First Year Uni...Conditional Probability (I think!)

This question is straight from an assignment I am doing.

I am trying to determine the probability that a good bean is rejected. i.e. P(Rejected | Good)

Question 5
You are the production manager in a large robusta coffee bean processing facility. An important stage in the production involves sorting to identify and remove defective beans. This is a manual process which is not only expensive (90 cents per thousand defective beans removed) but also a bottleneck that limits throughput. Research has revealed that 2.3% of beans are defective, and the remainder are good. The manual sorting cost associated with the good beans is zero.

Fortunately most defective robusta coffee beans can be identified by their colour, with most common being black or dark grey beans which have fermented or are over ripe. You rent a bean sorting machine which uses a CCD camera to classify beans on a darkness scale of 0 to 100. You can set a threshold value, and any bean darker than the threshold will be removed by a puff of air. The threshold is initially set at 54 (any bean with a darkness value greater than 54 will be treated as defective).

Research has provided the following pieces of information:
· Good beans have a roughly normal darkness distribution with an average of 27 and a standard deviation of 10.
· Defective beans also have a roughly normal darkness distribution with an average of 58 and a standard deviation of 12.

b. Determine the probability that a good bean is rejected. i.e. P(Rejected | Good)

So far I have the probabilities for:

(Good Bean) P(G)=.977
(defect Bean) P(D)=.023
(Rejected) P(R)=.55
(Kept) P(K)=.54

I made the assumption its out of 1000 beans.

This is where I get stuck, how do I go onto then find out P(Rejected | Good), I'm pretty sure the mean and std dev play into the equation but I have no clue.


Any help would be great.


Thanks

Assuming that "good bean" darkness is "normally distributed with an average of 27 and a standard deviation of 10", what is the probability a good bean will have darkness over 54 and so be rejected? What is the probability a good bean will have darkness under 54 and so not be rejected? Assuming that "bad bean" is "with an average of 58 and a standard deviation of 12", what is the probability a bad bean will have darkness under 54 and so not be rejected? Since 97.7 of the beans in the batch are good, out of 1000 beans, 977 are good and 23 are bad. How many of the good beans will be accepted and how many of the bad beans will be accepted? So how many beans total will be accepted and how many of them are good and how many bad?