= Uniform-count predictive property
{title2=$\Pr(Y=y)=1/(n+1)$}
A uniform <prior distribution> on the success chance makes the number of successes in any fixed number $n$ of <Bernoulli trials> uniform on $0,\ldots,n$, because $\binom ny\mathrm B(y+1,n-y+1)=1/(n+1)$. This is a statement about counts, not a uniform distribution over the $2^n$ ordered outcome sequences. It also makes predictions reversible between equally sized batches: their joint <probability distribution> is symmetric and their marginal count distributions are the same uniform distribution.
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