Transcript Introduction to Probability

Quantitative Methods
Quantitative methods
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Subjects
(1st meeting)
Introduction to Quantitative Methods
Probability (Introduction)
(2nd Meeting)
Probability Distribution
(3rd meeting)
Decision Analysis
(4th Meeting)
Linear Programming
(5th meeting)
Multicriteria Decision Aid
Introduction to Quantitative Methods
Decision
Goal
Decision
Problem
Strategic
Planning
Plan
(Balasubramanian, 1999)
Management
Control
Operational
Control
Unstructured
E-Commerce Carreer paths
Grievances
Semi-structured
Forecasting
Budgeting
Assignments
Structured
Dividends
Purchasing
Billing
Decision Types (Gorry and Scott Morton, 1971)
Decision-making process
Problem Recognition
Implementation
Problem Definition
Choice
Alternative Generation
Model Development
Alternative Analysis
(Courtney, 2001)
The problem
• How can organizations improve the quality of their
decision-making processes?
Problem Recognition
Implementation
Choice
Alternative Analysis
Problem Definition
Alternative Generation
Model Development
Simplified Example – Break Even Point
The level of sales where income=cost
euro
Income
T: prise, A:Fixed Cost,
B:production cost per unit
Income = Cost
Break even point
Total Cost
Τ * Χ = Α + Β* Χ ,
(Τ – Β) * Χ = Α ,
A
Χ = Α/(Τ-Β)
Sales (Number of products)
Probability
Probability
• Probability is a numerical measure of the likehood
that an event will occur
• Scale : from 0 to 1
• Very useful to decision making – Example
• Most of the airlines overbooks Why?
0
0.5
The occurrence of an event is just as likely as
it is unlikely
1
Experiment and the sample space
• Experiment: any process that generates well
defined outcomes. (ex. Toss a coin)
• Sample space: the set of all possible experiment
outcomes
• Assigning probabilities to experiment outcomes –
the probability of an experiment outcome is a
numerical measure of the likehood that the
experimental outcome will occur.
0  P(Ei)  1, P(E1) + P(E2) +…+ P(En) =1
Relative Frequency Method
An experiment occurs n times. The outcome a occurs
na times. The relatice frequency is
P= na/n
The probability to have the output a is P(A)=na/n
This method is referred as the relative frequency
method
Example.From a set of 400 potential customers of a
shop the 100 purchased product while 300 did not.
The relative frequency = 100/400 = 0.25. This is the
probability a potential customer to purchase a
product.
Subjective Method
• When little relevant data are available.
• We use all the available information in order to
express a degree of belief (on a scale from 0 to 1).
High degree of subjectivity.
Example:
E1: proposal accepted,
E2: Proposal not accepted
P(E1)=0.2, P(E2)=0.8 – belief of A
P(E1)=0.4, P(E2)=0.6 – belief of B
Events and their Probabilities
Event is a collection of sample points. Example:
rolling a die. Event-the even numbers {2,4,6}.
The probability to have an even number is the sum of
the probabilities to have 2, 4, 6.
P=1/6+1/6+1/6 = 3/6=0.5
Basic Relations
Complement of an Event A: All sample points that are not in
A. Denoted A’.
P(A) +P(A’)=1
P(A) = 1 – P(A’)
A
A’
Additional Law
Union of Events A and B is the events containing all
sample points belonging to A and B or both. Denoted
AUB.
Intersection of events A and B is the event containing
the sample points belonging to both a and B. Denoted
A B.
A
A B
AUB
B
P(AUB) = P(A) + P(B) – P(A B)
P(AUB) = P(A) + P(B), A B =
Additional Law
Example: television viewers
H: husband is a regular viewer
W: wife is a regular viewer
P(H)=0.30, P(W) = 0.20, P(HW)=0.12
P(Husband or wife is a regular viewer)= 0.30+0.200.12 = 0.38
Conditional Probability
A and B two events.
Conditional Probability P(A\B) is the probability
of event A given the condition that event B has
already occurred.
P(AB)
P(B\A) = P(B)
P(A\B) = ---------------------P(B)
P(A\B)=P(A)
P(AB)
P(B\A) = ---------------------P(A)
when A, B are
independent
AB
A
B
Conditional Law
1200 officers. 960 men and 240 women. 324 were promoted
M: event an officer is a man
W: event an officer is a woman
B: event an officer is promoted
P(MB)=288/1200=0.24, P(MB’)=672/1200=0.56
P(WB)=36/1200=0.03, P(WB’)=204/1200=0.17
Joint Probabilities
Promoted
Men
288
Women
36
Total
324
Not
Promoted Total
672
960
204
240
876
Marginal
Probabilities
Promoted Not Promoted Total
Men
0,24
0,56
0,8
Women
0,03
0,17
0,2
Total
0,27
0,73
1
Conditional Law
• P(B\M)=P(MB)/P(M) = 0.24/0.80=0.30
• P(B\W)=P(WB)/P(W) = 0.03/0.20=0.15
Joint Probabilities
Promoted
Men
288
Women
36
Total
324
Not
Promoted Total
672
960
204
240
876
Marginal
Probabilities
Promoted Not Promoted Total
Men
0,24
0,56
0,8
Women
0,03
0,17
0,2
Total
0,27
0,73
1
Multiplicative Law
The probability of an intersection
P(AB) = P(A\B) P(B)
P(AB) = P(B\A) P(A)
When the events are independent
P(AB) = P(A) P(B)
Bayes Theorem
P(Ai) P(B\Ai)
P(Ai\B) = -----------------------------------------------------------P(A1) P(B\A1)+P(A2) P(B\A2)+…..+ P(An) P(B\An)
Prior
Probabilities
New
Information
Application
of the Bayes’
Theorem
Posterior
Probabilities
The Tabular Approach
• Step 1: Prepare three columns- the mutual exclusive events
for which posterior probabilities are desired, the prior
probabilities, the conditional probabilities of the new
information
• Step 2: Compute the joint probabilities for each event and
the new information using the multiplication law
• Step 3: Sum the joint probabilities to obtain the
probabilities of the new information
• Step 4: Compute the positive probabilities by using the
basic relationship of conditional probabilities.
Bayes Theorem: Example (1)
• Proportion of athletes who are using drugs =0,05
• Given that an athlete is not using drugs, that the
athletes will test positive (False positive) = 0,03
• Given that an athlete is using drugs, that the
athletes will test negative (False negative) = 0,07
Bayes Theorem: Example (2)
• P(Using Drugs) =0,05
• P(Not using Drugs) = 0,95
• P(Test Positive/Not Using Drugs) =0,03
• P(Test Negative/Not Using Drugs) = 0,97
• P(Test Negative/Using Drugs) =0,07
• P(Test Positive/Using Drugs) = 0,93
Bayes Theorem: Example (3)
Probability
0,05
0,95
Probability of
Given that
Positive Negative
Using Drugs
0,93 0,07
Not Using Drugs 0,03 0,97
Bayes Theorem: Example (4)
Join Probability Table
Probability of
Given that
Positive Negative
Using Drugs
0,0465 0,0035
Not Using Drugs 0,0285 0,9215
0,075
0,925
0,05
0,95
Bayes Theorem: Example (5)
• Given that the test result is positive:
- What is the probability that the athlete is actually
using drugs?
-
P(Using Drugs/Test Positive) = P(Using Drugs and Test Positive)/P(Test
Positive) = 0,0485/0,0750 = 0,62 or 62%
- What is the probability that the athlete is actually
not using drugs)
-
P(Not Using Drugs/Test Positive) = P(Not Using Drugs and Test
Positive)/P(Test Positive) = 0,0235 / 0,0750 = 0,38 or 38%
Bayes Theorem: Example (6)
Given that the test is
Probability that Positive Negative
Using Drugs
0,62 0,0038
Not Using Drugs
0,38 0,9962
Conclusions - Example
•The test is effective to verify that someone is not
taking drugs. Most of the time (99,62%), a negative
result indicates that the athlete is not taking drugs
•The test is not so effective to verify that someone is
taking drugs. In a large number of cases (38%), the
test gives a false positive
Decision Analysis
• A newspaper vendor purchases the paper at $0.15 and sells
them for $0.50. The demand for papers is as follows:
•
Demand
Probability
•
0
0.05
•
1
0.10
•
2
0.20
•
3
0.30
•
4
0.20
•
5
0.15
•The vendor is interested in determining how many
newspapers should be purchased in order to maximize
expected profit.
Analysis – Payoff Table
• . The minimum number of newspapers to be
purchased is 0.
•
• . The maximum number of newspapers to be
purchased is…?
• . Evaluation: Determine what happens when the
vendor purchases k newspapers.
Pay off Table
Probability Demand
0,05
0,1
0,2
0,3
0,2
0,15
Purchase Prise
Seling Prise
0
1
2
3
4
5
0
0
0
0
0
0
0
0,15
0,5
Number of Newspapers Purchased
1
2
3
4
-0,15
-0,3
-0,45
-0,8
0,35
0,2
0,05
-0,1
0,35
0,7
0,55
0,4
0,35
0,7
1,05
0,9
0,35
0,7
1,05
1,4
0,35
0,7
1,05
1,4
Marginal profit
Marginal Loss
0,35
0,15
5
-0,75
-0,25
0,25
0,75
1,25
1,75
Expected Profit
Probability Demand
0,05
0,1
0,2
0,3
0,2
0,15
0
1
2
3
4
5
Expected Profit
Purchase Prise
Seling Prise
0
0
0
0
0
0
0
Number of Newspapers Purchased
1
2
3
4
-0,15
-0,3
-0,45
-0,8
0,35
0,2
0,05
-0,1
0,35
0,7
0,55
0,4
0,35
0,7
1,05
0,9
0,35
0,7
1,05
1,4
0,35
0,7
1,05
1,4
0 0,325
0,15
0,5
0,6 0,775
Marginal profit
Marginal Loss
5
-0,75
-0,25
0,25
0,75
1,25
1,75
0,79 0,725
0,35
0,15
How Many to buy?
• . The previous table shows that if the vendor purchases 4
newspapers, then on average, the vendor would make $0.80
in profit. All others have lower expected values. Hence, the
vendor should purchase 4 newspapers.
• . Note that if the vendor purchases 4 newspapers, then on
any given day, the vendor would make {-$0.60, -$0.10, $0.40, $0.90, or $1.40}, not $0.80. However, in the long run,
the vendor would average $0.80.
• . Good homework assignment: Set up the Excel table so
that you can make changes to the parameters of the
problem (probabilities and prices)and the table would
automatically reflect the changes.
Exercises
The following table shows the distribution of blood types in the general
population.
What is the probability a person to have type 0 blood?
What is the probability a person will be Rh-?
What is the probability a married couple will both be Rh-?
What is the probability a married couple will both have type AB blood?
What is the probability a person will ne Rh– given she or he has type 0
blood
What is the probability a person will have type B blood given he or she is
Rh+
A
Rh+
Rh-
34%
6%
B
AB
9%
2%
4%
1%
O
38%
6%
Exercises
An oil company purchased an option on land in Alaska. Preliminary
geologic studies assign the following probabilities.
P(high-quality oil)=0.50
P(Medium quality oil)=0.20
P(no oil) = 0.30
a)What is the probability of finding oil?
b)After 200 feet of drilling on the first well, a soil test was made. The
probability of finding the particular typeof soil identified by the test are:
P(soil\high-quality oil)=0.20
P(soil\Medium-quality oil)=0.80
P(soil\no oil)=0.20
How should the firm interpret the soil test? What are the revised
probabilities, and what is the new probabilities of finding soil?