Plant A produces 40% of the company's pin volume and 67% of the company's defective pins

Plant B produces 60% of the company's total volume and 33% of the total defective pins

What is the probability a defective pin comes from plant A? and probability it comes from plant B?

Here is the exact language. Thanks again

The company has two factories, the older of which produces 40% of the total output. This means that a pin picked up at random has a 40% probability of coming from the old factory, whether it is defective or perfect; this is the prior probability. We find that the older factory's defective rate is twice that found in the newer factory. If a customer calls and complains about finding a defective pin, which of the two factories should the manager call?

The prior probability would suggest that the defective pin was most likely to have come from the new plant, which produces 60% of the total. On the other hand, that plant produces only one-third of the company's total of defective pins. When we revise the priors to reflect this additional information, the probability that the new plant made the defective pin turns out to be only 42.8%; there is a 57 .2% probability that the older plant is the culprit. This new estimate becomes the posterior probability.

Do you see that your paraphrase means something entirely different from the original?

You were right that plant A produces 40% of the pins and plant B produces 60%; but the first paragraph does

**not** say that plant A produces 67% of the total number of defective pins. It says that the rate of defective pins from each factory are in the ratio 2:1, but that does not make them 2/3 and 1/3 (much less the rounded values 0.67 and 0.33!), because they are not complements.

If the second paragraph here is from the book, then

the book is wrong. Plant B does

**not** produce 1/3 of the company's defective pins! (In fact, their conclusion means that 42.8% of the defective pins come from plant B, as I pointed out initially.) Who wrote this?

Rather, suppose that the defective rate for plant B is r, and for plant A is 2r. That is, P(defective | B) = r, and P(defective | A) = 2r.

Put that in the Bayes formula as given to you by pka, and see what you get. (The variable r will drop out.)