business statistics: a decision-making approach, 7th edition
TRANSCRIPT
Business Statistics
Department of Quantitative Methods & Information Systems
Dr. Mohammad Zainal QMIS 120
Chapter 4
Using Probability and
Probability Distributions
Chapter Goals
After completing this chapter, you should be able to:
Explain basic probability concepts and definitions
Use a Venn diagram or tree diagram to illustrate simple probabilities
Apply common rules of probability
Explain three approaches to assessing
probabilities
Apply common rules of probability, including the
Addition Rule and the Multiplication Rule
QMIS 120, by Dr. M. Zainal Chap 4-2
QMIS 120, by Dr. M. Zainal
Chapter Goals
Compute conditional probabilities
Determine whether events are statistically independent
Use Bayes’ Theorem for conditional probabilities
(continued)
Chap 4-3
Important Terms
Random Experiment – a process leading to an
uncertain outcome
Basic Outcome (Simple Event ) – a possible
outcome of a random experiment
Sample Space – the collection of all possible
outcomes of a random experiment
Event (Compound Event) – any subset of basic
outcomes from the sample space
Probability – the chance that an uncertain event
will occur (always between 0 and 1)
QMIS 120, by Dr. M. Zainal Chap 4-4
QMIS 120, by Dr. M. Zainal
Sample Space
The Sample Space is the collection of all
possible outcomes
e.g., All 6 faces of a die:
e.g., All 52 cards of a bridge deck:
Chap 4-5
QMIS 120, by Dr. M. Zainal
Sample Space
e.g., All 2 faces of a coin:
e.g., All 4 colors in the jar:
Chap 4-6
Examples of Sample Spaces
Rolling one die:
S = [1, 2, 3, 4, 5, 6]
Tossing one coin:
S = [Head, Tail]
QMIS 120, by Dr. M. Zainal Chap 4-7
Choosing one ball:
S = [Red, Blue, Green, Yellow]
Playing a game:
S = [Win, Lose, Tie]
Examples of Sample Spaces
QMIS 120, by Dr. M. Zainal Chap 4-8
QMIS 120, by Dr. M. Zainal
Events
Experimental outcome – An outcome from a
sample space with one characteristic
Example: A red card from a deck of cards
Event – May involve two or more outcomes
simultaneously
Example: An ace that is also red from a deck of
cards
Chap 4-9
QMIS 120, by Dr. M. Zainal
Visualizing Events
Contingency Tables
Tree Diagrams
Red 2 24 26
Black 2 24 26
Total 4 48 52
Ace Not Ace Total
Full Deck
of 52 Cards
Sample
Space
Sample
Space 2
24
2
24
Chap 4-10
QMIS 120, by Dr. M. Zainal
Experimental Outcomes
A automobile consultant records fuel type and vehicle type for a sample of vehicles
2 Fuel types: Gasoline, Diesel
3 Vehicle types: Truck, Car, SUV
6 possible experimental outcomes:
e1 Gasoline, Truck
e2 Gasoline, Car
e3 Gasoline, SUV
e4 Diesel, Truck
e5 Diesel, Car
e6 Diesel, SUV
Chap 4-11
Important Terms
Intersection of Events – If A and B are two
events in a sample space S, then the
intersection, A ∩ B, is the set of all outcomes in
S that belong to both A and B
A B AB
S
QMIS 120, by Dr. M. Zainal Chap 4-12
Important Terms
A and B are Mutually Exclusive Events if they
have no basic outcomes in common
i.e., the set A ∩ B is empty
If A occurs, then B cannot occur
A and B have no common elements
(continued)
A B
S
QMIS 120, by Dr. M. Zainal Chap 4-13
A ball cannot be
Green and Red
at the same
time.
Important Terms
Union of Events – If A and B are two events in a
sample space S, then the union, A U B, is the
set of all outcomes in S that belong to either
A or B
(continued)
A B
The entire shaded area
represents A U B
S
QMIS 120, by Dr. M. Zainal Chap 4-14
Important Terms
Events E1, E2, … Ek are Collectively Exhaustive
events if E1 U E2 U . . . U Ek = S
i.e., the events completely cover the sample space
The Complement of an event A is the set of all
basic outcomes in the sample space that do not
belong to A. The complement is denoted
= +
(continued)
A
A S
A AA
A
QMIS 120, by Dr. M. Zainal Chap 4-15
Examples
Rolling a die
S = [1, 2, 3, 4, 5, 6]
Let A be the event “Number rolled is even”
Let B be the event “Number rolled is at least 4”
Then:
QMIS 120, by Dr. M. Zainal Chap 4-16
(continued)
Complements:
Intersections:
Unions:
Examples
QMIS 120, by Dr. M. Zainal Chap 4-17
Mutually exclusive:
Collectively exhaustive:
(continued)
Examples
QMIS 120, by Dr. M. Zainal Chap 4-18
Probability
Probability – the chance that
an uncertain event will occur
(always between 0 and 1)
Also, it is a numerical
measure of the likelihood
that an event will occur
0 ≤ P(A) ≤ 1 For any event A
Certain
Impossible
.5
1
0
QMIS 120, by Dr. M. Zainal Chap 4-19
Assessing Probability
There are three approaches to assessing the
probability of an uncertain event:
1. classical probability
Assumes all outcomes in the sample space are
equally likely to occur
QMIS 120, by Dr. M. Zainal Chap 4-20
spacesampletheinoutcomesofnumbertotal
eventthesatisfythatoutcomesofnumber
N
NAeventofyprobabilit A
S = [1, 2, 3, 4, 5, 6] A = [2, 4, 6] C = [5, 6]
Classical Probability Examples
QMIS 120, by Dr. M. Zainal Chap 4-21
Example: Find the probability of obtaining an
even number in one roll of a die and find the
probability of obtaining a number greater than
4.
Counting the Possible Outcomes
Use the Combinations formula to determine the
number of distinct combinations of n distinct objects
that can be formed, taking them r at a time (No
ordering)
where
n! = n(n-1)(n-2)…(1)
0! = 1 by definition
!( )
!( )!
n n
n r r r
nC C
r n r
QMIS 120, by Dr. M. Zainal Chap 4-22
Counting the Possible Outcomes
Combinations: If one has 5 different objects (e.g. A, B,
C, D, and E), how many ways can they be grouped as
3 objects when position does not matter (e.g. ABC,
ABD, ABE, ACD, ACE, ADE are correct but CBA is
not ok as is equal to ABC)
QMIS 120, by Dr. M. Zainal Chap 4-23
Counting the Possible Outcomes
Use the Permutations formula to determine the
number of ways we can arrange n distinct objects,
taking them r at a time (with ordering)
where
n! = n(n-1)(n-2)…(1)
0! = 1 by definition
!( )
( )!
n n
n r r r
nP P
n r
QMIS 120, by Dr. M. Zainal Chap 4-24
Counting the Possible Outcomes
Permutations: Given that position (order) is important,
if one has 5 different objects (e.g. A, B, C, D, and E),
how many unique ways can they be placed in 3
positions (e.g. ADE, AED, DEA, DAE, EAD, EDA,
ABC, ACB, BCA, BAC etc.)
QMIS 120, by Dr. M. Zainal Chap 4-25
Assessing Probability
Three approaches (continued)
2. relative frequency probability
the limit of the proportion of times that an event A occurs in a large
number of trials, n
3. subjective probability
an individual opinion or belief about the probability of occurrence
populationtheineventsofnumbertotal
Aeventsatisfythatpopulationtheineventsofnumber
n
nAeventofyprobabilit A
QMIS 120, by Dr. M. Zainal Chap 4-26
Relative Frequency Probability
Examples
QMIS 120, by Dr. M. Zainal Chap 4-27
Example: In a group of 500 women, 80 have played
football at least once in their life. Suppose one of these
500 woman is selected. What is the probability that she
played football at least once?
Solution: n = 500 and f = 80
Relative frequency f Women
420 / 500 = 0.84 420 Did not play football
80 / 500 = 0.16 80 Played football
Sum = 1.0 n = 500
Probability Postulates
1. If A is any event in the sample space S, then
2. Let A be an event in S, and let Ei denote the basic
outcomes. Then
(the notation means that the summation is over all the basic outcomes in A)
3. P(S) = 1
1P(A)0
i
A
P(A) P(E )
QMIS 120, by Dr. M. Zainal Chap 4-28
Probability Rules
The Complement rule:
1)AP(P(A)i.e., P(A)1)AP(
A
A
QMIS 120, by Dr. M. Zainal Chap 4-29
Joint and Marginal Probabilities
The probability of a joint event, A ∩ B:
Computing a marginal probability:
Where B1, B2, …, Bk are k mutually exclusive and
collectively exhaustive events
outcomeselementaryofnumbertotal
BandAsatisfyingoutcomesofnumberB)P(A
)BP(A)BP(A)BP(AP(A) k21
QMIS 120, by Dr. M. Zainal Chap 4-30
QMIS 120, by Dr. M. Zainal
Marginal Probability
Marginal probability is the probability of a single
event without consideration of any other event
Suppose one employee is selected out of a
sample of 100, he/she maybe classified either on
the bases of gender alone or on the bases of a
marital status.
Total Single Married
60 45 15
40 36 4
Male
Female
Total 100 81 19
Chap 4-31
QMIS 120, by Dr. M. Zainal
Marginal Probability Example
The probability of each of the following event is
called marginal probability
Total Single Married
60 45 15
40 36 4
Male
Female
Total 100 81 19
P(male) = 60/100
P(female) = 40/100
P(married) = 19/100
P(single) = 81/100
Chap 4-32
QMIS 120, by Dr. M. Zainal
Conditional Probability
This probability is called conditional probability and
is written as “the probability that the employee
selected is married given that he is a male.”
P(married | male )
Read as “given”
This event has
already occurred
The event whose probability
is to be determined
Chap 4-33
Conditional Probability
A conditional probability is the probability of one
event, given that another event has occurred:
P(B)
B)P(AB)|P(A
P(A)
B)P(AA)|P(B
The conditional
probability of A
given that B has
occurred
The conditional
probability of B
given that A has
occurred
QMIS 120, by Dr. M. Zainal Chap 4-34
QMIS 120, by Dr. M. Zainal
Example: Find the conditional probability P(married | male)
for the data on 100 employees
Solution
Total Single Married
60 45 15 Male
40 36 4 Female
100 81 19 Total
Conditional Probability Example
Chap 4-35
QMIS 120, by Dr. M. Zainal
(continued)
Conditional Probability Example
Chap 4-36
QMIS 120, by Dr. M. Zainal
What is the probability that a car has a CD player, given that it has AC ?
i.e., we want to find P(CD | AC)
Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both.
Conditional Probability Example
Chap 4-37
QMIS 120, by Dr. M. Zainal
No CD CD Total
AC
No AC
Total 1.0
Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD).
20% of the cars have both.
(continued)
Conditional Probability Example
Chap 4-38
Using a Tree Diagram
All
Cars
Given AC
or no AC:
QMIS 120, by Dr. M. Zainal Chap 4-39
QMIS 120, by Dr. M. Zainal
Conditional Probability Example
Example: Consider the experiment of tossing a fair die.
Denote by A and B the following events:
A={Observing an even number}, B={Observing a number of
dots less than or equal to 3}. Find the probability of the
event A, given the event B.
Solution
Chap 4-40
Multiplication Rule
Multiplication rule for two events A and B:
also
P(B)B)|P(AB)P(A
P(A)A)|P(BB)P(A
QMIS 120, by Dr. M. Zainal Chap 4-41
QMIS 120, by Dr. M. Zainal
Independent and Dependent Events
Independent: Occurrence of one does not
influence the probability of
occurrence of the other
Probability Concepts
E1 = heads on one flip of fair coin
E2 = heads on second flip of same coin
Result of second flip does not depend on the
result of the first flip.
Chap 4-42
QMIS 120, by Dr. M. Zainal
Independent and Dependent Events
Dependent: Occurrence of one affects the
probability of the other
Probability Concepts
E1 = rain forecasted on the news
E2 = take umbrella to work
Probability of the second event is affected by the
occurrence of the first event
Chap 4-43
Statistical Independence
Two events are statistically independent
if and only if:
Events A and B are independent when the probability of one
event is not affected by the other event
If A and B are independent, then
P(A)B)|P(A
P(B)P(A)B)P(A
P(B)A)|P(B
if P(B)>0
if P(A)>0
QMIS 120, by Dr. M. Zainal Chap 4-44
QMIS 120, by Dr. M. Zainal
Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both.
Statistical Independence Example
No CD CD Total
AC .2 .5 .7
No AC .2 .1 .3
Total .4 .6 1.0
Are the events AC and CD statistically independent?
Chap 4-45
No CD CD Total
AC .2 .5 .7
No AC .2 .1 .3
Total .4 .6 1.0
(continued)
Statistical Independence Example
QMIS 120, by Dr. M. Zainal Chap 4-46
QMIS 120, by Dr. M. Zainal
For Independent Events:
Conditional probability for
independent events E1 , E2:
)P(E)E|P(E 121 0)P(Ewhere 2
)P(E)E|P(E 212 0)P(Ewhere 1
Chap 4-47
Probability Rules
The Addition rule:
The probability of the union of two events is
B)P(AP(B)P(A)B)P(A
A B
P(A or B) = P(A) + P(B) - P(A and B)
Don’t count common
elements twice!
A B + =
QMIS 120, by Dr. M. Zainal Chap 4-48
QMIS 120, by Dr. M. Zainal
Addition Rule for Mutually Exclusive Events
If A and B are mutually exclusive, then
P(A ∩ B) = 0
So
P(A or B) = P(A) + P(B) - P(A and B)
P(A U B) = P(A) + P(B)
A B
Chap 4-49
A Probability Table
B
A
A
B
)BP(A
)BAP( B)AP(
P(A)B)P(A
)AP(
)BP(P(B) 1.0P(S)
Probabilities and joint probabilities for two
events A and B are summarized in this table:
QMIS 120, by Dr. M. Zainal Chap 4-50
Law of Total Probability
Consider the following Venn diagram
Can we find the area (probability) of A assuming
that we know the probability of each Bi & P(A|Bi) ?
B1 B2 Bn-1 Bn
B
A
A1 A2 An-1 An
QMIS 120, by Dr. M. Zainal Chap 4-51
Bayes’ Theorem
where:
Bi = ith event of k mutually exclusive and collectively
exhaustive events
A = new event that might impact P(Bi)
i i
1 1 2 2
P(A|B )P(B )
P(A|B )P(B ) P(A|B )P(B )
i
i
P(A B )P(B |A)
P(A)
QMIS 120, by Dr. M. Zainal Chap 4-52
QMIS 120, by Dr. M. Zainal
Bayes’ Theorem Example
A drilling company has estimated a 40%
chance of striking oil for their new well.
A detailed test has been scheduled for more
information. Historically, 60% of successful
wells have had detailed tests, and 20% of
unsuccessful wells have had detailed tests.
Given that this well has been scheduled for a
detailed test, what is the probability that the well will be successful?
Chap 4-53
Let S = successful well
U = unsuccessful well
P(S) = , P(U) = (prior probabilities)
Define the detailed test event as D
Conditional probabilities:
Goal is to find
(continued)
Bayes’ Theorem Example
QMIS 120, by Dr. M. Zainal Chap 4-54
Apply Bayes’ Theorem:
(continued)
Bayes’ Theorem Example
QMIS 120, by Dr. M. Zainal Chap 4-55
QMIS 120, by Dr. M. Zainal
Given the detailed test, the revised probability
of a successful well has from the original
estimate of
Bayes’ Theorem Example (continued)
Chap 4-56
Complete Example
Consider a standard deck of card that we see normally
being played on computer or table. It consists of 52
cards with 4 suits and so 13 cards in a suit.
Consider a standard deck of 52 cards, with four suits:
Spade (S):♠ Heart (H): ♥
Club (C): ♣ Diamond (D): ♦
Spades and Clubs are traditionally
in black color, while Hearts and
Diamonds are in red.
Let event A = card is an Ace
Let event B = card is from a Red suit
QMIS 120, by Dr. M. Zainal Chap 4-57
QMIS 120, by Dr. M. Zainal
Black
Color Type Red Total
Ace 2 2 4
Non-Ace 24 24 48
Total 26 26 52
What is the probability of Red given it was Ace (Cond.)
Chap 4-58
Complete Example
Black
Color Type Red Total
Ace 2 2 4
Non-Ace 24 24 48
Total 26 26 52
What is the probability of Red and Ace (Intersection)
QMIS 120, by Dr. M. Zainal Chap 4-59
Complete Example
Black
Color Type Red Total
Ace 2 2 4
Non-Ace 24 24 48
Total 26 26 52
What is the probability of Red or Ace (Union)
QMIS 120, by Dr. M. Zainal Chap 4-60
Complete Example
Black
Color Type Red Total
Ace 2 2 4
Non-Ace 24 24 48
Total 26 26 52
What is the probability of Ace (Marginal)
QMIS 120, by Dr. M. Zainal Chap 4-61
Complete Example
Black
Color Type Red Total
Ace 2 2 4
Non-Ace 24 24 48
Total 26 26 52
Are the events Red and Ace statistically independent?
QMIS 120, by Dr. M. Zainal Chap 4-62
Complete Example
Black
Color Type Red Total
Ace 2 2 4
Non-Ace 24 24 48
Total 26 26 52
What is the complement of the event Ace?
QMIS 120, by Dr. M. Zainal Chap 4-63
Complete Example
Example: Manufacturing firm that receives shipment of parts from two different suppliers. Currently, 65 percent of the parts purchased by the company are from supplier 1 and the remaining 35 percent are from supplier 2. Historical Data suggest the quality rating of the two supplier are shown in the table:
Good Parts Bad Parts
Supplier 1 98 2
Supplier 2 95 5
a) Draw a tree diagram for this experiment with the probability
of all outcomes
b) Given the information the part is bad, What is the
probability the part came from supplier 1?
Bayes’ Theorem Example
QMIS 120, by Dr. M. Zainal Chap 4-64
Bayes’ Theorem Example
QMIS 120, by Dr. M. Zainal Chap 4-65
Apply Bayes’ Theorem:
(continued)
Bayes’ Theorem Example
QMIS 120, by Dr. M. Zainal Chap 4-66
Example: An insurance company rents 35% of the cars for
its customers from Avis and the rest from Hertz. From past
records they know that 8% of Avis cars break down and 5%
of Hertz cars break down.
A customer calls and complains that his rental car broke
down. What is the probability that his car was rented from
Avis?
Bayes’ Theorem Example
QMIS 120, by Dr. M. Zainal Chap 4-67
Bayes’ Theorem Example
QMIS 120, by Dr. M. Zainal Chap 4-68
Chapter Summary
Defined basic probability concepts
Sample spaces and events, intersection and union
of events, mutually exclusive and collectively
exhaustive events, complements
Examined basic probability rules
Complement rule, addition rule, multiplication rule
Defined conditional, joint, and marginal probabilities
Reviewed odds and the overinvolvement ratio
Defined statistical independence
Discussed Bayes’ theorem
QMIS 120, by Dr. M. Zainal Chap 4-69
Copyright
The materials of this presentation were mostly
taken from the PowerPoint files accompanied
Business Statistics: A Decision-Making Approach,
7e © 2008 Prentice-Hall, Inc.
QMIS 120, by Dr. M. Zainal Chap 4-70