Download 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

Tree Diagrams
Full
Deck
of Cards
© 2002 Prentice-Hall, Inc.
Not Ace
Total
Black
Red
2
2
24
24
26
26
Total
4
48
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.
© 2002 Prentice-Hall, Inc.
P(Red Card or Ace)
4 Aces + 26 Red Cards - 2 Red Aces

52 total number of cards
28 7


52 13
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  A | Bi  P  Bi 
P  Bi | A  
P  A | B1  P  B1       P  A | Bk  P  Bk 
P  Bi and A 

P  A
Same
Event
© 2002 Prentice-Hall, Inc.
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
PR | 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
PR | 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  P X j  1
© 2002 Prentice-Hall, Inc.
P X  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"
Tails in 2 Tosses of Coin
n : sample size
© 2002 Prentice-Hall, Inc.
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 .11  .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.
e3.6 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)
  EX   
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 

P  2 
 .30
10 
A : number of "successes" in population
 
3 
X : number of "successes" in sample
n : sample size
N : population size
X
© 2002 Prentice-Hall, Inc.
 0,1, 2,
, n
Chap 4-47
Hypergeometric Distribution
Characteristics

Mean


A
  EX   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