To answer the questions, we need to calculate the expected value and variance of the profits associated with the two expansion alternatives based on the given probability distribution.

Data from the Table:

• Demand Levels: Baja (Low), Mediana (Medium), Alta (High)
• Probabilities: 0.20, 0.50, 0.30
• Profit at Medium Scale Expansion (in thousands):
  • Low Demand (50), Medium Demand (150), High Demand (200)
• Profit at Large Scale Expansion (in thousands):
  • Low Demand (0), Medium Demand (100), High Demand (300)

Step 1: Calculate the Expected Value for Each Expansion

The expected value (E) for any scenario can be calculated as:

$E = \sum (Profit \times Probability)$

For medium scale (denote as $E_x$): $E_x = (50 \times 0.20) + (150 \times 0.50) + (200 \times 0.30)$

For large scale (denote as $E_y$): $E_y = (0 \times 0.20) + (100 \times 0.50) + (300 \times 0.30)$

Step 2: Calculate the Variance for Each Expansion

The variance ($\sigma^2$) is calculated as:

$\sigma^2 = \sum \left[(Profit - Expected\ Value)^2 \times Probability\right]$

Let's first calculate the expected values and then proceed with the variance calculations. I will do the calculations now.### Results:

1. Expected Values:

   • Medium Scale Expansion: $E_x = 145$ thousand dollars
   • Large Scale Expansion: $E_y = 140$ thousand dollars

2. Variance:

   • Medium Scale Expansion: $\sigma^2_x = 2725$
   • Large Scale Expansion: $\sigma^2_y = 12400$

Answering the Questions:

a) Value Expected and Variance:

• Medium Scale Expansion: Expected Value = 145, Variance = 2725
• Large Scale Expansion: Expected Value = 140, Variance = 12400

b) Maximizing Expected Utility: The Medium Scale Expansion has a higher expected value of 145 compared to 140 for the Large Scale Expansion. Therefore, for the objective of maximizing expected utility, Medium Scale Expansion is preferable.

c) Minimizing Risk or Uncertainty: Variance measures risk, and a lower variance implies lower risk. The Medium Scale Expansion has a significantly lower variance (2725) compared to the Large Scale Expansion (12400). Thus, for minimizing risk or uncertainty, Medium Scale Expansion is also preferable.

Would you like further details or have any questions?

Here are some related questions:

1. How do probability distributions impact expected values?
2. How does variance help in decision-making under uncertainty?
3. What is the significance of maximizing utility in business decisions?
4. Could a decision be optimal under one criterion but not another?
5. What other statistical measures could be used for risk assessment?

Tip: When evaluating options under uncertainty, balancing expected value and risk (variance) is essential for making informed decisions.