Explanation

To calculate the expected value of the VC firm's profit, we need to construct the discrete probability distribution of all possible outcomes:

Step 1: Calculate Probabilities

• Probability that only 1 company is successful:

  • This can happen in 2 ways (Company A succeeds & B fails, or Company A fails & B succeeds)
  • Probability = $0.30 \times (1 - 0.30) \times 2 = 0.30 \times 0.70 \times 2 = 0.42$

• Probability that both companies are successful:

  • $0.30 \times 0.30 = 0.09$

• Probability that neither company is successful:

  • $(1 - 0.30) \times (1 - 0.30) = 0.70 \times 0.70 = 0.49$

Verification: $0.42 + 0.09 + 0.49 = 1.0$ ✓

Step 2: Calculate Expected Value

$$\begin{align*} E(X) &= (\text{Profit from 1 success}) \times P(\text{1 success}) \\ &+ (\text{Profit from 2 successes}) \times P(\text{2 successes}) \\ &+ (\text{Profit from 0 successes}) \times P(\text{0 successes}) \end{align*}$$ $$E(X) = \text{USD } 40 \text{ million} \times 0.42 + \text{USD } 70 \text{ million} \times 0.09 + \text{USD } (-15) \text{ million} \times 0.49$$ $$E(X) = 16.8 + 6.3 - 7.35 = \text{USD } 15.75 \text{ million}$$

Why Other Options are Incorrect:

• A (USD 6.30 million): Only considers the profit if both companies succeed ($0.09 \times 70 = 6.3$)
• B (USD 7.35 million): Fails to account for both ways that only 1 company can succeed
• D (USD 16.80 million): Only considers the profit if exactly 1 company succeeds ($0.42 \times 40 = 16.8$)

This calculation demonstrates the importance of considering all possible outcomes and their respective probabilities when calculating expected values in risk management contexts.