This is a problem of calculating the expected value (EV) of each investment to help determine which one is statistically the best option for the venture capitalist.

Let's calculate the expected value of each investment based on their respective outcomes and probabilities.

1. Software Company:

• 10% chance of returning a $5,000,000 profit
• 30% chance of returning a $1,000,000 profit
• 60% chance of losing the $1,000,000

Expected Value:

$\text{EV} = (0.10 \times 5,000,000) + (0.30 \times 1,000,000) + (0.60 \times (-1,000,000))$ $\text{EV} = 500,000 + 300,000 - 600,000 = 200,000$

2. Hardware Company:

• 20% chance of returning a $3,000,000 profit
• 40% chance of returning a $1,000,000 profit
• 40% chance of losing the $1,000,000

Expected Value:

$\text{EV} = (0.20 \times 3,000,000) + (0.40 \times 1,000,000) + (0.40 \times (-1,000,000))$ $\text{EV} = 600,000 + 400,000 - 400,000 = 600,000$

3. Biotech Firm:

• 10% chance of returning a $6,000,000 profit
• 65% chance of no profit or loss (i.e., $0)
• 25% chance of losing the $1,000,000

Expected Value:

$\text{EV} = (0.10 \times 6,000,000) + (0.65 \times 0) + (0.25 \times (-1,000,000))$ $\text{EV} = 600,000 + 0 - 250,000 = 350,000$

Comparison of Expected Values:

• Software Company EV: $200,000
• Hardware Company EV: $600,000
• Biotech Firm EV: $350,000

Conclusion:

The hardware company has the highest expected value of $600,000, so, based on this calculation, the venture capitalist should invest in the hardware company for the statistically best outcome.

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Would you like more details on any step of the calculation, or further questions about investment strategies? Here are 5 related questions:

1. How can expected value influence investment decisions beyond this example?
2. What are some limitations of using expected value in real-world investing?
3. How could risk tolerance affect the venture capitalist's decision?
4. Can expected value alone determine the best investment without considering risk?
5. How would different probabilities affect the expected value of each investment?

Tip: Expected value does not account for risk; it's purely statistical. You should also consider the potential magnitude of losses and whether you can tolerate them.