Counting & Probability · Medium problem

Three machines $A$, $B$, and $C$ produce $50\%$, $30\%$, and $20\%$ of a factory's output, respectively. Their defective rates are $2\%$, $3\%$, and $5\%$, respectively. An item chosen at random from the total output is found to be defective. Using the law of total probability and Bayes' theorem, find the probability that the defective item was produced by machine $C$. Round to 3 decimal places (an exact answer is also accepted).