# Probability question using Bayes rule

I have a probability question here which I believe I need to apply Bayes rule to solve it.

Here is the question:

A specific enzyme, QQ, which is designed to quickly help cows gain weight, is added to feed. In a recent experiment, it was noticed that some cows could have developed trotter rot because of having a diet containing this enzyme. The manufacturer producing this enzyme report that the probability of a cow developing trotter rot can increase from 0.1 (for a cow given standard feed) to 0.50 (for a cow given feed containing QQ enzyme). Out of 95 cows in an experiment, n cows were given feed containing the QQ enzyme.

(a) A vet randomly inspects a cow. If the selected cow has trotter rot, what is the probability it had been fed with the diet containing the enzyme?

(b) Compute the maximum number of cows which should be given feed containing the mentioned enzyme if we want to have the probability derived in part a) to be less than 0.65.

My working:

I defined the terms as

•TR = a cow has trotter rot

•TR*= a cow has no trotter rot

•QQ* = a cow given standard feed

•QQ = a cow given feed containing QQ enzyme

and extracted the probabilities from the question:

•P( TR | QQ*) = 0.1

•P( TR | QQ) = 0.5

Then let:

•P( TR) = p

•P( TR*) = 1-p

•P(QQ) = k

•P(QQ*) = 1-k

I believe (a) is asking to find P(QQ | TR) so I substitute the above into $$P(QQ | TR) = \frac{P(TR|QQ) P(QQ)}{P(TR|QQ) P(QQ) + P(TR|QQ*) P(QQ*)}$$

which I get $$P(QQ | TR) =\frac{0.5k}{0.4k+0.1}$$

And I have no way to find out the values for k and p, so it seems like an incorrect way to solve this question. Also, I'm not too sure whether I defined and extracted the information correctly ( especially P(QQ) and P(QQ*).

Any help would be appreciated.

Actually, $$k=n/95$$ based on the following sentence:
$$\frac{0.5k}{0.4k+0.1}<0.65 \rightarrow n<13\times 95/48$$, which yields a max value of $$25$$ for $$n$$.