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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using
Microsoft Excel
Basic Probability &
Discrete Probability Distributions
Chapter 4




4-1
Define experiment, outcome, event, sample
space, & probability
Use a contingency table to find probabilities
Describe 4 discrete probability distributions
Find the probability of discrete random
variables
Statistics for Managers Using Microsoft Excel, 1/e
Thinking Challenge
© 1998 Prentice-Hall, Inc.
What’s the probability of
getting a head on the
toss of a single fair coin?
Use a scale from 0 (no
way) to 1 (sure thing).
So toss a coin twice. Do
it! Did you get one head
& one tail? What’s it all
mean?
4-2
Statistics for Managers Using Microsoft Excel, 1/e
Many Repetitions!*
© 1998 Prentice-Hall, Inc.
Total Heads
Number of Tosses
1.00
0.75
0.50
0.25
0.00
0
25
50
75
100
125
Number of Tosses
4-3
Statistics for Managers Using Microsoft Excel, 1/e
Introduction to Probability
© 1998 Prentice-Hall, Inc.

Experiment


Outcome

4-4
Process of obtaining an observation,
outcome or simple event
Result of an experiment
Statistics for Managers Using Microsoft Excel, 1/e
Experiments & Outcomes
© 1998 Prentice-Hall, Inc.

Experiment


Outcome


Result of an experiment
Sample space (S)


4-5
Process of obtaining an observation,
outcome or simple event
Sample space
depends on
experimenter!
Collection of all possible outcomes
Defined by experimenter
Statistics for Managers Using Microsoft Excel, 1/e
Outcome Examples
© 1998 Prentice-Hall, Inc.
Experiment
Sample Space
Toss a coin, note face
Head, tail
Toss 2 coins, note faces
HH, HT, TH, TT
Select 1 card, note kind
2, 2, ..., A (52)
Select 1 card, note color
Red, black
Play a football game
Win, lose, tie
Inspect a part, note quality
Defective, good
Observe gender
Male, female
4-6
Statistics for Managers Using Microsoft Excel, 1/e
Outcome Properties
© 1998 Prentice-Hall, Inc.

Mutually exclusive


Experiment:
Observe gender
Collectively exhaustive

4-7
2 outcomes can not occur
at the same time
 Example: Both male &
female in same person
1 outcome in sample space
must occur
 Example: Male or female
© 1984-1994 T/Maker Co.
Statistics for Managers Using Microsoft Excel, 1/e
Events
© 1998 Prentice-Hall, Inc.

Any collection of outcomes

Simple event


Outcome with 1 characteristic
Compound event



Collection of outcomes or simple events
2 or more characteristics
Joint event: special case

4-8
2 events occurring simultaneously
Statistics for Managers Using Microsoft Excel, 1/e
Event Examples
© 1998 Prentice-Hall, Inc.
Experiment: Toss 2 coins. Note faces.
Event
Outcomes in Event
Sample space
HH, HT, TH, TT
1 head & 1 tail
HT, TH
Heads on 1st coin HH, HT
At least 1 head
HH, HT, TH
Heads on both
HH
4-9
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.

Listing


Visualizing
Sample Space
S = {Head, Tail}
Contingency Table
4 - 10
Statistics for Managers Using Microsoft Excel, 1/e
Contingency Table
© 1998 Prentice-Hall, Inc.
Experiment: Toss 2 coins. Note faces.
nd
2 Coin
st
Head
Tail
Total
Head
HH
HT
HH, HT
Tail
TH
TT
TH, TT
1 Coin
Simple
event
(head on
1st coin)
Total
HH, TH HT, TT
S = {HH, HT, TH, TT}
4 - 11
Outcome
(% or
count)
S
Sample space
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Joint Events
(Event A and Event B)
Experiment: Draw 1 card. Note kind, color & suit.
Sample
space (S):
2R, 2R,
2B, ..., AB
Joint event
Ace AND
Black:
AB, AB
4 - 12
Color
Type
Ace
Non-Ace
Total
Red
Black Total
Ace &
Red
Non &
Red
Red
Ace & Ace
Black
Non & NonBlack Ace
Black
S
Simple
event
Ace:
AR,
AR,
AB,
AB
Simple event black: 2B, ..., AB
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Either-Or Events
(Event A or Event B)
Experiment: Draw 1 card. Note kind, color & suit.
Sample
space (S):
2R, 2R,
2B, ..., AB
Color
Type
Ace
Ace &
Red
Non-Ace Non &
Red
Total
Red
Joint event
Ace OR
Black:
AR, ..., AB, 2B, ..., KB
4 - 13
Red
Black Total
Ace & Ace
Black
Non & NonBlack Ace
Black S
Simple
event
Ace:
AR,
AR,
AB,
AB
Simple event Black: 2B, ..., AB
Statistics for Managers Using Microsoft Excel, 1/e
Special Events
© 1998 Prentice-Hall, Inc.

Null event

Club & Diamond on
1 card draw
Null Event

Q

Complement of event

For event A, all
events not in A: A

Mutually exclusive event

4 - 14
Events do not occur
simultaneously
Q
Statistics for Managers Using Microsoft Excel, 1/e
What is Probability?
© 1998 Prentice-Hall, Inc.

Numerical measure
of likelihood that
event will occur



4 - 15
P(Event)
P(A)
Prob(A)
Statistics for Managers Using Microsoft Excel, 1/e
What is Probability?
© 1998 Prentice-Hall, Inc.

Numerical measure
of likelihood that
event will occur



P(Event)
P(A)
Prob(A)

Lies between 0 & 1

Sum of events is 1
1
Certain
.5
0
Impossible
© 1984-1994 T/Maker Co.
4 - 16
Statistics for Managers Using Microsoft Excel, 1/e
Assigning Event Probabilities
© 1998 Prentice-Hall, Inc.

a priori classical
method

Empirical classical
method

Subjective method
4 - 17
Statistics for Managers Using Microsoft Excel, 1/e
a priori Classical Method
© 1998 Prentice-Hall, Inc.

Prior knowledge of
process

Before experiment

P(Event) = X / T



4 - 18
X = No. of event outcomes
T = Total outcomes in sample space
Each of T outcomes is equally likely
 P(Outcome) = 1/T
© 1984-1994 T/Maker Co.
Statistics for Managers Using Microsoft Excel, 1/e
Empirical Classical Method
© 1998 Prentice-Hall, Inc.

Actual data collected

After experiment

P(Event) = X / T



Repeat experiment
T times
Event observed X times
Of 100 parts
inspected, only
2 defects!
Also called relative
frequency method
4 - 19
Statistics for Managers Using Microsoft Excel, 1/e
Subjective Method
© 1998 Prentice-Hall, Inc.

Individual knowledge
of situation

Before experiment

Unique process


© 1984-1994 T/Maker Co.
Not repeatable
Different probabilities
from different people
4 - 20
Statistics for Managers Using Microsoft Excel, 1/e
Thinking Challenge
© 1998 Prentice-Hall, Inc.
Which method should be used to find the
probability...

that a box of 24 bolts will be defective?

that a toss of a coin will be a tail?

that Tom will default on his PLUS loan?

that a student will earn an ‘A’ in this class?

that a new store on RTE. 1 will succeed?
4 - 21
Statistics for Managers Using Microsoft Excel, 1/e
Joint Event Probability
© 1998 Prentice-Hall, Inc.

Numerical measure of likelihood that
joint event will occur

Can often use contingency table


2 variables only
Formula methods



4 - 22
Addition rule
Conditional probability formula
Multiplication rule
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Event Probability Using
Contingency Table
Event
Event
B1
Total
A1
P(A1 and B1) P(A1 and B2)
P(A1)
A2
P(A2 and B1) P(A2 and B2)
P(A2)
Total
P(B1)
Joint Probability
4 - 23
B2
P(B2)
1
Marginal (Simple) Probability
Statistics for Managers Using Microsoft Excel, 1/e
Contingency Table Example
© 1998 Prentice-Hall, Inc.
Experiment: Draw 1 card. Note kind, color & suit.
Color
Type
Ace
Black
2/52
2/52
Total
4/52
Non-Ace 24/52
24/52 48/52
26/52
26/52 52/52
Total
P(Red)
4 - 24
Red
P(Ace)
P(Ace AND Red)
Statistics for Managers Using Microsoft Excel, 1/e
Thinking Challenge
© 1998 Prentice-Hall, Inc.
What’s the probability?
P(A) =
P(D) =
P(C and B) =
P(A or D) =
P(B and D) =
4 - 25
Event
A
Event
C
D
4
2
Total
6
B
1
3
4
Total
5
5
10
Statistics for Managers Using Microsoft Excel, 1/e
Solution*
© 1998 Prentice-Hall, Inc.
The probabilities are:
P(A) = 6/10
P(D) = 5/10
P(C and B) = 1/10
P(A or D) = 9/10
P(B and D) = 3/10
4 - 26
Event
A
Event
C
D
4
2
Total
6
B
1
3
4
Total
5
5
10
Statistics for Managers Using Microsoft Excel, 1/e
Addition Rule
© 1998 Prentice-Hall, Inc.

P(A or B) = P(A) + P(B) - P(A and B)

For mutually exclusive events:
P(A or B) = P(A) + P(B)
4 - 27
Statistics for Managers Using Microsoft Excel, 1/e
Addition Rule Example
© 1998 Prentice-Hall, Inc.
Experiment: Draw 1 card. Note kind, color & suit.
Color
Red
Black Total
Type
Ace
2
2
4
Non-Ace
24
24
48
Total
26
26
52
P(Ace OR Black) = P(Ace)+P(Black) - P(Ace AND Black)
4
26
2
28 ( 2  2  24)
 



52 52 52 52
52
4 - 28
Statistics for Managers Using Microsoft Excel, 1/e
Thinking Challenge
© 1998 Prentice-Hall, Inc.
Using the Addition Rule, what’s the
probability?
P(A or D) =
P(B or C) =
4 - 29
Event
A
Event
C
D
4
2
Total
6
B
1
3
4
Total
5
5
10
Statistics for Managers Using Microsoft Excel, 1/e
Solution*
© 1998 Prentice-Hall, Inc.
Using the Addition Rule, the probabilities
are:
P(A or D) = P(A) + P(D) - P(A and D)
6
5
2
9




10 10 10 10
P(B or C) = P(B) + P(C) - P(B and C)
4
5
1
8




10 10 10 10
4 - 30
Statistics for Managers Using Microsoft Excel, 1/e
Conditional Probability
© 1998 Prentice-Hall, Inc.

Event probability given that another
event occurred

Revise original sample space to account
for new information


Eliminates certain outcomes
P(A | B) = P(A and B)
P(B)
4 - 31
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Conditional Probability Using
Contingency Table
Experiment: Draw 1 card. Note kind,
color & suit.
Color
Type
Red
Black Total
Ace
2
2
4
Non-Ace
24
24
48
Total
26
26
52
Revised
sample
space
P(Ace and Black) 2 / 52
2
P(Ace | Black) =


P(Black)
26 / 52 26
4 - 32
Statistics for Managers Using Microsoft Excel, 1/e
Statistical Independence
© 1998 Prentice-Hall, Inc.

Event occurrence
does not affect
probability of another
event

e.g., toss 1 coin twice

Causality not implied

Tests for independence


4 - 33
P(A | B) = P(A)
P(A and B) = P(A)*P(B)
Statistics for Managers Using Microsoft Excel, 1/e
Thinking Challenge
© 1998 Prentice-Hall, Inc.
Using the table then the formula, what’s
the probability?
P(A|D) =
P(C|B) =
Are C & B
independent?
4 - 34
Event
A
Event
C
D
4
2
Total
6
B
1
3
4
Total
5
5
10
Statistics for Managers Using Microsoft Excel, 1/e
Solution*
© 1998 Prentice-Hall, Inc.
Using the formula, the probabilities are:
P(A | D) =
P(A and D) 2 / 10 2


P(D)
5 / 10 5
P(C and B) 1 / 10 1
P(C | B) =


P(B)
4 / 10 4
5
1
P(C) =

10 4
4 - 35
Dependent
Statistics for Managers Using Microsoft Excel, 1/e
Multiplication Rule
© 1998 Prentice-Hall, Inc.

P(A and B) = P(A)*P(B|A)
= P(B)*P(A| B)

For independent events:
P(A and B) = P(A)*P(B)
4 - 36
Statistics for Managers Using Microsoft Excel, 1/e
Multiplication Rule Example
© 1998 Prentice-Hall, Inc.
Experiment: Draw 1 card. Note kind,
color & suit.
Color
Type
Red
Black Total
Ace
2
2
4
Non-Ace
24
24
48
Total
26
26
52
P(Ace and Black) = P(Ace)  P(Black | Ace)
4   2
2


       
 52   4
 52
4 - 37
Statistics for Managers Using Microsoft Excel, 1/e
Thinking Challenge
© 1998 Prentice-Hall, Inc.
Using the Multiplication Rule, what’s the
probability?
P(C and B) =
P(B and D) =
P(A and B) =
4 - 38
Event
A
Event
C
D
4
2
Total
6
B
1
3
4
Total
5
5
10
Statistics for Managers Using Microsoft Excel, 1/e
Solution*
© 1998 Prentice-Hall, Inc.
Using the Multiplication Rule, the
probabilities are:
 5    1  1
P(C and B) = P(C)  P(B|C) =
 10  5 10
P(B and D) = P(B)  P(D|B) =
 4    3  3
 10  4 10
P(A and B) = P(A)  P(B|A)  0
4 - 39
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Discrete
Random Variables
4 - 40
Statistics for Managers Using Microsoft Excel, 1/e
Thinking Challenge
© 1998 Prentice-Hall, Inc.
You’re taking a 33
question multiple choice
test. Each question has 4
choices. Clueless on 1
question, you decide to
guess. What’s the
chance you’ll get it right?
If you guessed on all 33
questions, what would be
your grade? Pass?
4 - 41
Statistics for Managers Using Microsoft Excel, 1/e
Data Types
© 1998 Prentice-Hall, Inc.
Data
Discrete
4 - 42
Numerical
Categorical
(Quantitative)
(Qualitative)
Continuous
Statistics for Managers Using Microsoft Excel, 1/e
Random Variable
© 1998 Prentice-Hall, Inc.

A numerical outcome of an experiment

Number of tails in 2 coin tosses


Observe 0, 1, or 2 tails
Discrete random variable



Whole number (0, 1, 2, 3 etc.)
Obtained by counting
Usually finite number of values

4 - 43
Poisson random variable is exception ()
Statistics for Managers Using Microsoft Excel, 1/e
Discrete Random Variable Examples
© 1998 Prentice-Hall, Inc.
Experiment
Random
Variable
Possible
Values
Make 100 sales calls
# Sales
Inspect 70 radios
# Defective 0, 1, 2, ..., 70
Answer 33 questions
# Correct
Count cars at toll
# Cars
between 11:00 & 1:00 arriving
4 - 44
0, 1, 2, ..., 100
0, 1, 2, ..., 33
0, 1, 2, ..., 
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.

Discrete
Probability Distribution
List of all possible [ Xi, P(Xi) ] pairs


Xi = Value of random variable (outcome)
P(Xi) = Probability associated with value

Mutually exclusive (no overlap)

Collectively exhaustive (nothing left out)

0  P(Xi)  1

 P(Xi) = 1
4 - 45
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Discrete Probability Distribution
Example
Experiment: Toss 2 coins. Count # tails.
Probability Distribution
Values, Xi Probabilities, P(Xi)
0
1/4 = .25
1
2/4 = .50
2
1/4 = .25
© 1984-1994 T/Maker Co.
4 - 46
Statistics for Managers Using Microsoft Excel, 1/e
Visualizing Discrete Probability
Distributions
© 1998 Prentice-Hall, Inc.
Table
Listing
# Tails
f(Xi)
Count
P(Xi)
0
1
2
1
2
1
.25
.50
.25
{ (0, .25), (1, .50), (2, .25) }
Graph
P(X)
.50
Equation
n!
P( X ) 
p x (1  p)n  x
x !(n  x )!
.25
.00
X
0
4 - 47
1
2
Statistics for Managers Using Microsoft Excel, 1/e
Summary Measures
© 1998 Prentice-Hall, Inc.

Expected value




Notation for a Population
Mean of probability distribution
Weighted average of all possible values
 = E(X) = Xi P(Xi)
Variance


4 - 48
Weighted average squared deviation about
mean
2 = E[ (Xi (Xi  P(Xi)
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Summary Measures Calculation
Table
Xi P(Xi)
XiP(Xi)
Total
XiP(Xi)
4 - 49
Xi -  (Xi-)
2
(Xi-)2 P(Xi)
(Xi-)2 P(Xi)
Statistics for Managers Using Microsoft Excel, 1/e
Thinking Challenge
© 1998 Prentice-Hall, Inc.
You toss 2 coins.
You’re interested in the
number of tails. What
are the expected value
& standard deviation of
this random variable,
number of tails?
© 1984-1994 T/Maker Co.
4 - 50
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Expected Value & Variance
Solution*
2
Xi P(Xi) XiP(Xi) Xi -  (Xi-)
2
(Xi-) P(Xi)
0
.25
0
-1.00
1.00
.25
1
.50
.50
0
0
0
2
.25
.50
1.00
1.00
.25
 = 1.0
4 - 51
 = .50
2
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.

Discrete Probability Distribution
Function
Type of model

Representation of some
underlying phenomenon

Mathematical formula

Represents discrete
random variable

Used to get exact
probabilities
4 - 52
P( X  x )
-

e 
x!
x
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Discrete Probability Distribution
Models
Discrete
Probability
Distribution
Binomial
4 - 53
HyperGeometric
Negative
Binomial
Poisson
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Discrete Probability Distribution
Models
Discrete
Probability
Distribution
Binomial
4 - 54
HyperGeometric
Negative
Binomial
Poisson
Statistics for Managers Using Microsoft Excel, 1/e
Binomial Distribution
© 1998 Prentice-Hall, Inc.

Number of ‘successes’ in a sample
of n observations (trials)

# reds in 15 spins of roulette wheel

# defective items in a batch of 5 items

# correct on a 33 question exam

# customers who purchase out of 100
customers who enter store
4 - 55
Statistics for Managers Using Microsoft Excel, 1/e
Binomial Distribution Properties
© 1998 Prentice-Hall, Inc.

Two different sampling methods


Infinite population without replacement
Finite population with replacement

Sequence of n identical trials

Each trial has 2 outcomes

‘Success’ (desired outcome) or ‘failure’

Constant trial probability

Trials are independent
4 - 56
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Binomial Probability Distribution
Function
n!
x
n x
P( X ) 
p (1  p)
x !(n  x )!
P(X) = Probability of X ‘successes’
n = Sample size
p = Probability of ‘success’
x = Number of ‘successes’ in sample
(X = 0, 1, 2, ..., n)
4 - 57
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Binomial Probability Distribution
Example
Experiment: Toss 1 coin 4 times in a row.
Note # tails.
What’s the probability of 3 tails?
n!
x
n x
P( X ) 
p (1  p )
x !(n  x )!
4!
3
43
P ( X  3) 
.5 (1 .5)
3 !(4  3)!
 .25
4 - 58
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Binomial Distribution
Characteristics
Mean
  E ( X )  np
P(X)
.6
.4
.2
.0
0
n = 5 p = 0.1
X
1
2
3
4
5
Standard Deviation
  np (1  p)
P(X)
.6
.4
.2
.0
X
0
4 - 59
n = 5 p = 0.5
1
2
3
4
5
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Binomial Distribution Thinking
Challenge
You’re a telemarketer selling
service contracts for Macy’s.
You’ve sold 20 in your last
100 calls (p = .20). If you
call 12 people tonight,
what’s the probability of
A.
B.
C.
D.
4 - 60
No sales?
Exactly 2 sales?
At most 2 sales?
At least 2 sales?
Statistics for Managers Using Microsoft Excel, 1/e
Binomial Distribution Solution*
© 1998 Prentice-Hall, Inc.
A. P(0) = .0687
B. P(2) = .2835
C. P(at most 2) = P(0) + P(1) + P(2)
= .0687 + .2062 + .2835
= .5584
D. P(at least 2) = P(2) + P(3)...+ P(12)
= 1 - [P(0) + P(1)]
= 1 - .0687 - .2062
= .7251
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Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Discrete Probability Distribution
Models
Discrete
Probability
Distribution
Binomial
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HyperGeometric
Negative
Binomial
Poisson
Statistics for Managers Using Microsoft Excel, 1/e
Poisson Distribution
© 1998 Prentice-Hall, Inc.

Number of events that occur in an area
of opportunity

Events per unit


Example: Time, length, area, space
Examples



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# customers arriving in 20 minutes
# strikes per year in the U.S.
# defects per lot (group) of VCR's
Statistics for Managers Using Microsoft Excel, 1/e
Poisson Process
© 1998 Prentice-Hall, Inc.

Constant event probability


One event per interval


Average of 60/hr. is 1/min.
for 60 1-minute intervals
Don’t arrive together
Independent events

Arrival of 1 person does not
affect another’s arrival
© 1984-1994 T/Maker Co.
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Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Poisson Probability Distribution
Function
-
e 
P( X ) 
x!
x
P(X) = Probability of X ‘successes’
 = Expected (mean) number of ‘successes’
e = 2.71828 (base of natural logs)
x = Number of ‘successes’ per unit
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Statistics for Managers Using Microsoft Excel, 1/e
Poisson Distribution Characteristics
© 1998 Prentice-Hall, Inc.
= 0.5
P(X)
Mean
  E( X )  

.6
.4
.2
.0
X
N
0
 X i P( X i )
1
2
3
4
5
i 1
Standard Deviation
 
0
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= 6
P(X)
.6
.4
.2
.0
X
2
4
6
8
10
Statistics for Managers Using Microsoft Excel, 1/e
Poisson Distribution Example
© 1998 Prentice-Hall, Inc.
Customers arrive at a
rate of 72 per hour.
What is the
probability of 4
customers arriving in
3 minutes?
© 1995 Corel Corp.
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Statistics for Managers Using Microsoft Excel, 1/e
Poisson Distribution Solution
© 1998 Prentice-Hall, Inc.
72 per hr. = 1.2 per
min.
e -  x
P( X ) 
x!
4
= 3.6 per 3
-3.6
e  36
. 
P( X  4) 
min.
4!
interval
= .1912
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Statistics for Managers Using Microsoft Excel, 1/e
Thinking Challenge
© 1998 Prentice-Hall, Inc.
You work in Quality
Assurance for an
investment firm. A
clerk enters 75 words
per minute with 6
errors per hour. What
is the probability of 0
errors in a 255-word
bond transaction?
© 1984-1994 T/Maker Co.
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Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Poisson Distribution Solution:
Finding *
75 words/min = (75 words/min)(60 min/hr)
= 4500 words/hr
6 errors/hr = 6 errors/4500 words
= .00133 errors/word
In a 255-word transaction (interval):
 = (.00133 errors/word )(255 words)
= .34 errors/255-word transaction
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Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Poisson Distribution Solution:
Finding P(0)*
-
e 
P( X ) 
x!
P( X  0) 
e
x
-.34
.34
0
0!
= .7118
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Statistics for Managers Using Microsoft Excel, 1/e
Conclusion
© 1998 Prentice-Hall, Inc.

Defined experiment, outcome, event,
sample space, & probability

Used a contingency table to find
probabilities

Described 4 discrete probability
distributions

Found the probability of discrete
random variables
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Statistics for Managers Using Microsoft Excel, 1/e