Use Bayes's rule to find the indicated probability.
-Two stores sell a certain product. Store A has 40% of the sales, 3% of which are of defective items, and store B has 60% of the sales, 5% of which are of defective items. The difference in defective
Rates is due to different levels of pre-sale checking of the product. A person receives a defective Item of this product as a gift. What is the probability it came from store B?
A)0.75
B)0.3
C)0.2857
D)0.7143

Question

Use Bayes's rule to find the indicated probability.
-Two stores sell a certain product. Store A has 40% of the sales, 3% of which are of defective items, and store B has 60% of the sales, 5% of which are of defective items. The difference in defective
Rates is due to different levels of pre-sale checking of the product. A person receives a defective Item of this product as a gift. What is the probability it came from store B? 
A)0.75
B)0.3
C)0.2857
D)0.7143

Quizplus · Accepted Answer

The probability it came from store B is calculated using Bayes's rule. Let $P(A)$ be the probability the item is from store A, $P(B)$ the probability it's from store B, $P(D|A)$ the probability of a defective item given it's from A, and $P(D|B)$ the probability of a defective item given it's from B. We want $P(B|D)$, the probability the item is from B given it's defective. $$P(B|D) = \frac{P(D|B)P(B)}{P(D|A)P(A) + P(D|B)P(B)}$$$$P(B|D) = \frac{0.05 \times 0.6}{0.03 \times 0.4 + 0.05 \times 0.6} = \frac{0.03}{0.012 + 0.03} = \frac{0.03}{0.042} = 0.7143.$$

Use Bayes's Rule to Find the Indicated Probability