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

With a uniform beta distribution prior, the posterior predictive distribution for another batch of the same size isBut $\mathrm{B}(x+1,m-x+1)=1/[(m+1)(\genfrac{}{}{0ex}{}{m}{x})]$. HenceThe 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.