In a population of chips, 1% have a latent defect. A screening test judges 95% of the defective chips positive and wrongly judges 4% of the good chips positive. Which of the following is closest to the probability that a chip judged positive is actually defective?

Answer: B. About 19%

Considering 10,000 chips, 95 of the 100 defective chips test positive, and 396 of the 9,900 good chips also test positive. Of the 491 positives, only 95 are actually defective, so 95 ÷ 491 ≈ 0.19, or about 19%. Written with Bayes' theorem, P(defective | positive) = 0.01 × 0.95 ÷ (0.01 × 0.95 + 0.99 × 0.04) ≈ 0.19. When defects are rare, many good chips are mixed in with the positives even if the rate of wrongly judging them positive is small. This is called the effect of the base rate (prior probability).

• A. 0.01 × 0.95 + 0.99 × 0.04 = 0.0491 is the fraction of all chips judged positive.
• B. Correct. 95 ÷ (95 + 396) ≈ 0.19.
• C. 95% is the probability that a defective chip is judged positive, P(positive | defective), which reverses the condition and the outcome of the required P(defective | positive).