Challenge · Medium problem

A factory's output comes from three machines. Machine I makes $20\%$ of the items with a $5\%$ defect rate; Machine II makes $30\%$ with a $4\%$ defect rate; Machine III makes $50\%$ with a $2\%$ defect rate. An item chosen at random is found to be defective. Find the probability it was made by Machine III.

Express your answer as a common fraction.