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Chapter 2
Probability
Concepts and
Applications
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
2-1
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Learning Objectives
Students will be able to:
• Understand the basic foundations
of probability analysis
• Understand the difference
between mutually exclusive and
collectively exhaustive events
• Describe statistically dependent
and independent events
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
2-2
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Learning Objectives continued
• Describe and provide examples of
both discrete and continuous
random variables
• Explain the difference between
discrete and continuous
probability distributions
• Calculate expected values and
variances
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
2-3
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Introduction
• Life is uncertain!
• We must deal with risk!
• What is the chance that Hurricane
Frances will hit
New Orleans?
• Who will win the election?
• What is the chance that the oil
price will hit $50?
• Is it likely that SLU Faculty will
get a pay raise since the tuition
went up 7%?
• A probability is a numerical value
that measure the likelihood that an
event will occur
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
2-4
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
• How many Christmas trees
should be stored?
• How long will it take to
complete an activity?
• What should be the inventory
level?
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
2-5
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Hurricane Frances
To accompany Quantitative Analysis
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by Render/Stair/Hanna
2-6
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
• The word "statistics" is used in
several different senses. In the
broadest sense, "statistics" refers
to a range of techniques and
procedures for analyzing data,
interpreting data, displaying data,
and making decisions based on
data. This is what courses in
"statistics" generally cover. In a
second usage, a "statistic" is
defined as a numerical quantity
(such as the mean) calculated in a
sample. Such statistics are used to
estimate parameters.
• The term "statistics" sometimes
refers to calculated quantities
regardless of whether or not
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
2-7
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
• they are from a sample. For
example, one might ask about a
baseball player's statistics and be
referring to his or her batting
average, runs batted in, number of
home runs, etc. Or, "government
statistics" can refer to any
numerical indexes calculated by a
governmental agency.
• Although the different meanings
of "statistics" has the potential for
confusion, a careful consideration
of the context in which the word is
used should make its intended
meaning clear.
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
2-8
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Basic Statements About
Probability
1. The probability, P, of any
event or state of nature
occurring is greater than or
equal to 0 and less than or
equal to 1.
That is: 0  P(event)  1
2. The sum of the simple
probabilities for all possible
outcomes of an activity must
equal 1.
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
2-9
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Example 2.1
• Demand for white latex paint at
Diversey Paint and Supply has
always been 0, 1, 2, 3, or 4
gallons per day. (There are no
other possible outcomes; when
one outcome occurs, no other
can.) Over the past 200 days,
the frequencies of demand are
represented in the following
table:
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
2-10
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Example 2.1 - continued
Frequencies of Demand
Quantity
Number of Days
Demanded
(Gallons)
0
40
1
80
2
50
3
20
4
10
Total 200
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by Render/Stair/Hanna
2-11
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Example 2.1 - continued
Probabilities of Demand
Quant.
Freq.
Demand
(days)
Probability
0
40
(40/200) = 0.20
1
80
(80/200) = 0.40
2
50
(50/200) = 0.25
3
20
(20/200) = 0.10
4
10
(10/200) = 0.05
Total
Prob
= 1.00
Total days = 200
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by Render/Stair/Hanna
2-12
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Types of Probability
Objective probability:
P ( event ) =
Number of times event occurs
Total number of outcomes or occurrences
Determined by experiment or
observation:
• Probability of heads on coin flip
• Probably of spades on drawing card
from deck
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by Render/Stair/Hanna
2-13
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Types of Probability
Subjective probability:
Based upon judgement
Determined by:
• judgement of expert
• opinion polls
• etc.
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by Render/Stair/Hanna
2-14
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Mutually Exclusive Events
• Events are said to be mutually
exclusive if only one of the
events can occur on any one
trial
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by Render/Stair/Hanna
2-15
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Collectively Exhaustive
Events
• Events are said to be
collectively exhaustive if the list
of outcomes includes every
possible outcome: heads and
tails as possible outcomes of
coin flip
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by Render/Stair/Hanna
2-16
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Example 2
Rolling a
die has six
possible
outcomes
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
Outcome Probability
of Roll
1
1/6
2
1/6
3
1/6
4
1/6
5
1/6
6
1/6
Total = 1
2-17
© 2003 by Prentice Hall, Inc.,
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Example 2a
Outcome
Probability
Rolling two
of Roll = 5
dice results in Die 1 Die 2
a total of five
1
4
spots
2
3
showing.
3
2
There are a
total of 36
4
1
possible
outcomes.
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for Management, 8e
by Render/Stair/Hanna
2-18
1/36
1/36
1/36
1/36
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Example 3
Draw
Draw a space and a
club
Draw a face card
and a number
card
Draw an ace and a 3
Draw a club and a
nonclub
Draw a 5 and a
diamond
Draw a red card and
a diamond
To accompany Quantitative Analysis
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Mutually Collectively
Exclusive Exhaustive
Yes
No
Yes
Yes
Yes
No
Yes
Yes
No
No
No
No
2-19
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Probability :
Mutually Exclusive
P(event A or event B) =
P(event A) + P(event B)
or:
P(A or B) = P(A) + P(B)
i.e.,
P(spade or club) = P(spade) + P(club)
= 13/52 + 13/52
= 26/52 = 1/2 = 50%
To accompany Quantitative Analysis
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by Render/Stair/Hanna
2-20
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Probability:
Not Mutually Exclusive
P(event A or event B) =
P(event A) + P(event B) P(event A and event B both
occurring)
or
P(A or B) = P(A)+P(B) - P(A and B)
To accompany Quantitative Analysis
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by Render/Stair/Hanna
2-21
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
P(A and B)
(Venn Diagram)
P(A)
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
P(B)
2-22
© 2003 by Prentice Hall, Inc.,
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P(A or B)
-
+
P(A)
P(B)
=
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
P(A and B)
P(A or B)
2-23
© 2003 by Prentice Hall, Inc.,
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Statistical Dependence
• Events are either
• statistically independent (the
occurrence of one event has no
effect on the probability of
occurrence of the other) or
• statistically dependent (the
occurrence of one event gives
information about the occurrence
of the other)
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
2-24
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Which Are Independent?
• (a) Your education
(b) Your income level
• (a) Draw a Jack of Hearts from
a full 52 card deck
(b) Draw a Jack of Clubs from a
full 52 card deck
• (a) Chicago Cubs win the
National League pennant
(b) Chicago Cubs win the World
Series
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
2-25
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Probabilities Independent Events
• Marginal probability: the probability
of an event occurring:
[P(A)]
• Joint probability: the probability of
multiple, independent events,
occurring at the same time
P(AB) = P(A)*P(B)
• Conditional probability (for
independent events):
• the probability of event B given that
event A has occurred P(B|A) = P(B)
• or, the probability of event A given that
event B has occurred P(A|B) = P(A)
To accompany Quantitative Analysis
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by Render/Stair/Hanna
2-26
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Probability(A|B)
Independent Events
P(B)
P(B|A)
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by Render/Stair/Hanna
P(A|B)
2-27
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Statistically
Independent Events
A bucket
contains 3
black balls,
and 7 green
balls. We
draw a ball
from the
bucket,
replace it,
and draw a
second ball
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
1. P(black ball drawn
on first draw)
• P(B) = 0.30
(marginal
probability)
2. P(two green balls
drawn)
• P(GG) =
P(G)*P(G) =
0.70*0.70 =
0.49 (joint
probability for
two independent
events)
2-28
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Statistically Independent
Events - continued
1. P(black ball drawn on second
draw, first draw was green)
• P(B|G) = P(B) = 0.30
(conditional probability)
2. P(green ball drawn on second
draw, first draw was green)
• P(G|G) = 0.70
(conditional probability)
To accompany Quantitative Analysis
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by Render/Stair/Hanna
2-29
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Probabilities Dependent Events
• Marginal probability: probability of
an event occurring P(A)
• Conditional probability (for
dependent events):
• the probability of event B given that
event A has occurred P(B|A) =
P(AB)/P(A)
• the probability of event A given that
event B has occurred P(A|B) =
P(AB)/P(B)
To accompany Quantitative Analysis
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by Render/Stair/Hanna
2-30
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Probability(A|B)
/
P(A)
P(AB)
P(B)
P(A|B) = P(AB)/P(B)
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by Render/Stair/Hanna
2-31
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Probability(B|A)
/
P(B)
P(AB)
P(A)
P(B|A) = P(AB)/P(A)
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
2-32
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Statistically Dependent
Events
Then:
Assume that we
• P(WL) = 4/10 = 0.40
have an urn
• P(WN) = 2/10 = 0.20
containing 10 balls
of the following
• P(W) = 6/10 = 0.60
descriptions:
• P(YL) = 3/10 = 0.3
•4 are white (W)
and lettered (L)
•2 are white (W)
and numbered N
•3 are yellow (Y)
and lettered (L)
•1 is yellow (Y)
and numbered (N)
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by Render/Stair/Hanna
• P(YN) = 1/10 = 0.1
• P(Y) = 4/10 = 0.4
2-33
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Upper Saddle River, NJ 07458
Statistically Dependent
Events - Continued
Then:
• P(L|Y) = P(YL)/P(Y)
= 0.3/0.4 = 0.75
• P(Y|L) = P(YL)/P(L)
= 0.3/0.7 = 0.43
• P(W|L) = P(WL)/P(L)
= 0.4/0.7 = 0.57
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
2-34
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Joint Probabilities,
Dependent Events
Your stockbroker informs you that if
the stock market reaches the 10,500
point level by January, there is a
70% probability the Tubeless
Electronics will go up in value.
Your own feeling is that there is
only a 40% chance of the market
reaching 10,500 by January.
What is the probability that both the
stock market will reach 10,500
points, and the price of Tubeless
will go up in value?
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by Render/Stair/Hanna
2-35
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Joint Probabilities, Dependent
Events - continued
Let M represent
the event of the
Then:
stock market
P(MT) =P(T|M)P(M)
reaching the
= (0.70)(0.40)
10,500 point
= 0.28
level, and T
represent the
event that
Tubeless goes
up.
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by Render/Stair/Hanna
2-36
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Revising Probabilities:
Bayes’ Theorem
Bayes’ theorem can be used to
calculate revised or posterior
probabilities
Prior
Probabilities
Bayes’
Process
Posterior
Probabilities
New
Information
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by Render/Stair/Hanna
2-37
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
General Form of
Bayes’ Theorem
P( AB)
P( A | B) =
P( B)
or
P( B | A) P( A)
P( A | B) =
P( B | A) P( A)  P( B | A ) P( A )
where A = complement of the event A
For example, if the event A is " fair" die,
then the event A is " unfair" die.
To accompany Quantitative Analysis
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by Render/Stair/Hanna
2-38
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Posterior Probabilities
A cup contains two dice identical in
appearance. One, however, is fair
(unbiased), the other is loaded
(biased). The probability of rolling a
3 on the fair die is 1/6 or 0.166. The
probability of tossing the same
number on the loaded die is 0.60.
We have no idea which die is which,
but we select one by chance, and toss
it. The result is a 3.
What is the probability that the die
rolled was fair?
To accompany Quantitative Analysis
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by Render/Stair/Hanna
2-39
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Posterior Probabilities
Continued
• We know that:
P(fair) = 0.50
P(loaded) = 0.50
• And:
P(3|fair) = 0.166
P(3|loaded) =
0.60
• Then:
P(3 and fair) = P(3|fair)P(fair)
= (0.166)(0.50)
= 0.083
P(3 and loaded) = P(3|loaded)P(loaded)
= (0.60)(0.50)
= 0.300
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
2-40
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Posterior Probabilities
Continued
• A 3 can occur in combination with
the state “fair die” or in combination
with the state ”loaded die.” The sum
of their probabilities gives the
unconditional or marginal probability
of a 3 on a toss:
P(3) = 0.083 + 0.0300 = 0.383.
• Then, the probability that the die
rolled was the fair one is given by:
P(Fair | 3) =
P(Fair and 3) 0.083
=
= 0.22
P(3)
0.383
To accompany Quantitative Analysis
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by Render/Stair/Hanna
2-41
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Further Probability
Revisions
• To obtain further information as
to whether the die just rolled is
fair or loaded, let’s roll it again.
• Again we get a 3.
Given that we have now rolled
two 3s, what is the probability
that the die rolled is fair?
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by Render/Stair/Hanna
2-42
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Further Probability
Revisions - continued
P(fair) = 0.50, P(loaded) = 0.50 as before
P(3,3|fair) = (0.166)(0.166) = 0.027
P(3,3|loaded) = (0.60)(0.60) = 0.36
P(3,3 and fair) = P(3,3|fair)P(fair)
= (0.027)(0.05)
= 0.013
P(3,3 and loaded) = P(3,3|loaded)P(loaded)
= (0.36)(0.5)
= 0.18
P(3,3) = 0.013 + 0.18 = 0.193
To accompany Quantitative Analysis
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by Render/Stair/Hanna
2-43
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Further Probability
Revisions - continued
P(3,3 and Fair)
P(Fair | 3,3) =
P(3,3)
0.013
=
= 0.067
0.193
P(3,3 and Loaded)
P(Loaded | 3,3) =
P(3,3)
0.18
=
= 0.933
0.193
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by Render/Stair/Hanna
2-44
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Further Probability
Revisions - continued
To give the final comparison:
P(fair|3) = 0.22
P(loaded|3) = 0.78
P(fair|3,3) = 0.067
P(loaded|3,3) = 0.933
To accompany Quantitative Analysis
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by Render/Stair/Hanna
2-45
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Random Variables
• Discrete random variable - can
assume only a finite or limited
set of values- i.e., the number of
automobiles sold in a year
• Continuous random variable -
can assume any one of an
infinite set of values - i.e.,
temperature, product lifetime
To accompany Quantitative Analysis
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by Render/Stair/Hanna
2-46
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Random Variables
(Numeric)
Experiment
Outcome
Random Variable
Range of
Random
Variable
Stock 50
Xmas trees
Number of
trees sold
X = number of
trees sold
Inspect 600
items
Number
acceptable
Y = number
acceptable
0,1,2,…,
600
Send out
5,000 sales
letters
Number of
peoplee
responding
Z = number of
people responding
0,1,2,…,
5,000
Build an
apartment
building
%completed
after 4
months
R = %completed
after 4 months
0R 100
Test the
lifetime of a
light bulb
(minutes)
Time bulb
lasts - up to
80,000
minutes
S = time bulb
burns
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2-47
0,1,2,, 50
0S80,000
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Random
Variables (Non-numeric)
Experiment
Outcome
Random
Variable
Range of
Random
Variable
Students
Strongly agree (SA)
X = 5 if SA
1,2,3,4,5
respond to a Agree (A)
4 if A
questionnaire Neutral (N)
3 if N
Disagree (D)
2 if D
Strongly Disagree (SD)
1 if SD
One machine is Defective
Y = 0 if defective
0,1
inspected
Not defective
1 if not defective
Consumers
Good
Z = 3 if good
1,2,3
respond to how Average
2 if average
they like a
Poor
1 if poor
product
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by Render/Stair/Hanna
2-48
© 2003 by Prentice Hall, Inc.,
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Probability Distributions
Vehicles Owned
X
P(X)
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by Render/Stair/Hanna
0
0.005
1
0.435
2-49
2
0.555
3
0.206
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Expected Value of a
Discrete Probability
Distribution
n
E ( X ) =  X i P( X i )
i =1
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by Render/Stair/Hanna
2-50
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Variance of a Discrete
Probability Distribution
n
 =  X i  E  X  P X i 
2
2
i =1
 2 = 5  2.92 0.1  4  2.92 0.2
 3  2.9 0.3  (2 - 2.9) 2 (0.3)
2
 (1  2.9) 2 (0.1)
= 0.44 - 0.242  0.003  0.243  0.361
= 1.29
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by Render/Stair/Hanna
2-51
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Binomial Distribution
Assumptions:
1. Trials follow Bernoulli process
– two possible outcomes
2. Probabilities stay the same from
one trial to the next
3. Trials are statistically
independent
4. Number of trials is a positive
integer
To accompany Quantitative Analysis
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by Render/Stair/Hanna
2-52
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Binomial Distribution
n = number of trials
r = number of successes
p = probability of success
q = probability of failure
Probability of r successes
in n trials
n!
r nr
=
pq
r!(n - r)!
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by Render/Stair/Hanna
2-53
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Binomial Distribution
 = np
  = np( 1  p )
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by Render/Stair/Hanna
2-54
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Binomial Distribution
N = 5, p = 0.50
0.35
0.30
P(r)
0.25
0.20
0.15
0.10
0.05
0.00
1
2
3
4
5
6
(r) Number of Successes
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by Render/Stair/Hanna
2-55
© 2003 by Prentice Hall, Inc.,
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Example
Seventy percent of married couples
paid for their honeymoon
themselves. You randomly select
10 married couples and ask each
if they paid for their honeymoon
themselves. Find the probability
that the number of couples who
say they paid for their
honeymoon themselves is
a. Exactly seven
b. at least 7
c. Less than seven
To accompany Quantitative Analysis
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by Render/Stair/Hanna
2-56
© 2003 by Prentice Hall, Inc.,
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Probability Distribution
Continuous Random
Variable
Normal Distribution
Probability density function - f(X)
5
5.05
5.1
f (X ) =
5.15
5.2
5.3
5.35
1 / 2 ( X   ) 2
1
 2
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
5.25
2-57
e
2
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Upper Saddle River, NJ 07458
5.4
Normal Distribution for
Different Values of 
=40
=50
=60
40
50
60
0
30
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
2-58
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
70
Normal Distribution for
Different Values of 
=1
=0.1
=0.3
0
0.5
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
=0.2
1
2-59
1.5
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
2
Three Common Areas
Under the Curve
• Three Normal distributions with
different areas
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
2-60
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Three Common Areas
Under the Curve
Three
Normal
distributions
with
different
areas
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
2-61
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
The Relationship Between
Z and X
=100
=15
x
Z=

55
70
85
100
115
130
145
-3
-2
-1
0
1
2
3
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
2-62
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Haynes Construction
Company Example
Fig. 2.12
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
2-63
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Haynes Construction
Company Example
Fig. 2.13
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
2-64
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Haynes Construction
Company Example
Fig. 2.14
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
2-65
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
The Negative
Exponential
Distribution
f ( X ) = e
 x Expected value = 1/
Variance = 1/2
6
5
=5
4
3
2
1
0
0
0.2
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
0.4
0.6
2-66
0.8
1
1.2
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
The Poisson
Distribution
e
x 
P( X ) =
X!
Expected value = 
Variance = 
0.30
=2
0.25
0.20
0.15
0.10
0.05
0.00
1
2
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
3
4
2-67
5
6
7
8
9
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458
Hmwwork Assignment
September 7
2-30, 2-32, 2-34, 2-38, 2-41 0n
Page 70
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
2-68
© 2003 by Prentice Hall, Inc.,
Upper Saddle River, NJ 07458