Week Two Homework Assignment – Probability Distributions and Decision Analysis

A student has an important exam coming up and is contemplating not studying for the exam in order to attend a party with his friends. The student must earn a minimum score of 70% on the exam in order to successfully maintain his desired GPA. Suppose the student knows in advance that the exam will consist of twenty multiple choice questions with four possible answers for each question. Answer questions 1-3 using the preceding information and modeling this situation as a binomial distribution.

1. What is the probability that the student will successfully earn exactly the required minimum score of 70% on the exam based solely upon randomly guessing the correct answer for each question?

o 2.57

o 2.57E-02

o 2.57E-05

o 2.57E-04

2. What is the probability that the student will earn less than the required minimum score of 70% on the exam based solely upon randomly guessing the correct answer for each question?

o 0.74673

o 0.85198

o 0.99997

o 0.23499

3. What is the probability that the student will successfully earn no less than the required minimum score of 70% on the exam based solely upon randomly guessing the correct answer for each question?

o 3.51E-04

o 2.95E-05

o 6.87E-06

o 1.27E-03

The mean time required to complete a certain type of construction project is 52 weeks with a standard deviation of 3 weeks. Answer questions 4-7 using the preceding information and modeling this situation as a normal distribution.

4. What is the probability of completing the project in no more than 52 weeks?

o 0.25

o 0.50

o 0.75

o 0.05

5. What is the probability of completing the project in more than 55 weeks?

o 0.2743

o 0.5091

o 0.1587

o 0.0546

6. What is the probability of completing the project between 56 weeks and 64 weeks?

o 0.2587

o 0.3334

o 0.5876

o 0.0911

7. What is the probability of completing the project within plus or minus one standard deviation of the mean?

o 0.951

o 0.852

o 0.759

o 0.683

Customers arrive at a supermarket check-out counter with an average arrival rate of 9 customers per hour. Answer questions 8-10 using the preceding information and modeling this situation as a Poisson distribution.

8. What is the probability of less than 5 customers arriving at the supermarket check-out counter in a given one hour period?

o 0.054

o 0.446

o 0.359

o 0.612

9. What is the probability of exactly 12 customers arriving at the supermarket check-out counter in a given one hour period?

o 0.262

o 0.044

o 0.073

o 0.189

10. What is the probability of no less than 12 customers arriving at the supermarket check-out counter in a given one hour period?

o 0.115

o 0.197

o 0.381

o 0.686

A local commuter bus service advertises that buses run every twelve minutes along a certain route. Answer questions 11and 12 using the preceding information and modeling this situation as an exponential distribution.

11. What is the probability of a bus picking up the passengers at a given bus stop in less than or equal to 12 minutes following their arrival at the bus stop?

o 0.519

o 0.632

o 0.466

o 0.772

12. What is the probability of a bus picking up the passengers at a given bus stop in more than 15 minutes following their arrival at the bus stop?

o 0.287

o 0.343

o 0.541

o 0.119

Scores for a certain exam follow a normal distribution with a mean of 87 and a standard deviation of 4. Answer questions 13and 14 using the preceding information.

13. What is the standard Z-score associated with a score of 89?

o 0.0

o 1.1

o 0.8

o 0.5

14. What is the probability that a randomly selected student’s score will fall between a standard Z-score of -1.5 and a standard Z-score of 1.8?

o 0.599

o 0.682

o 0.761

o 0.897

Use the following data set and assumptions to create payoff and regret matrices in order to answer questions 15 through 18.

Alternative A Alternative B

Present value of costs incurred for selecting the alternative prior to knowing which state of nature actually occurs $7,000,000 $12,000,000

Present value of future revenues if the state of nature that actually occurs favors the selected alternative $75,000,000 $125,000,000

Present value of future recoverable costs if the state of nature that actually occurs does not favor the selected alternative $6,000,000 $5,000,000

Assume that there are four collectively exhaustive alternatives from which you must choose prior to knowing the actual state of nature that will occur.

· Select alternative A

· Select alternative B

· Select both alternatives A and B

· Select neither alternative A nor alternative B

Assume that there are two collectively exhaustive states of nature that could occur.

· The state of nature that occurs favors alternative A.

· The state of nature that occurs favors alternative B.

Assume that the estimated costs associated with selecting a given alternative must be incurred prior to knowing which state of nature will actually occur.

15. What is the payoff if alternative A is selected and the state of nature that occurs favors alternative A?

o -$7,000,000

o $0

o $61,000,000

o $68,000,000

16. What is the payoff if alternative B is selected and the state of nature that occurs favors alternative A?

o -$7,000,000

o $0

o $61,000,000

o $68,000,000

17. What is the regret if alternative A is selected and the state of nature that occurs favors alternative B?

o $0

o $1,000,000

o $113,000,000

o $114,000,000

18. What is the regret if alternative B is selected and the state of nature that occurs favors alternative A?

o $0

o $1,000,000

o $75,000,000

o $114,000,000

Use the payoff and regret data provided in the following payoff and regret matrices to answer questions 19 through 22.

Payoff Matrix

States of Nature

Alternatives State of Nature Favors A State of Nature Favors B State of Nature Favors Neither A nor B

A $88 ($5) ($5)

B ($8) $156 ($8)

A and B $80 $151 ($13)

Neither A nor B $0 $0 $0

Note: The values in the preceding table are shown in millions of dollars.

Regret Matrix

States of Nature

Alternatives State of Nature Favors A State of Nature Favors B State of Nature Favors Neither A nor B

A $0 $161 $5

B $96 $0 $8

A and B $8 $5 $13

Neither A nor B $88 $156 $0

Note: The values in the preceding table are shown in millions of dollars.

19. What is the optimal decision regarding which alternative to select using the Expected Monetary Value decision rule, assuming the probability of the state of nature that favors alternative A occurring is 0.50, the probability of the state of nature that favors alternative B occurring is 0.30, and the probability of the state of nature that favors neither location A nor location B occurring is 0.20?

o Alternative A

o Alternative B

o Alternative A and B

o Neither Alternative A nor Alternative B

20. What is the optimal decision regarding at which location(s) to purchase property using the Expected Opportunity Loss decision rule, assuming the probability of the state of nature that favors alternative A occurring is 0.50, the probability of the state of nature that favors alternative B occurring is 0.30, and the probability of the state of nature that favors neither location A nor location B occurring is 0.20?

o Alternative A

o Alternative B

o Alternative A and B

o Neither Alternative A nor Alternative B

21. A consulting firm has contacted your company claiming that their analysis conclusively indicates that the probability of the state of nature that favors location A occurring is 0.50, the probability of the state of nature that favors location B occurring is 0.30, and the probability of the state of nature that favors neither location A nor location B occurring is 0.20. The consultants have offered to share their analysis with your company for a fee of $25.0 million. What is the Expected Value of Perfect Information in this scenario?

o $27,200,000

o $8,100,000

o $19,500,000

o $0

22. Should your company accept the consultant’s offer?

o Yes

o No

For questions 23 through 25, use the following payoff matrix data to create a sensitivity analysis data table that summarizes the expected monetary value for each possible alternative relative to the probability of the state of nature that occurs favoring Alternative A being varied from 0.0 to 1.0 in increments of 0.01.

Payoff Matrix

Alternatives State of Nature 1 (Favors A) State of Nature 2 (Favors B)

A $105.0 ($5.0)

B ($25.0) $55.0

A and B $80.0 $50.0

Neither A nor B $0.0 $0.0

23. For what range of probability of the state of nature that favors alternative A occurring is selecting alternative A the optimal decision?

o 0.00 – 0.04

o 0.05 – 0.68

o 0.69 – 1.00

o No range

24. For what range of probability of the state of nature that favors alternative A occurring is selecting alternative B the optimal decision?

o 0.00 – 0.04

o 0.05 – 0.68

o 0.69 – 1.00

o No range

25. For what range of probability of the state of nature that favors alternative A occurring is selecting both alternatives A and B the optimal decision?

o 0.00 – 0.04

o 0.05 – 0.68

o 0.69 – 1.00

o No range