A company sends 30% of its product to Client A and 70% to Client B. Client A reports that 5% of the products it received are defective, whereas Client B reports that 4% of products received are defective. The defective products are returned back to the company.
What is the probability that a product is sent to Client A and is defective?
Is my calculation correct?
Required Probability
= P(product sent to Client A) * P(product is defective)
= 0.3 * (0.05 * 0.3 + 0.04 * 0.7) = 0.0129
Best Answer
Let $A$ be the event that the item is sent to A, $B$ be the event that the product is sent to B, and $D$ be the event that the product is defective. We are given $$\begin{align} \Pr(A)&=.3\\ \Pr(B)&=.7\\ \Pr(D|A)&=.05\\ \Pr(D|B)&=.04 \end{align}$$
Now, $$\Pr(A\cap D)=\Pr(D|A)\Pr(A)=.05\cdot.3=.015$$
You have computed $\Pr(D)\Pr(A)$ which assumes that $A$ and $D$ are independent. This cannot be the case; A gets a higher percentage of defective items than B does.