Answer: | Company A(X₁) Profits | -1 Million | 0 Million | 2 Million | 4 Million | |------------------------|------------|-----------|-----------|-----------| | P(X₁|X₂ = 100) | 0 | 0.1952 | 0.4144 | 0.3904 |

## Explanation The correct answer is **D** because: 1. **Conditional Probability Requirements**: For a valid probability distribution, all probabilities must sum to 1. In option D: - 0 + 0.1952 + 0.4144 + 0.3904 = 1.0000 ✓ 2. **Checking other options**: - **Option A**: 0.0697 + 0.4274 + 0.3436 + 0.1598 = 1.0005 (slightly above 1 due to rounding) - **Option B**: 0.0697 + 0.5274 + 0.6436 + 0.1598 = 1.4005 (clearly exceeds 1) - **Option C**: 0.0697 + 0.5274 + 0.3436 + 0.2598 = 1.2005 (exceeds 1) 3. **Conceptual Understanding**: The conditional distribution P(X₁|X₂ = 100) represents the probability distribution of Company A's profits given that Company B made exactly 100 million in profit. This must satisfy the properties of a probability distribution: - Each probability must be between 0 and 1 - The sum of all probabilities must equal 1 4. **Additional Validation**: Option D shows a more realistic distribution where probabilities are properly scaled and sum exactly to 1, indicating it's likely the correct conditional distribution calculated from the joint probability distribution of both companies' profits. **Key Takeaway**: When evaluating conditional distributions, always verify that the probabilities sum to 1, as this is a fundamental requirement for any valid probability distribution.

Author: Nikitesh Somanthe