Statistics for Managers Using Microsoft Excel, 3/e
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Transcript Statistics for Managers Using Microsoft Excel, 3/e
Statistics for Managers
Using Microsoft Excel
(3rd Edition)
Chapter 4
Basic Probability and Discrete
Probability Distributions
© 2002 Prentice-Hall, Inc.
Chap 4-1
Chapter Topics
Basic probability concepts
Conditional probability
Sample spaces and events, simple probability, joint
probability
Statistical independence, marginal probability
Bayes’s Theorem
© 2002 Prentice-Hall, Inc.
Chap 4-2
Chapter Topics
(continued)
The probability of a discrete random variable
Covariance and its applications in finance
Binomial distribution
Poisson distribution
Hypergeometric distribution
© 2002 Prentice-Hall, Inc.
Chap 4-3
Sample Spaces
Collection of all possible outcomes
e.g.: All six faces of a die:
e.g.: All 52 cards in a deck:
© 2002 Prentice-Hall, Inc.
Chap 4-4
Events
Simple event
Outcome from a sample space with one
characteristic
e.g.: A red card from a deck of cards
Joint event
Involves two outcomes simultaneously
e.g.: An ace that is also red from a deck of
cards
© 2002 Prentice-Hall, Inc.
Chap 4-5
Visualizing Events
Contingency Tables
Ace
Black
Red
Total
Tree Diagrams
Full
Deck
of Cards
© 2002 Prentice-Hall, Inc.
2
2
4
Not Ace
24
24
48
Total
26
26
52
Ace
Red
Cards
Black
Cards
Not an Ace
Ace
Not an Ace
Chap 4-6
Simple Events
The Event of a Triangle
There are 5 triangles in this collection of 18 objects
© 2002 Prentice-Hall, Inc.
Chap 4-7
Joint Events
The event of a triangle AND blue in color
Two triangles that are blue
© 2002 Prentice-Hall, Inc.
Chap 4-8
Special Events
Null Event
Impossible event
e.g.: Club & diamond on one card
draw
Complement of event
© 2002 Prentice-Hall, Inc.
For event A, all events not in A
Denoted as A’
e.g.: A: queen of diamonds
A’: all cards in a deck that are
not queen of diamonds
Chap 4-9
Special Events
Mutually exclusive events
Two events cannot occur together
e.g.: -- A: queen of diamonds; B: queen of clubs
(continued)
Events A and B are mutually exclusive
Collectively exhaustive events
One of the events must occur
The set of events covers the whole sample space
e.g.: -- A: all the aces; B: all the black cards; C: all the
diamonds; D: all the hearts
Events A, B, C and D are collectively
exhaustive
Events B, C and D are also collectively
exhaustive
© 2002 Prentice-Hall, Inc.
Chap 4-10
Contingency Table
A Deck of 52 Cards
Red Ace
Ace
Not an
Ace
Total
Red
2
24
26
Black
2
24
26
Total
4
48
52
Sample Space
© 2002 Prentice-Hall, Inc.
Chap 4-11
Tree Diagram
Event Possibilities
Full
Deck
of Cards
© 2002 Prentice-Hall, Inc.
Red
Cards
Ace
Not an Ace
Ace
Black
Cards
Not an Ace
Chap 4-12
Probability
Probability is the numerical
measure of the likelihood
that an event will occur
1
Certain
Value is between 0 and 1
Sum of the probabilities of
all mutually exclusive and
collective exhaustive events
is 1
© 2002 Prentice-Hall, Inc.
.5
0
Impossible
Chap 4-13
Computing Probabilities
The probability of an event E:
number of event outcomes
P( E )
total number of possible outcomes in the sample space
X
T
e.g. P(
) = 2/36
(There are 2 ways to get one 6 and the other 4)
Each of the outcomes in the sample space is
equally likely to occur
© 2002 Prentice-Hall, Inc.
Chap 4-14
Computing Joint Probability
The probability of a joint event, A and B:
P(A and B) = P(A B)
number of outcomes from both A and B
total number of possible outcomes in sample space
E.g. P(Red Card and Ace)
2 Red Aces
1
52 Total Number of Cards 26
© 2002 Prentice-Hall, Inc.
Chap 4-15
Joint Probability Using
Contingency Table
Event
B1
Event
Total
A1
P(A1 and B1) P(A1 and B2) P(A1)
A2
P(A2 and B1) P(A2 and B2) P(A2)
Total
Joint Probability
© 2002 Prentice-Hall, Inc.
B2
P(B1)
P(B2)
1
Marginal (Simple) Probability
Chap 4-16
Computing Compound
Probability
Probability of a compound event, A or B:
P( A or B) P( A B)
number of outcomes from either A or B or both
total number of outcomes in sample space
E.g.
P (Red Card or Ace)
4 Aces + 26 Red Cards - 2 Red Aces
52 total number of cards
28 7
52 13
© 2002 Prentice-Hall, Inc.
Chap 4-17
Compound Probability
(Addition Rule)
P(A1 or B1 ) = P(A1) + P(B1) - P(A1 and B1)
Event
Event
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
For Mutually Exclusive Events: P(A or B) = P(A) + P(B)
© 2002 Prentice-Hall, Inc.
Chap 4-18
Computing Conditional
Probability
The probability of event A given that event B
has occurred:
P( A and B)
P( A | B)
P( B)
E.g.
P (Red Card given that it is an Ace)
2 Red Aces 1
4 Aces
2
© 2002 Prentice-Hall, Inc.
Chap 4-19
Conditional Probability Using
Contingency Table
Color
Type
Red
Black
Total
Ace
2
2
4
Non-Ace
24
24
48
Total
26
26
52
Revised Sample Space
P(Ace and Red) 2 / 52
2
P(Ace | Red)
P(Red)
26 / 52 26
© 2002 Prentice-Hall, Inc.
Chap 4-20
Conditional Probability and
Statistical Independence
Conditional probability:
P( A and B)
P( A | B)
P( B)
Multiplication rule:
P( A and B) P( A | B) P( B)
P( B | A) P( A)
© 2002 Prentice-Hall, Inc.
Chap 4-21
Conditional Probability and
Statistical Independence
(continued)
Events A and B are independent if
P( A | B) P ( A)
or P ( B | A) P ( B )
or P ( A and B ) P ( A) P ( B )
Events A and B are independent when the
probability of one event, A, is not affected by
another event, B
© 2002 Prentice-Hall, Inc.
Chap 4-22
Bayes’s Theorem
P Bi | A
Same
Event
© 2002 Prentice-Hall, Inc.
P A | Bi P Bi
P A | B1 P B1 P A | Bk P Bk
P Bi and A
P A
Adding up
the parts
of A in all
the B’s
Chap 4-23
Bayes’s Theorem
Using Contingency Table
Fifty percent of borrowers repaid their loans. Out of those
who repaid, 40% had a college degree. Ten percent of
those who defaulted had a college degree. What is the
probability that a randomly selected borrower who has a
college degree will repay the loan?
P R .50
P C | R .4
P C | R .10
PR | C ?
© 2002 Prentice-Hall, Inc.
Chap 4-24
Bayes’s Theorem
Using Contingency Table
(continued)
Repay
Repay
Total
College
.2
.05
.25
College
.3
.45
.75
Total
.5
.5
1.0
PR | C
P C | R P R
P C | R P R P C | R P R
.4 .5
.2
.8
.4 .5 .1.5 .25
© 2002 Prentice-Hall, Inc.
Chap 4-25
Random Variable
Random Variable
Outcomes of an experiment expressed numerically
e.g.: Toss a die twice; count the number of times
the number 4 appears (0, 1 or 2 times)
© 2002 Prentice-Hall, Inc.
Chap 4-26
Discrete Random Variable
Discrete random variable
Obtained by counting (1, 2, 3, etc.)
Usually a finite number of different values
e.g.: Toss a coin five times; count the number of
tails (0, 1, 2, 3, 4, or 5 times)
© 2002 Prentice-Hall, Inc.
Chap 4-27
Discrete Probability
Distribution Example
Event: Toss two coins
Count the number of tails
Probability Distribution
Values
Probability
T
T
T
© 2002 Prentice-Hall, Inc.
0
1/4 = .25
1
2/4 = .50
2
1/4 = .25
T
Chap 4-28
Discrete Probability Distribution
List of all possible [Xj , p(Xj) ] pairs
Xj = value of random variable
P(Xj) = probability associated with value
Mutually exclusive (nothing in common)
Collectively exhaustive (nothing left out)
0 PX j 1
© 2002 Prentice-Hall, Inc.
PX 1
j
Chap 4-29
Summary Measures
Expected value (the mean)
Weighted average of the probability distribution
E X X jP X j
j
e.g.: Toss 2 coins, count the number of tails,
compute expected value
X jP X j
j
© 2002 Prentice-Hall, Inc.
0 2.5 1.5 2 .25 1
Chap 4-30
Summary Measures
(continued)
Variance
Weight average squared deviation about the mean
E X X j P X j
2
2
2
e.g. Toss two coins, count number of tails,
compute variance
X j P X j
2
2
0 1 .25 1 1 .5 2 1 .25 .5
2
© 2002 Prentice-Hall, Inc.
2
2
Chap 4-31
Covariance and its Application
N
XY X i E X Yi E Y P X iYi
i 1
X : discrete random variable
X i : i th outcome of X
Y : discrete random variable
Yi : i th outcome of Y
P X iYi : probability of occurrence of the i
th
outcome of X and the i th outcome of Y
© 2002 Prentice-Hall, Inc.
Chap 4-32
Computing the Mean for
Investment Returns
Return per $1,000 for two types of investments
P(XiYi)
Investment
Economic condition Dow Jones fund X Growth Stock Y
.2
Recession
-$100
-$200
.5
Stable Economy
+ 100
+ 50
.3
Expanding Economy
+ 250
+ 350
E X X 100.2 100.5 250.3 $105
E Y Y 200.2 50.5 350.3 $90
© 2002 Prentice-Hall, Inc.
Chap 4-33
Computing the Variance for
Investment Returns
P(XiYi)
Investment
Economic condition Dow Jones fund X Growth Stock Y
.2
Recession
-$100
-$200
.5
Stable Economy
+ 100
+ 50
.3
Expanding Economy
+ 250
+ 350
100 105 .2 100 105 .5 250 105 .3
2
2
X
2
2
X 121.35
14, 725
200 90 .2 50 90 .5 350 90 .3
2
2
Y
37,900
© 2002 Prentice-Hall, Inc.
2
2
Y 194.68
Chap 4-34
Computing the Covariance for
Investment Returns
P(XiYi)
Investment
Economic condition Dow Jones fund X Growth Stock Y
.2
Recession
-$100
-$200
.5
Stable Economy
+ 100
+ 50
.3
Expanding Economy
+ 250
+ 350
XY 100 105 200 90 .2 100 105 50 90 .5
250 105 350 90 .3 23,300
The Covariance of 23,000 indicates that the two investments are
positively related and will vary together in the same direction.
© 2002 Prentice-Hall, Inc.
Chap 4-35
Important Discrete
Probability Distributions
Discrete Probability
Distributions
Binomial
© 2002 Prentice-Hall, Inc.
Hypergeometric
Poisson
Chap 4-36
Binomial Probability Distribution
‘n’ identical trials
Two mutually exclusive outcomes on each
trials
e.g.: 15 tosses of a coin; ten light bulbs taken
from a warehouse
e.g.: Head or tail in each toss of a coin; defective
or not defective light bulb
Trials are independent
The outcome of one trial does not affect the
outcome of the other
© 2002 Prentice-Hall, Inc.
Chap 4-37
Binomial Probability Distribution
(continued)
Constant probability for each trial
e.g.: Probability of getting a tail is the same each
time we toss the coin
Two sampling methods
Infinite population without replacement
Finite population with replacement
© 2002 Prentice-Hall, Inc.
Chap 4-38
Binomial Probability
Distribution Function
n!
n X
X
P X
p 1 p
X ! n X !
P X : probability of X successes given n and p
X : number of "successes" in sample X 0,1,
, n
p : the probability of each "success"
n : sample size
© 2002 Prentice-Hall, Inc.
Tails in 2 Tosses of Coin
X
0
P(X)
1/4 = .25
1
2/4 = .50
2
1/4 = .25
Chap 4-39
Binomial Distribution
Characteristics
Mean
E X np
E.g. np 5 .1 .5
Variance and
Standard Deviation
2 np 1 p
np 1 p
P(X)
.6
.4
.2
0
n = 5 p = 0.1
X
0
1
2
3
4
5
E.g.
np 1 p 5 .11 .1 .6708
© 2002 Prentice-Hall, Inc.
Chap 4-40
Binomial Distribution in PHStat
PHStat | probability & prob. Distributions |
binomial
Example in excel spreadsheet
© 2002 Prentice-Hall, Inc.
Chap 4-41
Poisson Distribution
Poisson Process:
Discrete events in an “interval”
The probability of One Success
in an interval is stable
The probability of More than
One Success in this interval is 0
P( X x |
- x
e
x!
The probability of success is
independent from interval to
interval
e.g.: number of customers arriving in 15 minutes
e.g.: number of defects per case of light bulbs
© 2002 Prentice-Hall, Inc.
Chap 4-42
Poisson Probability
Distribution Function
e
P X
X!
P X : probability of X "successes" given
X
X : number of "successes" per unit
: expected (average) number of "successes"
e : 2.71828 (base of natural logs)
e.g.: Find the probability of 4
customers arriving in 3 minutes
when the mean is 3.6.
© 2002 Prentice-Hall, Inc.
e3.6 3.64
P X
.1912
4!
Chap 4-43
Poisson Distribution in PHStat
PHStat | probability & prob. Distributions |
Poisson
Example in excel spreadsheet
© 2002 Prentice-Hall, Inc.
Chap 4-44
Poisson Distribution
Characteristics
Mean
= 0.5
P(X)
EX
N
XiP Xi
.6
.4
.2
0
X
0
1
i 1
Standard Deviation
and Variance
© 2002 Prentice-Hall, Inc.
2
2
3
4
5
= 6
P(X)
.6
.4
.2
0
X
0
2
4
6
8
10
Chap 4-45
Hypergeometric Distribution
“n” trials in a sample taken from a finite
population of size N
Sample taken without replacement
Trials are dependent
Concerned with finding the probability of “X”
successes in the sample where there are “A”
successes in the population
© 2002 Prentice-Hall, Inc.
Chap 4-46
Hypergeometric Distribution
Function
A N A
X
n
X
P X
N
n
E.g. 3 Light bulbs were
selected from 10. Of the 10
there were 4 defective. What
is the probability that 2 of the
3 selected are defective?
P X : probability that X successes given n, N , and A
4 6
2 1
N : population size
P 2
.30
10
A : number of "successes" in population
3
X : number of "successes" in sample
n : sample size
X
© 2002 Prentice-Hall, Inc.
0,1, 2,
, n
Chap 4-47
Hypergeometric Distribution
Characteristics
Mean
A
EX n
N
Variance and Standard Deviation
nA N A N n
2
N2
N 1
nA N A N n
2
N
N 1
© 2002 Prentice-Hall, Inc.
Finite
Population
Correction
Factor
Chap 4-48
Hypergeometric Distribution in
PHStat
PHStat | probability & prob. Distributions |
Hypergeometric …
Example in excel spreadsheet
© 2002 Prentice-Hall, Inc.
Chap 4-49
Chapter Summary
Discussed basic probability concepts
Defined conditional probability
Sample spaces and events, simple probability,
and joint probability
Statistical independence, marginal probability
Discussed Bayes’s theorem
© 2002 Prentice-Hall, Inc.
Chap 4-50
Chapter Summary
(continued)
Addressed the probability of a discrete
random variable
Defined covariance and discussed its
application in finance
Discussed binomial distribution
Addressed Poisson distribution
Discussed hypergeometric distribution
© 2002 Prentice-Hall, Inc.
Chap 4-51