Yes he did, in two parts: easy non-brute-force ways exist (as in the Monte-Carlo integration in R, he argues why it isn't brute force) and also a non-easy analytical solution exists (when at least one set of parameters are integer, in the linked cross validated post).
He also explains that easy analytical solutions often don't exists to seemingly easy questions, such as this one. That's why Bayesian statistics needs Monte-Carlo.
You are right, I'm the idiot and didn't read the question carefully...
About the actual question, I don't think that's very well specified in this form. You could have A~Beta0 where Beta0 is very concentrated around let's say 0.5, and then find a B~Beta1 where Beta1 has 71% of its weight above 0.5. That would be a pretty good approximation for it, in fact as close as you want if you increase sample size of A. I can't verify this, but I think finding Beta1 should be easy. Obv this is not what you are looking for though, because this would require A to have a much larger sample size than B. I guess we should add some restriction, like similar sample sizes.
He also explains that easy analytical solutions often don't exists to seemingly easy questions, such as this one. That's why Bayesian statistics needs Monte-Carlo.