Explanation

This is a conditional probability problem using Bayes' theorem. We need to find P(Subprime | Late).

Given:

• Total subprime mortgages: 1,000
• Late subprime mortgages: 200
• Total prime mortgages: 600
• Late prime mortgages: 48
• Total mortgages: 1,000 + 600 = 1,600

Step 1: Calculate total late mortgages

Total late = Late subprime + Late prime = 200 + 48 = 248

Step 2: Apply Bayes' theorem

P(Subprime | Late) = P(Late | Subprime) × P(Subprime) / P(Late)

Where:

• P(Late | Subprime) = 200/1,000 = 0.20
• P(Subprime) = 1,000/1,600 = 0.625
• P(Late) = 248/1,600 = 0.155

Step 3: Calculate the probability

P(Subprime | Late) = (0.20 × 0.625) / 0.155 = 0.125 / 0.155 ≈ 0.8065 ≈ 81%

Alternatively, we can use the direct approach: P(Subprime | Late) = Late subprime / Total late = 200 / 248 ≈ 0.8065 ≈ 81%

Therefore, the probability that a randomly selected late mortgage is subprime is approximately 81%.