Answer-first summary for fast verification
Answer: 0.79
This is a Bayes' theorem problem. We need to find P(Standard | Loss). **Given:** - P(Standard) = 0.60 - P(Preferred) = 0.30 - P(Ultra-preferred) = 0.10 - P(Loss | Standard) = 0.01 - P(Loss | Preferred) = 0.005 - P(Loss | Ultra-preferred) = 0.001 **Using Bayes' theorem:** P(Standard | Loss) = [P(Standard) × P(Loss | Standard)] / [P(Standard) × P(Loss | Standard) + P(Preferred) × P(Loss | Preferred) + P(Ultra-preferred) × P(Loss | Ultra-preferred)] **Calculation:** Numerator = 0.60 × 0.01 = 0.006 Denominator = (0.60 × 0.01) + (0.30 × 0.005) + (0.10 × 0.001) = 0.006 + 0.0015 + 0.0001 = 0.0076 P(Standard | Loss) = 0.006 / 0.0076 = 0.7895 ≈ 0.79 **Interpretation:** Even though Standard projects have the highest probability of loss (0.01), they also make up the largest proportion of the portfolio (60%). When a loss occurs, there's a 79% chance it came from a Standard project, which aligns with option A (0.79).
Author: Nikitesh Somanthe
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An investment firm classifies capital projects into three different categories, depending on risk level: Standard, Preferred, and Ultra-preferred. Of the firm's projects, 60% is standard, 30% are preferred, and 10% are ultra-preferred. The probabilities of a project making a loss are 0.01, 0.005, and 0.001 for categories standard, preferred, and ultra-preferred respectively.
If a capital project makes a loss in the next year, then what is the probability that the project was standard (correct to 2 decimal places)?
A
0.79
B
0.73
C
0.22
D
0.15
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