Bayes rule for quality control. You're the foreman at a factory making ten million widgets per year. As a quality control step before shipment, you create a detector that tests for defective widgets before sending them to customers. The test is uniformly 95% accurate, meaning that the probability of testing positive given that the widget is defective is 0.95, as is the probability of testing negative given that the widget is not defective. Further, only one in 100,000 widgets is actually defective. (a) Suppose the test shows that a widget is defective. What are the chances that it's actually defective given the test result? (b) If we throw out all widgets that are test defective, how many good widgets are thrown away per year? How many bad widgets are still shipped to customers each year?