Question

A widget produced by a particular process has probability .1 of being defective. A test can be performed which has 99% accuracy. That is, if a defective widget is tested, the test will identify the widget as defective 99% of the time. And if a non-defective widget is tested, there is a 99% chance that the test will indicate that the widget is not defective. One widget is selected at random and is tested. If the test says that the widget is not defective, what is the probability that the widget actually is defective?

Answer 1

P(being defective) = 0.1

P(test identified the widget as defective | defective) = 0.99

P(test identified the widget as not defective | non- defective) = 0.99

P(test identified the widget as not defective | defective) = 1 - P(test identified the widget as defective | defective) = 1 - 0.99 = 0.01

P(test identified the widget as not defective) = P(test identified the widget as not defective | defective) * P( defective) + P(test identified the widget as not defective | non- defective) * P(non-defective)

                                                                     = 0.01 * 0.1 + 0.99 * (1 - 0.1)

                                                                     = 0.892

P(defective | test identified the widget as not defective) = P(test identified the widget as not defective | defective) * P( defective) / P(test identified the widget as not defective)

                                                                         = 0.01 * 0.1 / 0.892

                                                                         = 0.0011