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MS&E 252 Midterm Decision Analysis I October 26 th , 2004 Page 1 of 22 Midterm Midterm Examination MS&E 252: Decision Analysis I Please read the following instructions carefully! 1. This exam is closed book and closed notes, except for a single sheet (1 side). You may use a calculator and a foreign language dictionary. Please sit in alternate seats. 2. The answer sheet is stapled to the back of the exam. Please tear it off, print your name, SU ID#, Section TA, and sign the Honor Code and exam agreement. Your exam will not be graded unless you have printed your name and signed this statement. 3. There are three blank pages at the end for you to use as scratch paper. All exam questions carry the same weight. Don’t spend too much time on any one question. 4. Read over each question carefully. You may not need all the information given. 5. At the end of the exam, please make sure your probability assignments are written clearly in the appropriate boxes on the answer sheet. If the teaching team cannot read your probability assignment we will use our best judgment as to what you meant to write down and this decision will be final. You will not be allowed to rewrite your assignment after the exam has been graded. You will be graded solely on the basis of your probability assignments. Turn in your answer sheet ONLY. 6. The exam will end at 12:15 pm. You will have five minutes to turn in your answer sheet. In fairness to all members of the class, we will not accept exams after 12:20 pm. 7. You will receive a photocopy of your answer sheet. You will have until 11:59 pm on Thursday October 28 th , 2004 to enter your responses on the web. Please select “Midterm” from the pulldown menu on the answer sheet page. 8. In the spirit of the Honor Code, the Teaching Team will not remain in the classroom during the exam. If you have questions please ask. Clarifications will be announced to the class. 9. Unless stated otherwise, the characters in the exam prefer more money to less, and abide by the principles taught in this class. Do not make any additional assumptions. 10. If your probability assignments do not add up to one, the teaching team will normalize them. If you leave boxes blank and your assignments add up to less than one, we will distribute the remaining probability equally among the empty answer boxes. If your assignments add up to more than one and you have left boxes blank, we will normalize the non-blank boxes and place zeroes in the empty boxes. For example, the following assignments: Question Number a b c d i 0.3 0.5 0.3 0.3 j 0.7 k 0.9 0.2

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Page 1: Midterm

MS&E 252 Midterm Decision Analysis I October 26th, 2004

Page 1 of 22 Midterm

Midterm Examination

MS&E 252: Decision Analysis I

Please read the following instructions carefully! 1. This exam is closed book and closed notes, except for a single sheet (1 side). You may use a

calculator and a foreign language dictionary. Please sit in alternate seats. 2. The answer sheet is stapled to the back of the exam. Please tear it off, print your name, SU

ID#, Section TA, and sign the Honor Code and exam agreement. Your exam will not be graded unless you have printed your name and signed this statement.

3. There are three blank pages at the end for you to use as scratch paper. All exam questions carry the same weight. Don’t spend too much time on any one question.

4. Read over each question carefully. You may not need all the information given. 5. At the end of the exam, please make sure your probability assignments are written clearly in

the appropriate boxes on the answer sheet. If the teaching team cannot read your probability assignment we will use our best judgment as to what you meant to write down and this decision will be final. You will not be allowed to rewrite your assignment after the exam has been graded. You will be graded solely on the basis of your probability assignments. Turn in your answer sheet ONLY.

6. The exam will end at 12:15 pm. You will have five minutes to turn in your answer sheet. In fairness to all members of the class, we will not accept exams after 12:20 pm.

7. You will receive a photocopy of your answer sheet. You will have until 11:59 pm on Thursday October 28th, 2004 to enter your responses on the web. Please select “Midterm” from the pulldown menu on the answer sheet page.

8. In the spirit of the Honor Code, the Teaching Team will not remain in the classroom during the exam. If you have questions please ask. Clarifications will be announced to the class.

9. Unless stated otherwise, the characters in the exam prefer more money to less, and abide by the principles taught in this class. Do not make any additional assumptions.

10. If your probability assignments do not add up to one, the teaching team will normalize them. If you leave boxes blank and your assignments add up to less than one, we will distribute the remaining probability equally among the empty answer boxes. If your assignments add up to more than one and you have left boxes blank, we will normalize the non-blank boxes and place zeroes in the empty boxes. For example, the following assignments:

Question Number a b c d

i 0.3 0.5 0.3 0.3 j 0.7 k 0.9 0.2

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MS&E 252 Midterm Decision Analysis I October 26th, 2004

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would become:

Question Number a b c d i 0.214 0.357 0.214 0.214 j 0.1 0.1 0.7 0.1 k 0.82 0.18 0 0

If you leave a question blank it will be interpreted as an assignment of 0.25 across the board.

10. Your exam be will graded according to the following scoring function:

Question Score = )4ln

ln1(*

15100 p+

where p is the probability you assign to the correct answer.

11. The graph and table below show the number of points you will receive per question

depending on the probability you assign to the correct answer.

12. Relax and make good decisions!

-17.50

-15.00

-12.50

-10.00

-7.50

-5.00

-2.50

0.00

2.50

5.00

7.50

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Probability Assigned to Correct Answer (p)

Sco

re(p

)

p Score0 −∞

0.01 -15.480.05 -7.740.10 -4.410.15 -2.460.20 -1.070.25 0.000.30 0.880.35 1.620.40 2.260.45 2.830.50 3.330.55 3.790.60 4.210.65 4.600.70 4.950.75 5.280.80 5.590.85 5.890.90 6.160.95 6.421.00 6.67

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DO NOT TURN THIS PAGE UNTIL INSTRUCTED TO DO SO!

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1) Which of the following DOES NOT correspond to one of the steps of the Decision Analysis Cycle?

a) Appraise b) Evaluate

c) Clarify

d) Formulate

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2) Anthony is deciding which physical education class to take next quarter. Which of the following statements DOES NOT relate to the decision basis?

a) Anthony prefers to take an outdoor class. b) Anthony’s schedule only permits him to take beginning tennis and intermediate golf. c) Anthony asks fellow students for recommendations, and he learns that the tennis

instructor is great.

d) Anthony registers for beginning tennis and will not be able to drop the class.

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3) How many of the following statements about distinctions are true?

I. A clear distinction is always observable, because once it passes the clarity test we can ask the clairvoyant which degree of the distinction will occur;

II. Two distinctions X and Y can be relevant given &, but irrelevant given another distinction Z and &;

III. The events “this person has no children” and “this person has no grand-children” are not mutually exclusive, but they are collectively exhaustive.

a) 0 b) 1

c) 2 d) 3

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4) Elisa prefers “100% chance for an Infiniti G35” to “30% chance for an Audi A4 and 70% chance for a Honda Accord”. She learns that an Audi A4 costs $40,000 and that a Honda Accord costs $22,000. What can we say about her PIBP for an Infiniti G35?

a) It is impossible to tell from the information given. b) It is strictly greater than $22,000 and strictly less than $27,400. c) It is greater than or equal to $27,400 and strictly less than $40,000. d) It is greater than or equal to $40,000.

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5) Paul just acquired a certificate similar to the one discussed in class, with a bid of $50; he will call on the outcome of a thumbtack toss and if he is right, he will receive $100, otherwise nothing.

Suppose that Paul is allowed to sell the certificate to somebody else. Alice offers $60 for the certificate and Paul agrees to the transaction.

How many of the following statements must be true?

I. Paul’s PIBP for the deal at the time of his bid was greater than or equal to $50; II. Alice’s PIBP for the deal at the time she bought the certificate from Paul was

greater than or equal to $60; III. If Paul had not sold the certificate to Alice, he would have been willing to pay

at least $40 for the clairvoyant’s services.

a) 0 b) 1

c) 2 d) 3

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6) Suppose that the distinctions A, B, and C each have degrees of a and a’, b and b’, c and c’, respectively, and suppose you have the following beliefs:

• A is relevant to B given &, and B is relevant to C given &. • {a | &} = {a’ | &}

How many of the statements below must be true?

I. A is relevant to C, given &. II. B is relevant to A, given &. III. {c | b, &} � {c | b’, &}

a) 0 b) 1 c) 2 d) 3

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7) You friend Susan is unsure about her plans for Friday night, although she knows either John or Peter will ask her to see a movie (either Garden State or Shark Tale). John and Peter are friends, so only one of them will ask her to see the movie. You believe Peter is twice as likely as John to ask her. If John asks, you believe they will see Garden State with probability 1/4. If Peter asks, you believe they will see Garden State with probability 5/8. Given that Susan sees Shark Tale, what must be true?

a) You should assign a strictly greater probability that John asked her to the movie. b) You should assign a strictly greater probability that Peter asked her to the movie. c) You should assign an equal probability of John or Peter asking her to the movie. d) Susan ate popcorn at the movies.

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8) Examine the following relevance diagram: According to the above diagram, how many of the following statements are true (after performing any necessary and legitimate operations)?

I. B and C are irrelevant given & II. A and C are irrelevant given & III. A and B are irrelevant given C and & IV. A and B are relevant given &

a) 0 b) 1 c) 2 d) 3 or 4

A

C

B

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9) Examine the following relevance diagram: Consider the following manipulations on this diagram:

I. Adding an arrow from A to C II. Adding an arrow from C to A III. Flipping the arrow from A to B IV. Flipping the arrow from to B to C

How many of the above manipulations can be made without making any other changes to the relevance diagram?

a) 0 b) 1 c) 2 d) 3 or 4

A

C

B

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10) Alfie just had an interview with Super Decisive Guys (SDG), a leading strategic management-consulting firm. Before the interview, Alfie believed that he had a 50% chance of being hired. After the interview, Alfie found out that 90% of his friends who received a job offer from SDG had had a good interview. Of his friends who interviewed but did not receive a job offer from SDG, 69% had had a good interview. Alfie thinks that these numbers accurately represent his beliefs about his chances of receiving a job offer given his performance on the interview. Knowing that he managed to ace the interview, what probability should Alfie now assign to receiving a job offer?

a) Less than or equal to 0.25 b) Strictly higher than 0.25 but less than or equal to 0.5 c) Strictly higher than 0.5 but less than or equal to 0.75 d) Strictly higher than 0.75

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11) Consider the following relevance diagram:

Which of the following probability trees is best represented by this diagram?

a) b)

c) d) None of those trees are consistent with

the diagram.

X

Z Y

0.2

0.8

Y

Y’

0.1

0.9

Y

Y’

0.7

0.3

Z

Z’

1

0

Z

Z’

0.5

0.5

Z

Z’

0.5

0.5

Z

Z’

0.4

0.6

X

X’

0.2

0.8

Y

Y’

0.2

0.8

Y

Y’

0.7

0.3

Z

Z’

0.4

0.6

Z

Z’

0.7

0.3

Z

Z’

0.4

0.6

Z

Z’

0.5

0.5

X

X’

0.8

0.2

Y

Y’

0.3

0.7

Y

Y’

0.7

0.3

Z

Z’

0.7

0.3

Z

Z’

0.9

0.1

Z

Z’

0.9

0.1

Z

Z’

0.4

0.6

X

X’

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12) Consider the following tree about three distinctions A, B and C : A B C

C1

B1 0.2

0.1C2

0.8

C1

A1 B2 0.3

0.5 0.6C2

0.7

C1

B3 0.4

0.3C2

0.6

C1

B1 0.2

0.1C2

0.8

C1

A2 B2 0.3

0.5 0.5C2

0.7

C1

B3 0.4

0.4C2

0.6

How many of the following are valid statements about the tree above?

I. A is relevant to C given & II. A is relevant to B given & III. C is irrelevant to B given A and &

a) 0 b) 1 c) 2 d) 3

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13) Karen is installing a new alarm system in her house and is trying to draw a relevance diagram that best captures the relationships between the different uncertainties involved. Her beliefs are the following:

•••• She feels that the alarm system could be triggered by either a burglar or an earthquake, but there is a chance the alarm might not respond to either.

•••• She believes that the chances of a burglar robbing her house are lower if there is an earthquake at the same time.

•••• She believes that if there is an earthquake, it may be reported on the radio.

•••• She believes that if the alarm goes off, the security center which is linked to her alarm system may call back.

Which of these diagrams best captures Karen’s beliefs above?

a) b)

c) d)

Earthquake Burglar

Alarm

Radio Call

Earthquake Burglar

Alarm

Radio Call

Earthquake Burglar

Alarm

Radio Call

Earthquake Burglar

Alarm

Radio Call

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14) Examine the following relevance diagram: According to the above diagram, how many of the following statements are true (after performing any necessary and legitimate operations)?

I. A and D are irrelevant given &; II. A and D are irrelevant given C and &; III. A and D are irrelevant given B, C and &.

a) 0

b) 1

c) 2

d) 3

A

C

B

D

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15) You have just acquired a deal that entitles you to the following: •••• If you call correctly on the outcome of a coin toss and then call correctly again on the

outcome of a thumbtack toss, you will win $100; •••• Otherwise, you will win nothing.

Your PISP for the deal is $30. Fortunately, as you are trying to decide on what to call on both of those tosses, there are two people whose services you can hire to help you:

Claire – She has the exact same abilities as a clairvoyant but will only answer your questions regarding the outcome of the coin toss; Claire always tells the truth;

Cassandra – She has the exact same abilities as a clairvoyant and will answer your questions regarding both tosses; however, Cassandra always lies.

You cannot hire both Claire and Cassandra at the same time.

Which of the following statements must be true?

I. Your PIBP for Cassandra’s services should be strictly less than $70;

II. Your PIBP for Claire’s services should be strictly higher than your PIBP for Cassandra’s services.

a) Neither I nor II b) I only c) II only d) Both I and II

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MS&E 252: DECISION ANALYSIS I Midterm Answer Sheet

Last Name (print) ______________________________________SU ID# ________________ First Name: __________________________________________________ TA: (circle one):

Thomas Gael Elisa Susan Eric In recognition and in the spirit of the Honor Code, I certify that I will neither receive nor give unpermitted aid on this exam and that I will report, to the best of my ability, all Honor Code violations observed. I also understand that obtaining a score of negative infinity on a question will result in a zero for the question, and a decrement of two marks to my final grade. If I assign a zero probability to any answer, I declare that it is done consciously and against the advice of the teaching team. Signature: ________________________________________ Date ______________

Question Number

a b c d

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15