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Statistics for Managers
Using Microsoft® Excel
5th Edition
Chapter 5
Some Important Discrete Probability
Distributions
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-1
Learning Objectives
In this chapter, you will learn:
 The properties of a probability distribution
 To compute the expected value and variance
of a probability distribution
 To compute probabilities from the binomial,
Poisson, and hypergeometric distributions
 How to use these distributions to solve
business problems
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-2
Definitions
Random Variables
 A random variable represents a possible
numerical value from an uncertain event.
 Discrete random variables produce outcomes
that come from a counting process (i.e.
number of classes you are taking).
 Continuous random variables produce
outcomes that come from a measurement
(i.e. your annual salary, or your weight).
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-3
Definitions
Random Variables
Random
Variables
Ch. 5
Discrete
Random Variable
Continuous
Random Variable
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Ch. 6
Chap 5-4
Definitions
Probability Distribution
 A probability distribution for a discrete random
variable is a mutually exclusive listing of all possible
numerical outcomes for that variable and a particular
probability of occurrence associated with each outcome.
Number of Classes Taken
Probability
2
3
4
0.2
0.4
0.24
5
0.16
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-5
Discrete Random Variables
Expected Value (Mean)
N
  E(X)   X i P( X i )
i 1
Number of
Probability
Classes Taken
2
3
4
5
0.2
0.4
0.24
0.16
X*P(X)
0.40
1.20
0.96
0.80
Σ=3.36
So, the average number of classes taken E(X) is 3.36.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-6
Discrete Random Variables
Dispersion
 Variance of a discrete random variable
N
σ   [X i  E(X)] 2 P(X i )
2
i 1
 Standard Deviation of a discrete random variable
σ  σ2 
N
[X
i 1
i
 E(X)] 2 P(X i )
where:
E(X) = Expected value of the discrete random variable X
Xi = the ith outcome of X
P(Xi) = Probability of the ith occurrence of X
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-7
Variance of a Discrete
Probability Distribution
N
σ   [X i  E(X)] 2 P(X i )
2
i 1
X
P(X)
[X-E(X)]2P(X)
2
0.2
.36992
3
0.4
.05184
4
0.24
.09830
5
0.16
.43034
E(X)=3.36
σ2=.9504
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-8
Probability Distribution
Overview
Probability
Distributions
Ch. 5
Discrete
Probability
Distributions
Continuous
Probability
Distributions
Binomial
Normal
Poisson
Uniform
Hypergeometric
Exponential
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Ch. 6
Chap 5-9
The Binomial Distribution:
Properties
 A fixed number of observations, n
 ex. 15 tosses of a coin; ten light bulbs taken from a lot in
the warehouse
 Two mutually exclusive and collectively exhaustive
categories
 ex. head or tail in each toss of a coin; defective or not
defective light bulb; having a boy or girl
 Generally called “success” and “failure”
 Probability of success is p, probability of failure is 1 – p
 Constant probability for each observation
 ex. Probability of getting a tail is the same each time we
toss the coin
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-10
The Binomial Distribution:
Properties
 Observations are independent
 The outcome of one observation does not affect the
outcome of the other
 Two sampling methods
 Infinite population without replacement
 Finite population with replacement
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-11
Applications of the
Binomial Distribution
 A manufacturing plant labels items as either
defective or acceptable
 A firm bidding for contracts will either get a
contract or not
 A marketing research firm receives survey
responses of “yes I will buy” or “no I will
not”
 New job applicants either accept the offer or
reject it
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-12
Counting Techniques
Rule of Combinations
 The number of combinations of selecting X
objects out of n objects is:
n
n!
n CX  
 X   X! (n  X)!
 
where:
n! =n(n - 1)(n - 2) . . . (2)(1)
X! = X(X - 1)(X - 2) . . . (2)(1)
0! = 1 (by definition)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-13
Counting Techniques
Rule of Combinations
 How many possible 2 scoop combinations could you
create at an ice cream parlor if you have 3 flavors to
select from? (S, C, V) > SC, SV, CV
 The total choices is n = 3, and we select X = 2.
 3
3!
3!
3  2  1!
C






3 2
 2  2! (3  2)! 2!1! 2  1  1!  3
 
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-14
Counting Techniques
Rule of Combinations
 How many possible 3 scoop combinations could you
create at an ice cream parlor if you have 31 flavors
to select from?
 The total choices is n = 31, and we select X = 3.
 31
31!
31! 31  30  29  28!
31 C3  
 3   3!(31  3)!  3!28!  3  2  1  28!  31  5  29  4495
 
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-15
The Binomial Distribution
Formula
n!
P(X) 
p X (1  p) n X
X! (n  X)!
P(X) = probability of X successes in n trials,
with probability of success p on each trial
X = number of ‘successes’ in sample,
(X = 0, 1, 2, ..., n)
n
p
= sample size (number of trials
or observations)
= probability of “success”
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Example: Flip a coin four
times, let x = # heads:
n=4
p = 0.5
1 - p = (1 - .5) = .5
X = 0, 1, 2, 3, 4
Chap 5-16
The Binomial Distribution
Example
What is the probability of one success in five
observations if the probability of success is .1?
X = 1, n = 5, and p = .1
n!
P(X  1) 
p X (1  p ) n  X
X! (n  X)!
5!

(.1)1 (1  .1) 51
1!(5  1)!
 (5)(.1)(.9 ) 4
 .32805
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-17
The Binomial Distribution
Example
Suppose the probability of purchasing a defective
computer is 0.02. What is the probability of
purchasing 2 defective computers is a lot of 10?
X = 2, n = 10, and p = .02
n!
p X (1  p) n  X
X! (n  X)!
10!

(.02) 2 (1  .02)10 2
2!(10  2)!
 (45)(.0004 )(.8508)
 .01531
P(X  2) 
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-18
The Binomial Distribution
Shape
 The shape of the binomial
distribution depends on the
values of p and n
 Here, n = 5 and p = .1
n = 5 p = 0.1
P(X)
.6
.4
.2
0
0
1
3
4
5
X
5
X
n = 5 p = 0.5
P(X)
 Here, n = 5 and p = .5
2
.6
.4
.2
0
0
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
1
2
3
4
Chap 5-19
The Binomial Distribution
Using Binomial Tables
n = 10
x
…
p=.20
p=.25
p=.30
p=.35
p=.40
p=.45
p=.50
0
1
2
3
4
5
6
7
8
9
10
…
…
…
…
…
…
…
…
…
…
…
0.1074
0.2684
0.3020
0.2013
0.0881
0.0264
0.0055
0.0008
0.0001
0.0000
0.0000
0.0563
0.1877
0.2816
0.2503
0.1460
0.0584
0.0162
0.0031
0.0004
0.0000
0.0000
0.0282
0.1211
0.2335
0.2668
0.2001
0.1029
0.0368
0.0090
0.0014
0.0001
0.0000
0.0135
0.0725
0.1757
0.2522
0.2377
0.1536
0.0689
0.0212
0.0043
0.0005
0.0000
0.0060
0.0403
0.1209
0.2150
0.2508
0.2007
0.1115
0.0425
0.0106
0.0016
0.0001
0.0025
0.0207
0.0763
0.1665
0.2384
0.2340
0.1596
0.0746
0.0229
0.0042
0.0003
0.0010
0.0098
0.0439
0.1172
0.2051
0.2461
0.2051
0.1172
0.0439
0.0098
0.0010
10
9
8
7
6
5
4
3
2
1
0
…
p=.80
p=.75
p=.70
p=.65
p=.60
p=.55
p=.50
x
Examples:
n = 10, p = .35, x = 3:
P(x = 3|n =10, p = .35) = .2522
n = 10, p = .75, x = 2:
P(x = 2|n =10, p = .75) = .0004
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-20
The Binomial Distribution
Characteristics
 Mean
μ  E(x)  np
 Variance and Standard Deviation
σ  np(1 - p)
2
σ
np(1 - p)
Where n = sample size
p = probability of success
(1 – p) = probability of failure
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-21
The Binomial Distribution
Characteristics
Examples
μ  np  (5)(.1)  0.5
σ  np(1 - p)  (5)(.1)(1  .1)
 0.6708
μ  np  (5)(.5)  2.5
σ  np(1 - p)  (5)(.5)(1  .5)
 1.118
n = 5 p = 0.1
P(X)
.6
.4
.2
0
0
1
2
3
4
5
X
5
X
n = 5 p = 0.5
P(X)
.6
.4
.2
0
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
0
1
2
3
4
Chap 5-22
Using PHStat

Select PHStat / Probability & Prob. Distributions /
Binomial…
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-23
Using PHStat
(continued)
 Enter desired values in dialog box
Here: n = 10
p = .35
Output for X = 0
to X = 10 will be
generated by PHStat
Optional check boxes
for additional output
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-24
PHStat Output
P(X = 3 | n = 10, p = .35) = .2522
P(X > 5 | n = 10, p = .35) = .0949
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-25
The Poisson Distribution
Definitions
 An area of opportunity is a continuous unit
or interval of time, volume, or such area in
which more than one occurrence of an event
can occur.
 ex. The number of scratches in a car’s paint
 ex. The number of mosquito bites on a person
 ex. The number of computer crashes in a day
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-26
The Poisson Distribution
Formula
λ
e λ
P(X) 
X!
x
where:
X = the probability of X events in an area of opportunity
 = expected number of events
e = mathematical constant approximated by 2.71828…
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-27
Poisson Distribution
Characteristics
 Mean
μλ
 Variance and Standard Deviation
σ λ
2
σ λ
where
 = expected number of successes per unit
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-28
The Poisson Distribution
Shape
0.70
 = .50
0.60
0.50
0
1
2
3
4
5
6
7
P(X)
0.6065
0.3033
0.0758
0.0126
0.0016
0.0002
0.0000
0.0000
P(x)
X
0.40
0.30
0.20
0.10
0.00
0
1
2
3
4
5
6
7
x
P(X = 2) = .0758
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-29
The Poisson Distribution
Shape
 The shape of the Poisson Distribution depends
on the parameter  :
 = 0.50
 = 3.00
0.70
0.25
0.60
0.20
0.15
0.40
P(x)
P(x)
0.50
0.30
0.10
0.20
0.05
0.10
0.00
0.00
0
1
2
3
4
5
6
7
1
x
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
2
3
4
5
6
7
8
9
10
11
12
x
Chap 5-30
The Poisson Distribution
Example
 Suppose that, on average, 5 cars enter a parking
lot per minute. What is the probability that in a
given minute, 7 cars will enter?
 So, X = 7 and λ = 5
e  λ λ x e 5 57
P(7) 

 0.104
X!
7!
 So, there is a 10.4% chance 7 cars will enter the
parking in a given minute.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-31
The Poisson Distribution
Using Poisson Tables

X
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
0
1
2
3
4
5
6
7
0.9048
0.0905
0.0045
0.0002
0.0000
0.0000
0.0000
0.0000
0.8187
0.1637
0.0164
0.0011
0.0001
0.0000
0.0000
0.0000
0.7408
0.2222
0.0333
0.0033
0.0003
0.0000
0.0000
0.0000
0.6703
0.2681
0.0536
0.0072
0.0007
0.0001
0.0000
0.0000
0.6065
0.3033
0.0758
0.0126
0.0016
0.0002
0.0000
0.0000
0.5488
0.3293
0.0988
0.0198
0.0030
0.0004
0.0000
0.0000
0.4966
0.3476
0.1217
0.0284
0.0050
0.0007
0.0001
0.0000
0.4493
0.3595
0.1438
0.0383
0.0077
0.0012
0.0002
0.0000
0.4066
0.3659
0.1647
0.0494
0.0111
0.0020
0.0003
0.0000
Example: Find P(X = 2) if  = .50
e  λ λ X e 0.50 (0.50) 2
P(X  2) 

 .0758
X!
2!
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-32
Poisson Distribution
in PHStat
 Select:
PHStat / Probability & Prob. Distributions / Poisson…
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-33
Poisson Distribution
in PHStat
(continued)
 Complete dialog box entries and get output …
P(X = 2) = 0.0758
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-34
The Hypergeometric
Distribution
 The binomial distribution is applicable
when selecting from a finite population with
replacement or from an infinite population
without replacement.
 The hypergeometric distribution is
applicable when selecting from a finite
population without replacement.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-35
The Hypergeometric
Distribution
 “n” trials in a sample taken from a finite
population of size N
 Sample taken without replacement
 Outcomes of trials are dependent
 Concerned with finding the probability of “X”
successes in the sample where there are “A”
successes in the population
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-36
The Hypergeometric
Distribution
 A  N  A 
 

X n X 
P( X )   
N
 
n 
Where
N = population size
A = number of successes in the population
N – A = number of failures in the population
n = sample size
X = number of successes in the sample
n – X = number of failures in the sample
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-37
The Hypergeometric Distribution
Characteristics
 The mean of the hypergeometric distribution is:
μ  E(x) 
nA
N
 The standard deviation is:
σ
Where
nA(N - A) N - n

2
N
N -1
N-n
N - 1 is called the “Finite Population Correction Factor”
from sampling without replacement from a finite population
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-38
The Hypergeometric Distribution
Example
 Different computers are checked from 10 in the
department. 4 of the 10 computers have illegal
software loaded. What is the probability that 2 of the
3 selected computers have illegal software loaded?
 So, N = 10, n = 3, A = 4, X = 2
 A  N  A   4  6 
 
   
 X  n  X   2 1  (6)(6)
     
P(X  2)   
 0.3
N
10
120
 
 
 
 
n 
3 
 
 
 The
probability that 2 of the 3 selected computers
have illegal software loaded is .30, or 30%.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-39
Hypergeometric Distribution
in PHStat
 Select:
PHStat / Probability & Prob. Distributions / Hypergeometric …
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-40
Hypergeometric Distribution
(continued)
in PHStat
 Complete dialog box entries and get output …
N = 10
A=4
n=3
X=2
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
P(X = 2) = 0.3
Chap 5-41
Chapter Summary
In this chapter, we have
 Addressed the probability of a discrete random
variable its mean and standard deviation
 Discussed the Binomial distribution
 Reviewed the Poisson distribution
 Discussed the Hypergeometric distribution
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 5-42