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Let's start by addressing the first problem:

### Problem 1: Optimal Investment Strategy
#### Given Information:
- The investor has three choices: **Company A**, **Company B**, or **Company C**.
- The investor's expected net earnings for each company under three economic conditions (recession, stability, growth) are given in the matrix.
- Probabilities for these economic conditions are:
  - Recession: 0.2
  - Stability: 0.3
  - Growth: 0.5

#### Payoff Matrix (in Rs '000):
|                | **Recession** | **Stability** | **Growth** |
|----------------|---------------|---------------|------------|
| **Company A**  | -15           | 6             | 20         |
| **Company B**  | 4             | 7.5           | 8          |
| **Company C**  | -1.5          | 6             | 15         |

#### Expected Payoff Calculation:
For each company, the expected return is the weighted sum of payoffs across all three scenarios, using the given probabilities.

1. **Company A**:
   \[
   \text{Expected return} = (0.2 \times -15) + (0.3 \times 6) + (0.5 \times 20)
   \]
   \[
   = -3 + 1.8 + 10 = 8.8
   \]

2. **Company B**:
   \[
   \text{Expected return} = (0.2 \times 4) + (0.3 \times 7.5) + (0.5 \times 8)
   \]
   \[
   = 0.8 + 2.25 + 4 = 7.05
   \]

3. **Company C**:
   \[
   \text{Expected return} = (0.2 \times -1.5) + (0.3 \times 6) + (0.5 \times 15)
   \]
   \[
   = -0.3 + 1.8 + 7.5 = 9.0
   \]

#### Conclusion:
- The expected returns for the three companies are:
  - **Company A**: 8.8
  - **Company B**: 7.05
  - **Company C**: 9.0

Thus, **Company C** offers the highest expected return, and the optimal investment strategy is to invest in **Company C**.

#### Decision Tree:
To visualize the decision-making process, we can create a decision tree where each branch corresponds to an investment option, and each node shows the economic conditions (recession, stability, growth) and their associated probabilities and payoffs. From the tree, the investor would select the branch with the highest expected return (i.e., Company C).

Would you like more details on how to draw the decision tree? Let me know if you have any questions on this part, and I'll move on to Problem 2.

---

Here are some related questions for further exploration:

1. How does a change in the probability of growth (say, from 0.5 to 0.6) affect the optimal strategy?
2. What is the effect of adding a new investment option with a similar risk-return profile as Company B?
3. How would the optimal investment strategy change if the investor could invest in multiple companies?
4. If the investor is risk-averse, how would you adjust the decision-making process?
5. How could the use of diversification (investing in more than one company) alter the expected returns?

**Tip:** Expected value calculations help weigh future payoffs based on their likelihood, an essential tool in decision-making under uncertainty.

The uploaded image contains three decision-making problems involving investment strategies, drilling for water supply, and choosing between two investments.

Learn how to solve decision-making problems involving investment strategies, drilling decisions, and choosing between two projects. This guide covers probability, expected value calculations, decision trees, and cost-benefit analysis. Suitable for advanced high school and college students.

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