= Solution
The next <Beta distribution> has conditional <expected value> and <variance>
$$
\mathbb E[p_T\mid X]=\frac{\alpha+X}{s+n},\qquad
\operatorname{Var}(p_T\mid X)=\frac{(\alpha+X)(\beta+n-X)}{(s+n)^2(s+n+1)}.
$$
For large $n$, its <expected value> approaches $X/n$, while its <variance> is at most $1/[4(s+n+1)]$. The final draw therefore stays close to the first. These are <prior distributions> for the cup probabilities: the auxiliary count is a device for constructing dependence, rather than additional observed tea data.
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