A company has 2 machines that produce widgets.An older machine produces 23% defective widgets,while the new machine produces only 8% defective widgets.In addition,the new machine produces 3 times as many widgets as the older machine does.Given a randomly chosen widget was tested and found to be defective,what is the probability it was produced by the new machine?
A)0.08
B)0.15
C)0.489
D)0.511

Question

Quizplus · Accepted Answer

The probability that a defective widget came from the new machine can be calculated using Bayes' theorem. Let's denote the older machine as O and the new machine as N. The probability of a widget being defective given it's from the new machine is P(D|N) = 0.08, and the probability of a widget being defective given it's from the old machine is P(D|O) = 0.23. Since the new machine produces three times as many widgets as the old one, let's assume the old machine produces x widgets, then the new machine produces 3x widgets. The total production is x + 3x = 4x. The probability of picking a widget from the new machine is P(N) = 3x / 4x = 3/4, and from the old machine is P(O) = x / 4x = 1/4.The overall probability of getting a defective widget, P(D), is the sum of the probabilities of getting a defective widget from each machine, weighted by how many widgets each machine produces: P(D) = P(D|N)P(N) + P(D|O)P(O) = 0.08(3/4) + 0.23(1/4) = 0.06 + 0.0575 = 0.1175.The probability that a defective widget came from the new machine is P(N|D) = [P(D|N)P(N)] / P(D) = [0.08(3/4)] / 0.1175 = 0.06 / 0.1175 ≈ 0.511.

A Company Has 2 Machines That Produce Widgets