Help with Baye's Theorem?

Question

Jansen Electronics has four machines that produce an identical component for use in its videocassette players. The proportions of the components produced by each machine and the probability of one of that component being defective are as follows:

Machine 1- Proportion of components produced= .15 Probability of defective component= .04

Machine 2- Proportion of components produced= .30 Probability of defective component= .02

Machine 3- Proportion of components produced= .35 Probability of defective component= .02

Machine 4- Proportion of components produced= .20 Probability of defective component= .03

What is the probability that the component selected at random is defective?

* Our book only teaches us how to do the problem if they tell us what Machine we're using, not which one we're selecting at random...

Answer 1

The way you've posed the question has nothing to do with Bayes' Theorem. Had the question been

If a component picked at random is found to be defective what is the probability that machine x produced it?

then Bayes' Theorem would have been useful.

Anyways, I'll answer your question: The total proportion of defective component is

0.15*0.04 + 0.3*0.02 + 0.35*0.02 + 0.2*0.03 = 0.025

this means that if you just picked a component at random there is a 0.025 (2.5%) chance that it'll be defective.

/m

Answer 2

P(defective) = .15*.04 + .30*.02 + .35*.02 + .20*.03

= .006 + .006 + .007 + .006 = .025