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Start with the Beta distribution draw in the Beta-binomial exchangeable coupling. Write $s=\alpha +\beta$ andThese 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.

For each auxiliary count $x$, the correctly normalized joint expression above is symmetric in $p_M,p_T$. Summing it over $x=0,\dots ,n$ preserves that symmetry, so these are exchangeable random variables. Their marginal distributions are therefore identical. Since $p_M$ was generated from a Beta distribution,The Beta-binomial exchangeable coupling changes the dependence, while leaving both prior distributions unchanged.