Tutorial D

A car consists of 40000 parts. Suppose that the probability that a part breaks down is independent of the probability that other parts break down. The probability that a part breaks down during a ride is 0.0001.

What is the probability that all parts work during a ride?

• 83.4%

• 1.8%

• 74.2%

• None of the above

Answer explanation

0,9999^40000 = 1,8%

Suppose that for women of the age 34 and younger the probability to have a pregnancy of a baby with Down syndrome is 1 in 1400. Suppose that this probability is 1 in 140 when the woman is of the age 35 or older. Of all baby’s, the probability that the mother is 34 or younger is 80%.

What is the probability that a mother, with a baby with Down syndrome, has an age of 34 or younger?

• 8.2%

• 28.6%

• 0.06%

• 0.067%

Answer explanation

The correct answer is b. You have to use Bayes Rule to calculate this. For more information, see the lecture slides on the class on ‘Judgement under risk and uncertainty’. The calculation is as follows:

P(34 and younger|Down) = P(Down|34 and younger)*P(34 and younger) / [P(Down|34 and younger)*P(34 and younger) + P(not Down|not 34 and younger)*P(not 34 and younger)] = (1/1400)*0.8 / [ (1/1400)*0.8 + (1/140)*0.2] = 28.6%

Paul can choose between the following options:

A: receiving €20 immediately

B: receiving €25 in 1 week

C: receiving €20 in 3 weeks

D: receiving €25 in 4 weeks

 

Paul is indifferent between A and B. He is also indifferent between C and D. Paul satisfies discounted utility with quasi-hyperbolic discounting (βδ^t), where time is denoted in weeks. His utility function is u(x) = x.

 

Which of the following is true?

• δ = 0.8 and β = 1.

• δ = 0.8 and we do not have sufficient information to determine β.

• Without knowing β we cannot determine δ.

• a, b, and c are all false.

Answer explanation

The correct answer is a. The calculation is as follows:

DU(A) = u(20) = 20

DU(B) = βδ¹ u(25) = 25*βδ

DU(C) = βδ³ u(20) = 20*βδ³ 

DU(D) = βδ⁴ u(25) = 25*βδ⁴ 

From DU(C) = DU(D) it follows that δ = 20/25 = 0.8.

Then from DU(A) = DU(B) it follows that βδ = 20/25 = 0.8, so it must be the case that β = 1.

You are offered the choice between four options, A, B, C, and D, which give different payoffs in different states of the world. There are four possible states of the world: S1, S2, S3, and S4. The utility levels that are obtained with each of these options are given in the following table.

Which of the following combinations of preferences violate the sure-thing principle?

• A ≻ B and C ≻ D

• C ≻ A and D ≻ B

• B ≻ C and A ≻ D

• None of the above violate the sure-thing principle

Answer explanation

The correct answer is d. The argumentation is as follows:

If A≻B, then we must have C≻D by applying the sure thing twice (once to S₁ and once to S₃).

If C≻A, then we must have D≻B by applying the sure thing twice (once to S₂ and once to S₄).

If B≻C, then we cannot apply the sure thing principle, because there is no state with a common outcome. So that also means that B≻C and A≻D does not violate the sure thing principle.

 

To learn more about the sure thing principle, we refer to the lecture slides of the course. The class ‘Choice under risk and uncertainty’ tells more about it.

 

• A ≻ B ≻ C

• C ≻ B ≻ A

• B ≻ A ≻ C

• C ∼ B ≻ A

If you ask people how many groups of 3 out of 9 you can make, they typically give an answer that is greater than when asking how many groups of 6 out of 9 you can create.

 

Of what heuristic or bias is this an example?

• Retrievability of instances

• Effectiveness of a search set

• Hindsight bias

• None of the above

Answer explanation

The correct answer is d. This is an example of a bias called ‘imaginability’. This bias belongs to the category ‘availability heuristic’ (‘retrievablity of instances’ and ‘effectiveness of a search set’ also belong to the availability heuristic).

You bought a ticket with a €30 discount to go to the Efteling (an amusement park in the province of Brabant). The ticket now cost you €10. Instead of going to the Efteling, you could also choose to go to your favorite amusement park, Walibi (in the province of Flevoland). A ticket for Walibi costs €30, but you are willing to pay €50 for it.

 

What are the opportunity costs of going to the Efteling?

• €40

• €50

• €20

• €10

Answer explanation

The correct answer is c. Opportunity costs are the utility that is not generated. This utility is ‘revenues minus costs’ and thus 50 - 30 = 20.

Peter makes the following claim: “I do not like prospect theory, because people should behave according to expected utility, not according to prospect theory.” 

 

Ann makes another claim: “I like prospect theory, because it is a more realistic theory of how people behave than expected utility theory.”

 

According to these claims…

• … Peter finds prospect theory a normative theory.

• … Peter finds expected utility a normative theory.

• … Ann finds prospect theory a normative theory.

• … Ann finds expected utility a descriptive theory.

Answer explanation

The correct answer is b. Peter says he does not like prospect theory, because he thinks expected utility is the normative theory. So he thinks prospect theory is not normative. Ann says that she likes prospect theory, because it is descriptively better than expected utility.

You are offered the choice between three options A, B, and C, which give different payoffs in different states of the world. There are only two possible states of the world: S1 and S2. The utility levels that are obtained with each of these options are given in the following table.

Which option would you choose according to the minimax-regret criterion?

• According to the minimax-regret criterion you would choose A

• According to the minimax-regret criterion you would choose B

• According to the minimax-regret criterion you would choose C

• According to the minimax-regret criterion you would be indifferent between B and C

Answer explanation

The correct answer is b. The regret levels are as follows:

 

            S₁              S₂

A          0          10-5=5

B          4-2=2   10-8=2

C          4-1=3   0

 

The maximum regret is 5 for A, 2 for B and 3 for C. You want to minimize this maximum regret, so you Choose B.

Paul satisfies expected utility with utility function

(𝑥) = 𝑥^0.8 for gains 𝑥 ≥ 0.

He is offered the lottery L which gives €100 with 10% probability and €1 otherwise. In which range falls

 

Paul’s certainty equivalent of lottery L?

• Between 3.4 and 3.7

• Between 4.7 and 5.0

• Between 7.2 and 7.5

• Between 10.8 and 11.1

Answer explanation

The correct answer is c. The calculation is as follows:

EU(L) = 0.1*u(100) + 0.9*u(1)

By definition we have u(ce) = EU(L).

It follows that ce^0.8 = 0.1*100^0.8 + 0.9*1^0.8, so ce = (0.1*100^0.8 + 0.9*1^0.8)^(1/0.8) ≈ 7.25.