We had a (prior) hypothesis that these two binary groups were 30% different. So two categorical variables that are binary. We have the data and they are 11% different. I did not know if there was a Bayesian approach I could take in examining if the groups are different. I saw the following (I have not completely read it), but I don't think it is quite what I am looking for. Also, there are approximately 50 people in each group and I think the percentages were 53 and 64.

Any suggestions on how I may look at these data given a prior study suggested a possible prior difference of 35%. Traditionally I would use Fisher's Exact, but I thought this may be a nice hoorah for me to try something new and perhaps incorporate the 30% difference.