Solution
= Solution
Start with the <Beta distribution> draw in the <Beta-binomial exchangeable coupling>. Write $s=\alpha+\beta$ and
$$
m=\frac\alpha s,\qquad v=\frac{\alpha\beta}{s^2(s+1)}.
$$
These are the <expected value> and <variance> of $p_M$. The fixed $n$ in this coupling is a dependence parameter chosen by the investigator, not necessarily the observed number of cups. The four numbered stages together address the single printed correlation request.