Consider the following payoff table with amounts in $ millions. The column with investments lists mutually exclusive investment decisions.If you use the expected value criterion and you choose investment d₂,then the probability that you assign to the state of nature S₁ is __________. A)more than 0.25 and less than 0.67 B)more than 0.33 and less than 0.67 C)more than 0.25 and less than 0.5 D)more than 0.33 and less than 0.5

Question

Quizplus · Accepted Answer

Using the expected value criterion, we need to find the expected value of each investment option. Expected value of investment d: (0.4 x 8) + (0.6 x 4) = 5.2 Expected value of investment d_{: (0.3 x 15) + (0.7 x 3) = 6.6
Expected value of investment d₁: (0.5 x 12) + (0.5 x 0) = 6

Therefore, investment d₁ has the highest expected value of 6 million. To choose this investment, we must assign a probability to the state of nature S₁, which corresponds to the outcome of either a favorable or unfavorable business climate.

Since d₁ is the best choice, we know that the expected value of this investment is greater than or equal to the expected value of the other options. Therefore, using the expected value criterion again, we can set up the following inequality:

(0.5 x 12) + (0.5 x 0) ≥ (0.4 x 8) + (0.6 x 4) and (0.5 x 12) + (0.5 x 0) ≥ (0.3 x 15) + (0.7 x 3)

Simplifying these inequalities, we get:

6 ≥ 4.8 and 6 ≥ 5.4

Both of these inequalities are true. Therefore, the probability assigned to S₁ must be greater than the probabilities assigned to the other two states of nature, which correspond to the outcomes of either a strong or weak business climate.

To find the range of probabilities that satisfy these inequalities, we can use the formula:

(EV of d₁) - (EV of other investment) / (difference in EVs) ≤ P ≤ 1 - [(EV of other investment) - (EV of d₁)] / (difference in EVs)

Applying this formula to the first inequality, we get:

(6 - 4.8) / (6 - 5.2) ≤ P ≤ 1 - (5.2 - 6) / (6 - 5.2)

0.4 ≤ P ≤ 0.6667

Similarly, for the second inequality, we get:

0.3333 ≤ P ≤ 0.5

Therefore, the probability that you assign to the state of nature S₁ is more than 0.33 and less than 0.5. Answer choice D is the only one within this range.}

Consider the Following Payoff Table with Amounts in $ Millions