1

Business System Analysis & Decision Making- Lecture 10

Zhangxi Lin

ISQS 5340

July 2006

2

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

3

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)

4

Venn Diagrams

Female(30)

ForeignStudent(20)

Female foreign student (10)

(10)

30+10 = 40 20+10 = 30

Male non-foreign student(40)

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Probability Review

Complement )(1)( APAP

Female

Foreignstudent

Non Female

Non ForeignStudent

)(1)( BPBP

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Bayes’ Theorem

)&()()|()(

)&()|(

)&()()|()(

)&()|(

BAPAPABPAP

BAPABP

BAPBPBAPBP

BAPBAP

)()|()()|(

)()|(

)(

)()|()|(

)()|()()|(

BPBAPBPBAP

BPBAP

AP

BPBAPABP

APABPBPBAP

So:

The above formula is referred to as Bayes’ theorem. It is extremely Useful in decision analysis when using information.

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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.

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Histograms

Bin Math Video

<=2 4 0

3-5 6 1

6-8 9 10

>8 3 11

0

2

4

6

8

10

12

<=2 3-5 6-8 >8

Math

Video

The histogram is based on the survey dataFrom the ISQS 5340 class

The scale: 1-10 indicatingStrong negative to strong positive response to the survey questions.

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Descriptive Statistics

  Math Video  Note

Mean 5.59091 8.59091  The average of the data

Median 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  

Sum 123 189  

Count 22 22  

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The Relationship between Data

Chart Title

0

2

4

6

8

10

12

0 2 4 6 8 10 12

Case

Vid

eo

Case-Video

Linear (Case-Video)

Math-Video

0

2

4

6

8

10

12

0 2 4 6 8 10 12

Math

Vid

eo

Math-Video

Linear (Math-Video)

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Regression

Y = a + b*X Example: Video_point = a + b*Math_point

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Regression: Math - Video

Regression Statistics

R Square 0.0248

Standard Error 1.487

Observations 22

 Coefficie

ntsStandard

Error t Stat P-valueLower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

Intercept 9.086 0.7632 11.9051.56E-

10 7.494 10.678 7.494 10.678

X Variable 1 -0.0885 0.1242 -0.713 0.4843 -0.3475 0.1705 -0.348 0.1705

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Regression: Case - Video

Regression Statistics

R Square 0.227

Standard Error 1.323

Observations 22

 Coefficie

ntsStandard

Error t StatP-

value 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

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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.

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Use insurancePay $100+$2 = $102

Not use insurancePay $100

Buyer

$18Good

Bad- $100

Bad

0.99

0.01

0.99

0.01

Good

Expected Value of Perfect Information (EVPI)

$20

- $2

Revisit the previous question: An buyer is to buy something online

EMV = $18.8

EMV = $17.8

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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

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Decision Tree

Buyer

Insurance

No Ins

Insurance

No Ins

Bad (?)

Good (?)Predict: Good (?)

Predict: Bad (?)

Questions:

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 aSuggestion?

Bad (?)

Good (?)

Bad (?)

Good (?)

Bad (?)

Good (?)

$18

- $100

$20

- $2

$18

$20

- $2

- $100

Extended from the previous online trading question

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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

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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

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Decision Tree

Buyer

Insurance

No Ins

Insurance

No Ins

Bad (0.0022)

Good (0.9978)Predict: GoodP(G) = 0.893

Predict: Bad P(W) = 0.107

Bad (0.0748)

Good (0.9252)

Bad (0.0748)

Good (0.9252)

Bad (0.0022)

Good (0.9978) $18

- $100

$20

- $2

$18

$20

- $2

- $100

EMV = $19.87Your choice

EMV = $17.78

EMV = $11.03

EMV = $16.50Your choice

With the help of other people’s suggestion your decision making accuracy is improved

P(Good) = 0.99, P(Bad) = 0.01

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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)

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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.