Solution

Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 36 / 1 / f / Solution

Conditional on the pooled number $s$ of successes in $m+n$ <exchangeable random variables> that are binary, all placements of those successes are equally likely under a common success-chance mixture. The count in a chosen batch of size $n$ therefore has a <hypergeometric distribution>, with mass $\binom sy\binom{m+n-s}{n-y}/\binom{m+n}n$. Under the <uniform-count predictive property>, $s$ is uniform on $0,\ldots,m+n$. Dividing the joint pooled-and-batch probability by the first batch's uniform marginal gives the <posterior predictive distribution>, with factor $(m+1)/(m+n+1)$. Substituting $s=x+y$ in the hypergeometric expression does not give a single normalized hypergeometric mass as $y$ varies, since its population success total then varies too.

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 Incoming links: Hypergeometric allocation of exchangeable Bernoulli counts