Answer: 7.89%

## Explanation This question requires applying **Bayes' theorem** to calculate a conditional probability. Let's define the events: - **A** = Bond defaults - **B** = Market decreases by 20% We need to find: **P[A|B]** = Probability that bond defaults given market decreases by 20% ### Step 1: Identify the relevant probabilities from the table From the probability matrix: - **P[A ∩ B]** = Probability that bond defaults AND market decreases by 20% = **3%** (0.03) - **P[B]** = Total probability that market decreases by 20% = 35% + 3% = **38%** (0.38) ### Step 2: Apply Bayes' theorem Bayes' theorem formula: \[ P[A|B] = \frac{P[A \cap B]}{P[B]} \] Substitute the values: \[ P[A|B] = \frac{0.03}{0.38} = 0.0789 \rightarrow 7.89\% \] ### Step 3: Why other options are incorrect - **A (3.00%)**: This is P[A ∩ B] - the joint probability, not the conditional probability - **B (4.00%)**: This is the unconditional probability of default P[A] = 1% + 3% = 4% - **D (10.53%)**: This incorrectly uses P[A] in the numerator: 0.04/0.38 = 0.1053 ### Key Concept This demonstrates how to calculate conditional probabilities using joint and marginal probabilities from a contingency table. The conditional probability P[A|B] represents the probability of event A occurring given that event B has already occurred.

Author: LeetQuiz Editorial Team