Items used in production process in a factory come from three suppliers S1, S2 and S3. S1 provides 20%, S2 provides 30% and S3 provides 50%. However, S1 produces 5% (0.05) defective items, S2 produces 2% (0.02) while S3 produces 1% (0.01). If an item sampled at random was found to be defective, what is the prior probability it came from Supplier S1?
A) 20%.
B) 5%.
C) 2%.
D) 1%.

Question

Quizplus · Accepted Answer

The prior probability that a randomly selected defective item came from Supplier S1 can be calculated using Bayes' theorem. However, for this question, we can directly calculate the probability of selecting a defective item from S1 out of all defective items supplied by S1, S2, and S3. This is done by multiplying the probability of selecting an item from S1 (20% or 0.20) by the probability that the item is defective (5% or 0.05), and then dividing by the total probability of selecting a defective item from all suppliers. The calculation is as follows:$$P(\text{Defective from S1}) = \frac{0.20 \times 0.05}{(0.20 \times 0.05) + (0.30 \times 0.02) + (0.50 \times 0.01)}$$This calculation shows the proportion of defective items that come from S1 compared to all defective items, which directly answers the question about the prior probability of a defective item coming from S1.

Items Used in Production Process in a Factory Come from Three