A venture capitalist, willing to invest $1,000,000, has three investments to choose from. The first investment, a software company, has a 10% chance of returning $5,000,000 profit, a 30% chance of returning $1,000,000 profit, and a 60% chance of losing the million dollars. The second company, a hardware company, has a 20% chance of returning $3,000,000 profit, a 40% chance of returning $1,000,000 profit, and a 40% chance of losing the million dollars. The third company, a biotech firm, has a 10% chance of returning $6,000,000 profit, a 65% of no profit or loss, and a 25% chance of losing the million dollars.

Construct a PDF for each investment.
Software company:
x	                          P(X = x)	         x · P(X = x)
$5,000,000
$1,000,000
−$1,000,000

Let’s construct the **probability distribution function (PDF)** for each investment. A **PDF** gives the probability associated with each outcome. For each investment, we multiply each outcome $x$ by its respective probability $P(X = x)$ to calculate the **expected value** component $x \cdot P(X = x)$.

---

### **1. Software Company PDF**  
- **Outcomes**: \$5,000,000, \$1,000,000, and $-\$1,000,000$.  
- **Probabilities**: 0.10, 0.30, and 0.60, respectively.

| **x**              | **P(X = x)** | **x · P(X = x)**   |
|--------------------|--------------|--------------------|
| \$5,000,000        | 0.10         | \$500,000          |
| \$1,000,000        | 0.30         | \$300,000          |
| $-\$1,000,000$   | 0.60         | $-\$600,000$     |

#### **Expected Value for Software Company:**
\[
E(X) = 500,000 + 300,000 - 600,000 = 200,000
\]

---

### **2. Hardware Company PDF**  
- **Outcomes**: \$3,000,000, \$1,000,000, and $-\$1,000,000$.  
- **Probabilities**: 0.20, 0.40, and 0.40, respectively.

| **x**              | **P(X = x)** | **x · P(X = x)**   |
|--------------------|--------------|--------------------|
| \$3,000,000        | 0.20         | \$600,000          |
| \$1,000,000        | 0.40         | \$400,000          |
| $-\$1,000,000$   | 0.40         | $-\$400,000$     |

#### **Expected Value for Hardware Company:**
\[
E(X) = 600,000 + 400,000 - 400,000 = 600,000
\]

---

### **3. Biotech Firm PDF**  
- **Outcomes**: \$6,000,000, \$0, and $-\$1,000,000$.  
- **Probabilities**: 0.10, 0.65, and 0.25, respectively.

| **x**              | **P(X = x)** | **x · P(X = x)**   |
|--------------------|--------------|--------------------|
| \$6,000,000        | 0.10         | \$600,000          |
| \$0                | 0.65         | \$0                |
| $-\$1,000,000$   | 0.25         | $-\$250,000$     |

#### **Expected Value for Biotech Firm:**
\[
E(X) = 600,000 + 0 - 250,000 = 350,000
\]

---

### Summary of Expected Values:
1. **Software Company**: \$200,000  
2. **Hardware Company**: \$600,000  
3. **Biotech Firm**: \$350,000

---

**Interpretation:**  
Based on the expected values, the **hardware company** offers the highest expected return, making it the most attractive option in terms of expected profit. However, further considerations (like risk tolerance) might affect the decision.

---

Would you like more details or have any questions?  
Here are some related questions to explore:  
1. What is the variance and standard deviation for each investment?  
2. How does the probability of loss affect the expected value?  
3. What would happen if we adjusted the probabilities?  
4. How would a risk-averse investor evaluate these options?  
5. What is the probability of at least breaking even in each case?  

**Tip:** The **expected value** only captures the average return, not the risk or variability of the investment. Calculating variance can provide more insight into risk.

A venture capitalist, willing to invest $1,000,000, has three investments to choose from. The first investment, a software company, has a 10% chance of returning $5,000,000 profit, a 30% chance of returning $1,000,000 profit, and a 60% chance of losing the million dollars. The second company, a hardware company, has a 20% chance of returning $3,000,000 profit, a 40% chance of returning $1,000,000 profit, and a 40% chance of losing the million dollars. The third company, a biotech firm, has a 10% chance of returning $6,000,000 profit, a 65% of no profit or loss, and a 25% chance of losing the million dollars.

Analyze a venture capitalist's investment decision between three companies using probability and expected value calculations. Understand key concepts like expected value and risk analysis to determine the best option. Suitable for high school students (Grades 10-12).

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