Simple statistics exercise... Too simple? I have to solve the following exercise:
There are three different parts needed to construct a machine: part $1$ ($1000$ pieces), part $2$ ($400$ pieces), part $3$ ($600$ pieces).
The probability for a part $1$ or part $2$ piece to be defective is $2 \%$. The probability for a part $3$ piece to be defective is $2.8\%$.
An engineer takes one piece and it turns out to be defective. The question is what is the probability that the engineer took a part $2$ piece?
My approach is just dividing the number of defective part 2 pieces by the number of all defective pieces like that:
$$\frac{400\times 0.02}{1000\times 0.2+400\times 0.02+600\times 0.028}$$
But it seems too simple. Does it seem correct to you?
Thanks in advance

Nice intuition. To make your argument more rigourous, consider using Bayes rule.
\begin{align}P(\text{part 2}|\text{defective})=\frac{P(\text{part 2})}{P(\text{defective})}P(\text{defective}|\text{part 2}) \end{align}
$$P(\text{part 2})=\frac{400}{2000}$$
$$P(\text{defective})=\frac{2}{100}\frac{1000}{2000}+\frac{2}{100}\frac{400}{2000}+\frac{2.8}{100}\frac{600}{2000}$$
$$P(\text{defective}|\text{part 2}) = \frac{2}{100} $$
