Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 35 / 2 / c / Solution

Independent Jeffreys priors and the two independent binomial distribution likelihood factors give, by Beta-binomial conjugacy,The Bayesian posteriors remain independent because each observation factor involves only its own probability. Their posterior means are respectively $0.7$ and $0.3$. Notice that $p_T$ is the probability of saying milk first when tea was first, rather than the probability of a correct tea identification.