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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Statistics for Managers Using Microsoft Excel
Basic Probability & Discrete Probability Distributions
Chapter 4 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
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Thinking Challenge
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?
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Many Repetitions!*
0.00
0.25
0.50
0.75
1.00
0 25 50 75 100 125
Number of Tosses
Total Heads Number of Tosses
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Introduction to Probability
Experiment Process of obtaining an observation,
outcome or simple event
Outcome Result of an experiment
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Sample space depends on experimenter!
Experiment Process of obtaining an observation,
outcome or simple event
Outcome Result of an experiment
Sample space (S) Collection of all possible outcomes Defined by experimenter
Experiments & Outcomes
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Outcome Examples
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
Experiment Sample Space
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Outcome Properties
Mutually exclusive 2 outcomes can not occur
at the same time Example: Both male &
female in same person
Collectively exhaustive 1 outcome in sample space
must occur Example: Male or female
Experiment: Observe gender
© 1984-1994 T/Maker Co.
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Events
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
2 events occurring simultaneously
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Event Examples
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
Experiment: Toss 2 coins. Note faces.Experiment: Toss 2 coins. Note faces.
Event Outcomes in Event
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Visualizing Sample Space
Listing S = {Head, Tail}
Contingency Table
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
2nd Coin1st Coin Head Tail Total
Head HH HT HH, HT
Tail TH TT TH, TT
Total HH, TH HT, TT S
Contingency Table
Experiment: Toss 2 coins. Note faces.Experiment: Toss 2 coins. Note faces.
S = {HH, HT, TH, TT} Sample space
Outcome (% or
count) Simpleevent (head on1st coin)
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
ColorType Red Black Total
Ace Ace &Red
Ace &Black
Ace
Non-Ace Non &Red
Non &Black
Non-Ace
Total Red Black S
Joint Events (Event A and Event B)
Sample space (S):
2R, 2R, 2B,
..., AB
Experiment: Draw 1 card. Note kind, color & suit.
Joint event Ace AND Black:
AB, AB
Simple event Ace:
AR,
AR,
AB,
ABSimple event black: 2B, ..., AB
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
ColorType Red Black Total
Ace Ace &Red
Ace &Black
Ace
Non-Ace Non &Red
Non &Black
Non-Ace
Total Red Black S
Either-Or Events (Event A or Event B)
Sample space (S):
2R, 2R,
2B, ..., AB
Experiment: Draw 1 card. Note kind, color & suit.
Joint eventAce ORBlack:
AR, ..., AB,2B, ..., KB
Simple event Ace:
AR,
AR,
AB,
AB
Simple event Black: 2B, ..., AB
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Special Events
Null event Club & Diamond on
1 card draw
Complement of event For event A, all
events not in A: A Mutually exclusive event
Events do not occur simultaneously
Null Event
Q
Q
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
What is Probability?
Numerical measure of likelihood that event will occur P(Event) P(A) Prob(A)
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
What is Probability?
Numerical measure of likelihood that event will occur P(Event) P(A) Prob(A)
Lies between 0 & 1
Sum of events is 1
1
.5 0
Certain
Impossible© 1984-1994 T/Maker Co.
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Statistics for Managers Using Microsoft Excel, 1/e
Assigning Event Probabilities
a priori classical method
Empirical classical method
Subjective method
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
a priori Classical Method
Prior knowledge of process
Before experiment
P(Event) = X / T 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.
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Empirical Classical Method
Actual data collected
After experiment
P(Event) = X / T Repeat experiment
T times Event observed X times
Also called relative frequency method
Of 100 parts inspected, only 2 defects!
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Subjective Method
Individual knowledge of situation
Before experiment
Unique process Not repeatable
Different probabilities from different people
© 1984-1994 T/Maker Co.
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Thinking Challenge
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?
Which method should be used to find the probability...
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Statistics for Managers Using Microsoft Excel, 1/e
Joint Event Probability
Numerical measure of likelihood that joint event will occur
Can often use contingency table 2 variables only
Formula methods Addition rule Conditional probability formula Multiplication rule
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Statistics for Managers Using Microsoft Excel, 1/e
EventEvent B1 B2 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) P(B2) 1
Event Probability Using Contingency Table
Joint Probability Marginal (Simple) Probability
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Statistics for Managers Using Microsoft Excel, 1/e
ColorType Red Black Total
Ace 2/52 2/52 4/52
Non-Ace 24/52 24/52 48/52
Total 26/52 26/52 52/52
Contingency Table Example
Experiment: Draw 1 card. Note kind, color & suit.
P(Ace)
P(Ace AND Red)P(Red)
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Thinking Challenge
What’s the probability?
P(A) =
P(D) =
P(C and B) =
P(A or D) =
P(B and D) =
EventEvent C D Total
A 4 2 6
B 1 3 4
Total 5 5 10
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Solution*
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
EventEvent C D Total
A 4 2 6
B 1 3 4
Total 5 5 10
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Addition Rule
P(A or B) = P(A) + P(B) - P(A and B)
For mutually exclusive events:
P(A or B) = P(A) + P(B)
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
ColorType Red Black Total
Ace 2 2 4
Non-Ace 24 24 48
Total 26 26 52
Addition Rule Example
Experiment: Draw 1 card. Note kind, color & suit.
P(Ace OR Black) = P(Ace)+P(Black) - P(Ace AND Black)
4
522652
252
2852
2 2 2452
( )
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Thinking Challenge
Using the Addition Rule, what’s the probability?
P(A or D) =
P(B or C) =Event
Event C D Total
A 4 2 6
B 1 3 4
Total 5 5 10
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Solution*
Using the Addition Rule, the probabilities are:
P(A or D) = P(A) + P(D) - P(A and D)
610
5
102
109
10
P(B or C) = P(B) + P(C) - P(B and C)
4
105
101
108
10
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Conditional Probability
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)
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
ColorType Red Black Total
Ace 2 2 4
Non-Ace 24 24 48
Total 26 26 52
Conditional Probability Using Contingency Table
Experiment: Draw 1 card. Note kind, Experiment: Draw 1 card. Note kind, color & suit.color & suit.
P(Ace | Black) = P(Ace and Black)
P(Black)
2 5226 52
226
//
Revised sample
space
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Event occurrence does not affect probability of another event e.g., toss 1 coin twice
Causality not implied Tests for independence
P(A | B) = P(A) P(A and B) = P(A)*P(B)
Statistical Independence
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Thinking Challenge
Using the table then the formula, what’s the probability?
P(A|D) =
P(C|B) =
Are C & B independent?
EventEvent C D Total
A 4 2 6
B 1 3 4
Total 5 5 10
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Solution*
Using the formula, the probabilities are:
P(A | D) = P(A and D)P(D)
2 105 10
25
//
P(C | B) = P(C and B)P(B)
P(C) = 510
1 104 10
14
14
//
Dependent
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Multiplication Rule
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)
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
ColorType Red Black Total
Ace 2 2 4
Non-Ace 24 24 48
Total 26 26 52
Multiplication Rule Example
Experiment: Draw 1 card. Note kind, color & suit.
P(Ace and Black) = P(Ace) P(Black |Ace)4
52
2
4
2
52
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Thinking Challenge
Using the Multiplication Rule, what’s the probability?
P(C and B) =
P(B and D) =
P(A and B) =
EventEvent C D Total
A 4 2 6
B 1 3 4
Total 5 5 10
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Solution*
Using the Multiplication Rule, the probabilities are:
P(A and B) = P(A) P(B|A) 0
P(C and B) = P(C) P(B|C) =
P(B and D) = P(B) P(D|B) =
510
15
110
410
34
310
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Discrete Random Variables
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Thinking Challenge
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?
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Data Types
Data
Numerical(Quantitative)
Categorical(Qualitative)
Discrete Continuous
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Random Variable
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
Poisson random variable is exception ()
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Discrete Random Variable Examples
Experiment RandomVariable
PossibleValues
Make 100 sales calls # Sales 0, 1, 2, ..., 100
Inspect 70 radios # Defective 0, 1, 2, ..., 70
Answer 33 questions # Correct 0, 1, 2, ..., 33
Count cars at tollbetween 11:00 & 1:00
# Carsarriving
0, 1, 2, ...,
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
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
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Discrete Probability Distribution Example
Probability Distribution
Values, Xi Probabilities, P(Xi)
0 1/4 = .25
1 2/4 = .50
2 1/4 = .25
Experiment: Toss 2 coins. Count # tails.
© 1984-1994 T/Maker Co.
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Visualizing Discrete Probability Distributions
.00
.25
.50
0 1 2
X
P(X)
# Tails f(Xi)Count
P(Xi)
0 1 .251 2 .502 1 .25
P Xn
x n xp px n x( )
!!( )!
( )
1
{ (0, .25), (1, .50), (2, .25) }{ (0, .25), (1, .50), (2, .25) }
ListingTable
Graph Equation
4 - 48
© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Summary MeasuresNotation for a Population
Expected value Mean of probability distribution Weighted average of all possible values = E(X) = Xi P(Xi)
Variance Weighted average squared deviation about
mean 2 = E[ (Xi (Xi P(Xi)
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Summary Measures Calculation Table
Xi P(Xi) XiP(Xi) Xi - (Xi-)2 (Xi-)2 P(Xi)
Total XiP(Xi) (Xi-)2 P(Xi)
4 - 50
© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Thinking Challenge
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.
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Expected Value & Variance Solution*
Xi P(Xi) XiP(Xi) Xi - (Xi-)2 (Xi-)2 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 2 = .50
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Discrete Probability Distribution Function
Type of model Representation of some
underlying phenomenon
Mathematical formula Represents discrete
random variable Used to get exact
probabilities
P X x
x
x
( )
!
e-
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Discrete Probability Distribution Models
DiscreteProbabilityDistribution
BinomialHyper-
GeometricNegativeBinomial
Poisson
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Discrete Probability Distribution Models
DiscreteProbabilityDistribution
BinomialHyper-
GeometricNegativeBinomial
Poisson
4 - 55
© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Binomial Distribution
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
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Binomial Distribution Properties
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 - 57
© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Binomial Probability Distribution Function
P Xn
x n xp px n x( )
!!( )!
( )
1
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 - 58
© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Binomial Probability Distribution Example
P Xn
x n xp p
P X
x n x( )!
!( )!( )
( )!
!( )!. ( . )
1
34
3 4 35 1 53 4 3
.25
Experiment: Toss 1 coin 4 times in a row. Note # tails.
What’s the probability of 3 tails?
4 - 59
© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
.0
.2
.4
.6
0 1 2 3 4 5
X
P(X)
.0
.2
.4
.6
0 1 2 3 4 5
X
P(X)
Binomial Distribution Characteristics
.0
.2
.4
.6
0 1 2 3 4 5
X
P(X)
.0
.2
.4
.6
0 1 2 3 4 5
X
P(X)
n = 5 p = 0.1
n = 5 p = 0.5
E X np
np p
( )
( )1
Mean
Standard Deviation
4 - 60
© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
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 ofA. No sales?
B. Exactly 2 sales?
C. At most 2 sales?
D. At least 2 sales?
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Binomial Distribution Solution*
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
4 - 62
© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Discrete Probability Distribution Models
DiscreteProbabilityDistribution
BinomialHyper-
GeometricNegativeBinomial
Poisson
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Poisson Distribution
Number of events that occur in an area of opportunity Events per unit
Example: Time, length, area, space
Examples # customers arriving in 20 minutes # strikes per year in the U.S. # defects per lot (group) of VCR's
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Poisson Process
Constant event probability Average of 60/hr. is 1/min.
for 60 1-minute intervals
One event per interval 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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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Poisson Probability Distribution Function
P Xx
x
( )!
e-
P(X) = Probability of X ‘successes’
= Expected (mean) number of ‘successes’
e = 2.71828 (base of natural logs)
x = Number of ‘successes’ per unit
4 - 66
© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Poisson Distribution Characteristics
.0
.2
.4
.6
0 1 2 3 4 5
X
P(X)
.0
.2
.4
.6
0 1 2 3 4 5
X
P(X)
.0
.2
.4
.6
0 2 4 6 8 10
X
P(X)
.0
.2
.4
.6
0 2 4 6 8 10
X
P(X)
= 0.5
= 6
E X
X P Xii
N
i
( )
( )1
Mean
Standard Deviation
4 - 67
© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Poisson Distribution Example
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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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Poisson Distribution Solution
72 per hr. = 1.2 per min.
= 3.6 per 3 min.
interval
P Xx
P X
x
( )!
( ).
!
e
e
= .1912
-
- .6
436
4
3 4
4 - 69
© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Thinking Challenge
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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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
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
4 - 71
© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Poisson Distribution Solution: Finding P(0)*
P Xx
P X
x
( )!
( ).
!
.
e
e
= .7118
-
- 34
034
0
0
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© 1998 Prentice-Hall, Inc.
Statistics for Managers Using Microsoft Excel, 1/e
Conclusion
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