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 that a randomly chosen widget was tested and was found to be defective,what is the probability that it was produced by the new machine?
A)0.08
B)0.15
C)0.489
D)0.511

Question

Quizplus · Accepted Answer

Let A be the event that the widget was produced by the new machine and B be the event that the widget is defective. We are asked to find P(A|B), the probability that the widget was produced by the new machine given that it is defective. We can use Bayes' theorem to calculate this:

P(A|B) = P(B|A)P(A) / P(B)

We can calculate each of these probabilities as follows:

P(B|A) = 0.08 (the probability that a widget produced by the new machine is defective)
P(A) = 3/4 (since the new machine produces 3 times as many widgets as the old machine, the probability that a randomly chosen widget was produced by the new machine is 3/4)
P(B) = P(B|A)P(A) + P(B|not A)P(not A) = 0.08 * 3/4 + 0.23 * 1/4 = 0.115

Plugging into Bayes' theorem, we get:

P(A|B) = 0.08 * 3/4 / 0.115 = 0.511

Therefore, the answer is D.

A Company Has 2 Machines That Produce Widgets