TV?
P(S) = 320/500 = .64.
P(T) = 175/500 = .35.
P(S and T) = 100/500 = .20.
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22. 5- 22
EXAMPLE 5 continued
 If a student is selected at random, what is the probability
that the student has either a stereo or a TV in his or her
room?
P(S or T) = P(S) + P(T) - P(S and T)
= .64 +.35 - .20 = .79.
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23. 5- 23
Joint Probability
A joint probability measures the likelihood that two or
more events will happen concurrently.
 An example would be the event that a student has both a
stereo and TV in his or her dorm room.
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24. 5- 24
Special Rule of Multiplication
The special rule of multiplication requires that two
events A and B are independent.
Two events A and B are independent if the
occurrence of one has no effect on the probability of
the occurrence of the other.
This rule is written: P(A and B) = P(A)P(B)
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25. 5- 25
EXAMPLE 6
Chris owns two stocks, IBM and General Electric
(GE). The probability that IBM stock will increase
in value next year is .5 and the probability that GE
stock will increase in value next year is .7.
Assume the two stocks are independent. What is
the probability that both stocks will increase in
value next year?
P(IBM and GE) = (.5)(.7) = .35.
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26. 5- 26
EXAMPLE 6 continued
 What is the probability that at least one of these
stocks increase in value during the next year?
(This means that either one can increase or both.)
P(at least one) = (.5)(.3) + (.5)(.7) +(.7)(.5)
= .85.
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27. 5- 27
Conditional Probability
A conditional probability is the probability of a
particular event occurring, given that another
event has occurred.
The probability of the event A given that the
event B has occurred is written P(A|B).
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28. 5- 28
General Multiplication Rule
The general rule of multiplication is used to find
the joint probability that two events will occur.
It states that for two events A and B, the joint
probability that both events will happen is found by
multiplying the probability that event A will happen
by the conditional probability of B given that A has
occurred.
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29. 5- 29
General Multiplication Rule
The joint probability, P(A and B) is given by
the following formula:
P(A and B) = P(A)P(B/A)
or
P(A and B) = P(B)P(A/B)
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30. 5- 30
EXAMPLE 7
The Dean of the School of Business at Owens
University collected the following information
about undergraduate students in her college:
MAJOR Male Female Total
Accounting 170 110 280
Finance 120 100 220
Marketing 160 70 230
Management 150 120 270
Total 600 400 1000
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31. 5- 31
EXAMPLE 7 continued
If a student is selected at random, what is the
probability that the student is a female (F)
accounting major (A)
P(A and F) = 110/1000.
Given that the student is a female, what is the
probability that she is an accounting major?
P(A|F) = P(A and F)/P(F)
= [110/1000]/[400/1000] = .275
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32. 5- 32
Tree Diagrams
A tree diagram is useful for portraying conditional
and joint probabilities. It is particularly useful for
analyzing business decisions involving several
stages.
EXAMPLE 8: In a bag containing 7 red chips and 5
blue chips you select 2 chips one after the other without
replacement. Construct a tree diagram showing this
information.
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33. 5- 33
EXAMPLE 8 continued
6/11 R2
7/12 R1
5/11 B2
7/11 R2
5/12 B1
4/11 B2
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34. 5- 34
Bayes’ Theorem
 Bayes’ Theorem is a method for revising a probability
given additional information.
 It is computed using the following formula:
P( A1 ) P( B / A1 )
P( A1 | B) =
P( A1 ) P( B / A1 ) + P( A2 ) P( B / A2 )
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35. 5- 35
EXAMPLE 9
Duff Cola Company recently received several
complaints that their bottles are under-filled. A
complaint was received today but the production
manager is unable to identify which of the two
Springfield plants (A or B) filled this bottle.
What is the probability that the under-filled bottle
came from plant A?
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36. 5- 36
EXAMPLE 9 continued
The following table summarizes the Duff
production experience.
% of Total % of under-
Production filled bottles
A 55 3
B 45 4
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37. 5- 37
Example 9 continued
P ( A) P (U / A)
P( A / U ) =
P ( A) P (U / A) + P ( B ) P (U / B )
.55(.03)
= = .4783
.55(.03) +.45(.04)
The likelihood the bottle was filled in Plant A
is reduced from .55 to .4783.
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38. 5- 38
Some Principles of Counting
The multiplication formula indicates that if there
are m ways of doing one thing and n ways of doing
another thing, there are m x n ways of doing both.
Example 10: Dr. Delong has 10 shirts and 8
ties. How many shirt and tie outfits does he
have?
(10)(8) = 80
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39. 5- 39
Some Principles of Counting
A permutation is any arrangement of r objects selected
from n possible objects.
 Note: The order of arrangement is important in
permutations.
n!
n Pr =
(n − r )!
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40. 5- 40
Some Principles of Counting
A combination is the number of ways to choose r
objects from a group of n objects without regard
to order.
n!
nCr =
r! (n − r )!
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41. 5- 41
EXAMPLE 11
There are 12 players on the Carolina Forest High
School basketball team. Coach Thompson must
pick five players among the twelve on the team to
comprise the starting lineup. How many different
groups are possible?
12!
12C 5 = = 792
5! (12 − 5)!
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42. 5- 42
Example 11 continued
Suppose that in addition to selecting the group, he
must also rank each of the players in that starting
lineup according to their ability.
12!
12 P 5 = = 95,040
(12 − 5)!
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