 Solution

The law of iterated expectation yieldsOne may obtain $\mathrm{Var}(p_T)=v$ either from its unchanged marginal distribution proved below or directly from the law of total variance: the variance of its conditional mean is $nv/(s+n)$ and its expected conditional variance is $sv/(s+n)$. Hence the correlation coefficient is exactlyAlso $\mathbb{E}[(p_T-p_M{)}^2]=2vs/(s+n)\to 0$. The two probabilities become close while keeping their original beta marginals.