How to calculate posterior probabilities (Bayes related, I think)

I graduated Uni 16 years ago and generally have no problems with the math and stats I need in my life ... until today. I am breeding horses and there has been a very interesting study on a specific gene. I have been granted access to the source data but need help with calculating some posterior probabilities.

In essense, for the gene in question, each horse is either AA, CA or CC. (As usual, the foal gets one gene from each parent) Several houndred horses have been tested so there are a lot of observations. 30 foals (children) of one stallion (male horse) has been tested, and 10 are CA while 20 are CC. Since that stallion has CC foals I know he must be CA or CC. Since there are so many CC foals, that seems most likely. Unfortunately we don't know the genes of the mothers, but the population is 11 % AA, 20 % CC and 69 % CA so I guess we can just assume the 30 mothers are distributed according to this.

So how do I calculate the posterior probability that this stallion is CC? I suspect I have to use Bayes' Theorem in one form or another but I will reluctantly admit that I am not sure how to apply it and how to define the terms correctly. The strange part is that at some point I was actually relatively comfortable with this, but I have forgotten quite a bit as I never use this part. So any and all input is greatly appreciated!


TS Contributor
I think this can be looked at from a classical POV. Lets us take the Null Hypothesis that your horse is CA. What is the probability of having a AA offspring?

P(AA)=P(AA|mate AA)*P(mate AA) +P(AA|mate CA)*P(mateCA)+P(AA|mateCC)*P(mateCC)=0.5*0.11+0.25*0.69+0*0.2=0.2275

The probability of not having a single AA in 30 offspring is (1-0.2275)^30=0.0004 IF your horse is CA.
Either you are in this very unprobable situation or your horse is CC in which case the probability of not having any AA in 30 offspring is 1 :)

Excellent, I see I was going about it the wrong way as I kept focusing on Bayes and how to apply it - and got a bit stuck that way. This works well, thanks :)