= Solution
With a uniform <Beta distribution> <prior>, the <posterior predictive distribution> for another batch of the same size is
$$
p(y\mid x)=\binom my
\frac{\mathrm B(x+y+1,2m-x-y+1)}{\mathrm B(x+1,m-x+1)}.
$$
But $\mathrm B(x+1,m-x+1)=1/[(m+1)\binom mx]$. Hence
$$
\boxed{p(y\mid x)=(m+1)\binom mx\binom my\,
\mathrm B(x+y+1,2m-x-y+1)=p(x\mid y).}
$$
The displayed expression is symmetric in $x,y$. This is the <uniform-count predictive property>: the joint batch counts are exchangeable and both marginal counts are uniform. Conditional reversal is not a general property of arbitrary priors; it holds here because those marginal probabilities coincide.
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