Business System Analysis &
Decision Making
- Lecture 10
Zhangxi Lin
ISQS 5340
July 2006
1
Modeling Uncertainty
 Probability Review
 Using Data



Histograms
Descriptive Statistics
Regression
 Value of Information



Conditional Probability and Bayes’ Theorem
Expected Value of Perfect Information
Expected Value of Imperfect Information
2
Probability Review
 P(A|B) = P(A and B) / P(B)
 “Probability of A given B”
 Example, there are 40 female students in a class of
100. 10 of them are from some foreign countries. 20
male students are also foreign students.
 Even A: student from a foreign country
 Even B: a female student
 If randomly choosing a female student to present in
the class, the probability she is a foreign student:
P(A|B) = 10 / 40 = 0.25, or P(A|B) = P (A & B) / P (B)
= (10 /100) / (40 / 100) = 0.1 / 0.4 = 0.25
 That is, P(A|B) = # of A&B / # of B = (# of A&B / Total)
/ (# of B / Total) = P(A & B) / P(B)
3
Venn Diagrams
30+10 = 40
Female
(30)
Male non-foreign student
(40)
20+10 = 30
Foreign
(10) Student
(20)
Female foreign student (10)
4
Probability Review
 Complement
P( A )  1  P( A)
P( B )  1  P( B)
Non Female
Female
Non Foreign
Student
Foreign
student
5
Bayes’ Theorem
P( A & B)
 P( A | B) P( B)  P( A & B)
P( B)
P( A & B)
P( B | A) 
 P( B | A) P( A)  P( A & B)
P( A)
P( A | B) 
So:
P( A | B) P( B)  P( B | A) P( A)
P( B | A) 
P( A | B) P( B)
P( A | B) P( B)

P( A)
P( A | B) P( B)  P( A | B ) P( B )
The above formula is referred to as Bayes’ theorem. It is extremely
Useful in decision analysis when using information.
6
Using Data
 We have addressed briefly behavioral judgments and
theoretical probability issues under certainty and
uncertainty. We now consider how to use data to
conduct our decision analysis.
 Why need data?

No data no decision. Think about why search engine is
so hot.
 How to make data useful
 IT helps us to cope with information explosion
 Models and methods are important to guide us how to
analyze data.
7
Histograms
Bin
Math
Video
<=2
4
0
3-5
6
1
6-8
9
10
>8
3
11
The histogram is based on the survey data
From the ISQS 5340 class
12
10
The scale: 1-10 indicating
Strong negative to strong
positive response to the
survey questions.
8
Math
6
Video
4
2
0
<=2
3-5
6-8
>8
8
Descriptive Statistics
Math
Mean
Video
Note
5.59091
8.59091
6
8.5
Standard Deviation
2.61241
1.46902
Dev = V0.5
Sample Variance
6.82468
2.15801
V = (x- mean)2 / # of obs
Range
9
5
Minimum
1
5
Maximum
10
10
123
189
22
22
Median
Sum
Count
The average of the data
9
The Relationship between Data
Math-Video
12
8
6
4
Math-Video
2
Linear (Math-Video)
0
0
2
4
6
8
10
12
Math
Chart Title
12
10
8
Video
Video
10
6
4
2
Case-Video
0
0
2
4
6
8
Linear (Case-Video)
10
12
Case
10
Regression
 Y = a + b*X
 Example: Video_point = a + b*Math_point
11
Regression: Math - Video
Regression Statistics
R Square
Standard Error
Observations
0.0248
1.487
22
Coefficie
nts
Intercept
X Variable 1
Standard
Error
t Stat
P-value
Lower
95%
Upper
95%
Lower
95.0%
Upper
95.0%
9.086
0.7632
11.905
1.56E10
7.494
10.678
7.494
10.678
-0.0885
0.1242
-0.713
0.4843
-0.3475
0.1705
-0.348
0.1705
12
Regression: Case - Video
Regression Statistics
R Square
0.227
Standard Error
1.323
Observations
22
Coefficie
nts
Standard
Error
t Stat
Pvalue
Lower 95%
Upper
95%
Lower
95.0%
Upper
95.0%
Intercept
6.623
0.859
7.708
2E-07
4.83
8.415
4.83
8.415
X Variable 1
0.297
0.122
2.425
0.025
0.042
0.552
0.042
0.552
13
Value of Information
 When facing uncertain prospects we need
information in order to reduce uncertainty
 Information gathering includes consulting
experts, conducting surveys, performing
mathematical or statistical analyses, etc.
14
Expected Value of Perfect Information
(EVPI)
Revisit the previous question: An buyer is to buy something online
Bad
Not use insurance
Pay $100
- $100
0.01
EMV = $18.8
0.99
Buyer
Good
$20
Bad
- $2
0.01
EMV = $17.8
Use insurance
Pay $100+$2 = $102
Good
0.99
$18
15
Expected Value of Imperfect
Information (EVII)
 We rarely access to perfect information, which is common. Thus
we must extend our analysis to deal with imperfect information.
 Now suppose we can access the online reputation to estimate
the risk in trading with a seller.
 Someone provide their suggestions to you according to their
experience. Their predictions are not 100% correct:
 If the product is actually good, the person’s prediction is 90%
correct, whereas the remaining 10% is suggested bad.
 If the product is actually bad, the person’s prediction is 80%
correct, whereas the remaining 20% is suggested good.
 Although the estimate is not accurate enough, it can be used to
improve our decision making:
 If we predict the risk is high to buy the product online, we
purchase insurance
16
Decision Tree
Extended from the previous online trading question
No Ins
Predict: Good (?)
Insurance
Buyer
No Ins
- $100
Good (?)
$20
Bad (?)
- $2
Good (?)
$18
Bad (?)
- $100
Good (?)
$20
Bad (?)
Predict: Bad (?)
Insurance
Questions:
Bad (?)
Good (?)
1. Given the
suggestion
What is your
decision?
2. What is the
probability
wrt the decision you
made?
3. How do you
estimate
The accuracy of a
Suggestion?
- $2
$18
17
Applying Bayes’ Theorem





Let “Good” be even A
Let “Bad” be even B
Let “Suggest Good” be event G
Let “Suggest Bad” be event W
According to the previous information, we know:



P(G|A) = 0.9, P(W|A) = 0.1
P(W|B) = 0.8, P(G|B) = 0.2
P(A) = 0.99, P(B) = 0.01
 We want to learn the probability the outcome is good providing the
suggestion is “good”. i.e.

P(A|G) = ?
 We want to learn the probability the outcome is bad providing the
suggestion is “bad”. i.e.

P(B|W) = ?
 We may apply Bayes’ theorem to solve this with imperfect information
18
Applying Bayes’ Theorem
 According to previous formula, we have
 P(A|G) = P(G|A)P(A) / P(G)
= P(G|A)P(A) / [P(G|A)P(A) + P(G|B)P(B)]
= P(G|A)P(A) / [P(G|A)P(A) + P(G|B)(1 - P(A))]
= 0.9 * 0.99 / [0.9 * 0.99 + 0.2 * 0.01]
= 0.9978 > 0.99
 P(B|W) = P(W|B)P(B) / P(W)
= P(W|B)P(B) / [P(W|B)P(B) + P(W|A)P(A)]
= P(W|B)P(B) / [P(W|B)P(B) + P(W|A)(1 - P(B))]
= 0.8 * 0.01 / [0.8 * 0.01 + 0.1 * 0.99]
= 0.0748 > 0.01
 Apparently, the suggestion provide better information
than the original probability
19
Decision Tree
P(Good) = 0.99, P(Bad) = 0.01
Bad (0.0022)
Predict: Good
P(G) = 0.893
No Ins
- $100
Good (0.9978)
$20
Bad (0.0022)
- $2
EMV = $19.87
Your choice
EMV = $17.78
Insurance
Buyer
No Ins
Good (0.9978)
$18
Bad (0.0748)
- $100
EMV = $11.03
Good (0.9252)
Bad (0.0748)
Predict: Bad
P(W) = 0.107
Insurance
Good (0.9252)
$20
- $2
$18
With the help of other people’s suggestion your decision making
accuracy is improved
EMV = $16.50
Your choice
20
Exercise 3


There is only two events in a scenario: A and B. If P(A) = 0.7, P(B) = 0.5, P(A|B)
= 0.4, and P(A & B) = 0.2, calculate P(B|A).
You are to buy a new digital camera. It costs $400 (but worth $600 to you). You
are offered to buy a 3-year warrantee for $50, which allows you to exchange for
a brand new camera if your camera get any problem. Otherwise, your camera
could be useless if it stops working. To decide if this is necessary, you ask your
friend for advice. You friend can provide a correct advice with 80% probability if
the camera will be in good quality. He can also identify the possible quality
problem with 70% probability, which will encourage you to buy the warrantee.
You know the probability that the camera will have problems in a period of 3
years is 10%.

(1)





Draw a decision tree
Calculate the conditional probability that you buy a good camera given that your friend
provide a positive advice.
Calculate the conditional probability you buy a camera in poor quality given that your
friend provide a negative advice.
Calculate EMVs under different situations
(2)


If you have a utility function U(x) = x0.6, without the advice, what will be your choice?
Compare the difference between the solution with the one from (1)
21
Homework #3

Suppose you have three choices of investment:







High risk stock with a 0.5 probability of making $10,000 if the market
will be up, a 0.3 probability of making $100 if the market is flat, and a
0.2 probability of losing $1,600 if the market is down.
Low risk stock with a 0.5 of probability making $3,600 if the market
will be up, a 0.3 probability of making $900 if the market is flat, and a
0.2 probability of losing $625 if the market is down.
You can also save the money in the saving account making $2500
Draw a decision tree
Calculate the EMV and make your decision
If the utility function is U(x) = +x0.6, what are expected utilities of the
choices? Which one should be your choice?
Explain why the decision outcomes are different wrt different criteria.
22