Answer: 0.94

## Detailed Explanation Let's calculate the probability step by step: **Step 1: First Line of Defense** - First Line detects 70% of all incidents - Probability detected by First Line: 0.70 - Probability remaining undetected: 1 - 0.70 = 0.30 **Step 2: Second Line of Defense** - Second Line detects 60% of remaining undetected incidents - Probability detected by Second Line: 0.30 × 0.60 = 0.18 - Probability remaining undetected: 0.30 × (1 - 0.60) = 0.30 × 0.40 = 0.12 **Step 3: Third Line of Defense** - Third Line detects 50% of remaining undetected incidents - Probability detected by Third Line: 0.12 × 0.50 = 0.06 - Probability remaining undetected: 0.12 × (1 - 0.50) = 0.12 × 0.50 = 0.06 **Step 4: Total Probability of Detection** - Total detected = First Line + Second Line + Third Line - Total detected = 0.70 + 0.18 + 0.06 = 0.94 **Step 5: Probability of Detection by At Least One Line** - This is equivalent to 1 minus the probability that no line detects the incident - Probability no detection = 0.06 - Probability at least one detection = 1 - 0.06 = 0.94 Wait, let me recalculate this carefully: **Correct Calculation:** - First Line detects: 70% = 0.70 - Remaining undetected: 30% = 0.30 - Second Line detects: 60% of 30% = 0.60 × 0.30 = 0.18 - Remaining undetected: 0.30 - 0.18 = 0.12 - Third Line detects: 50% of 12% = 0.50 × 0.12 = 0.06 - Remaining undetected: 0.12 - 0.06 = 0.06 **Total detected = 0.70 + 0.18 + 0.06 = 0.94** But the question asks for "probability that an operational risk incident is detected by at least one of the three lines of defense" which should be 1 - probability(no detection) = 1 - 0.06 = 0.94 However, looking at the options, 0.94 is option A, but let me verify if there's an error in my calculation. **Alternative approach using multiplication:** - Probability NOT detected by First Line: 1 - 0.70 = 0.30 - Probability NOT detected by Second Line given First Line missed: 1 - 0.60 = 0.40 - Probability NOT detected by Third Line given both missed: 1 - 0.50 = 0.50 - Probability NO detection = 0.30 × 0.40 × 0.50 = 0.06 - Probability AT LEAST ONE detection = 1 - 0.06 = 0.94 This confirms that the correct answer should be 0.94, which corresponds to option A. However, let me double-check the interpretation of the problem. The Second Line detects 60% of the remaining undetected incidents after the First Line, and the Third Line detects 50% of the remaining undetected incidents after the Second Line. This is exactly what I calculated. Therefore, the correct probability is 0.94, which is option A. **Answer: A (0.94)**

Author: Tanishq Prabhu