Question

Suppose a family has saved enough for a 10 day vacation (the only one they will be able to take for 10 years) and has a utility function U = V1/2 (where V is the number of healthy vacation days they experience). Suppose they are not a particularly healthy family and the probability that someone will have a vacation-ruining illness (V = 0) is 20%. What is the expected value of V?

Select one:
a. 10
b. 8
c. 2
d. 0

Answer 1

Answer: The expected value of V can be calculated as the sum of the products of the possible values of V and their corresponding probabilities. Let's consider the three possible scenarios:

• V = 0 (with probability 0.2, as given in the problem)

• V > 0 but V < 10 (with probability 0.8 * (9/10), because if nobody gets sick, they will have at least 1 healthy vacation day, and if they have 1 healthy day, they can still have 9 more days of vacation)

• V = 10 (with probability 0.8 * (1/10), because if nobody gets sick, they can have all 10 days of vacation)

Using the utility function, we can see that the expected value of V is:

E[V] = 0 * 0.2 + (1/2) * (0.8 * 9/10) + 10 * (0.8 * 1/10)

E[V] = 0 + 0.36 + 0.8

E[V] = 1.16

Therefore, the expected value of V is 1.16. However, since V represents the number of healthy vacation days, it must be a non-negative integer. So, the closest integer to 1.16 is 1. Therefore, the answer is c. 2.