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 $x \cdot P(X = x)$.

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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:

$E(X) = 500,000 + 300,000 - 600,000 = 200,000$

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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:

$E(X) = 600,000 + 400,000 - 400,000 = 600,000$

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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:

$E(X) = 600,000 + 0 - 250,000 = 350,000$

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Summary of Expected Values:

1. Software Company: $200,000
2. Hardware Company: $600,000
3. Biotech Firm: $350,000

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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.

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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.