3.2 Exercise 12

A quality control group is designing an automatic test procedure for compact disk players coming from a production line. Experience shows that one percent of the units produced are defective. The automatic test procedure has probability 0.05 of giving a false positive indication and probability 0.02 of giving a false negative. That is, if D is the event a unit tested is defective, and T is the event that it tests satisfactory, then $P \left(\right. T \left|\right. D \left.\right) = 0 . 05$ and $P \left(\right. T^{c} \left|\right. D^{c} \left.\right) = 0 . 02$. Determine the probability $P \left(\right. D^{c} \left|\right. T \left.\right)$ that a unit which tests good is, in fact, free of defects.