Suppose one needs to prepare for a final exam. Passing the class will be determined by their passing of the final. The student plans to devote either four, eight, or twelve hours to studying. They obviously want to pass the class, but also have other finals to prepare for.Thus, multiple objectives exist.These will need to be combined into a single valuation for the branches.

The main uncertainty from the perspective of the student is the difficulty level of the test.It could be either difficult or easy, and the student really has little indication of which it will be. Because of this complete uncertainty, the student will simply estimate a 50% chance of each option. In either case, the student will have a probability of passing that is in accordance with the number of hours spent studying. If the test is difficult, Table A applies, and if the test is easy, Table B applies.

Table A: difficult exam probabilities

hours studying	probability of passing
4	0.2
8	0.7
12	0.75

Table B: easy exam probabilities

hours studying	probability of passing
4	0.6
8	0.8
12	0.9

Question 1: Construct the influence diagram for the problem

For now, we can assume that all of our objectives can be consolidated into a single octagonal node we will call "Value." The specific manner in which we will accomplish this consolidation is detailed in question 2. Carefully construct the influence diagram for this scenario.

Question 2: Constructing the Tree

In order to evaluate the decision tree, we will need valuations for all possible paths through the tree.For our scenario, this will involve 2 3 = 6 different valuations, though the number of actual paths through the tree is greater.

Of course, the challenge is that we are attempting to deal with two different objectives at the same time (maximizing probability of passing while minimizing study time), and those two different objectives are also on two different measurement scales (time and probability). We will need to consolidate these to evaluate the tree. We will do so by attempting to attach monetary values to each measure, which is, in essence, incorporating a sort of utility.

For the class, the student is only concerned with successfully passing or not (one could assume either that it is just a pass/fail class or that the student is not very focused on GPA). We will (initially) evaluate the value of passing the class at $2500, the cost of tuition for the class. Because of the time crunch and numerous obligations, the student is evaluating their time at $50 per hour. There are actually multiple correct ways to handle this, but when consolidating, I would recommend handling passing the class as positive value, failing the class as no value, and the future time investment as negative value.

For example, if the student were to spend 10 hours studying for the test (not at actual amount under consideration), the value for passing would be 2500 - 10(5) = 2000 and the value for failing would be 0 - 10(5) = -500.

After determining the valuations for all possible paths, construct and evaluate the tree.Also, clearly identify which alternative should be chosen. This will be our base tree.

Question 3: Sensitivity analysis, Part I (sensitivity to valuation/utility)

Suppose upon further reflection the student realizes that passage of the class has possibly been undervalued. Consideration should be made not only for tuition cost, but also the fact that more time would need to be exerted again to pass, plus degree pursuit and lifelong goals could be delayed. The student decides that they have probably worked an average of 5 hours a week on the course for the past 15 weeks. However, because the time crunch is not as significant during the normal semester, the value on each hour only needs to be $30. This yields a value for passing the class of $2500 (tuition) + $30 5 15 (study time) = $4750, which the student decides to round up to $5000 to additionally account for the (apparently small) psychological benefit that would be experienced if they pass. Thus, $5000 should be used for the value of passing the class instead of $2500 in determining the values of the paths (it still needs to be combined with the study times). Re-evaluate the tree under these conditions, and discuss what (if any) changes to the tree evaluation will exist when comparing with the original tree.

Question 4: Sensitivity analysis, Part II (sensitivity to probabilities)

Using the tree from Question 2, we now address the main component of the problem that was a complete guess: the probability that the professor gives a difficult final.We will contrast our base case (50%) with two other cases (unsurprisingly, one high, and one low). For each of the following cases, construct a new tree and evaluate.

Case 1: The probability of an easy exam is 95%.

Case 2: The probability of a difficult exam is 95%

After determining the value of the new trees, identify the spread of values, as well as any differences in decisions. If the decisions are different, identify any probabilities at which the decision strategy changes.

Note: Should the two cases result in different extremes (4 hours and 12 hours) the change in strategy will likely occur twice, once where the strategy changes from 4 hours to 8 hours, and once again where it changes from 8 hours to 12 hours.Both cutoffs would be needed in this case.

Question 5: Value of perfect information

We are back to the base tree from Question 2 (with 50% probabilities) for all remaining questions.

Suppose the professor lets slip what type of final he is intending to give, removing this uncertainty. What is the value of this perfect information? (The completed new decision tree must be part of the answer.)

Question 6: Value of perfect control

Suppose the student is (unethically) considering bribing the professor into giving the easier exam. What would be the maximum amount the student would be willing to pay?