= Solution
Independent <Jeffreys priors> and the two independent <binomial distribution> likelihood factors give, by <Beta-binomial conjugacy>,
$$
\boxed{p_M\mid\mathcal D\sim\operatorname{Beta}(7/2,3/2),\qquad p_T\mid\mathcal D\sim\operatorname{Beta}(3/2,7/2).}
$$
The <Bayesian posteriors> remain independent because each observation factor involves only its own probability. Their <posterior means> are respectively \b[$0.7$ 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.
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