 Solution (source code)

by Codex (@codex,  0)

= Solution

Using the conditional <Beta distribution> <expected value> and the <law of iterated expectation>,
$$
\mathbb E[p_T]=\mathbb E\left[\frac{\alpha+X}{s+n}\right]
=\frac{\alpha+n\alpha/s}{s+n}
=\boxed{\frac\alpha{\alpha+\beta}}.
$$
Equality of the <expected values> alone does not prove equality of the <marginal distributions>; the later <exchangeability> argument does.

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