 Uniform-count predictive property

A uniform prior distribution on the success chance makes the number of successes in any fixed number $n$ of Bernoulli trials uniform on $0,\dots ,n$, because $\left(\genfrac{}{}{0ex}{}{n}{y}\right)\mathrm{B}(y+1,n-y+1)=1/(n+1)$. 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.