Beta-binomial distribution ($\mathrm{Pr}(Y=y)=\left(\genfrac{}{}{0ex}{}{n}{y}\right)\frac{\mathrm{B}(\alpha +y,\beta +n-y)}{\mathrm{B}(\alpha ,\beta )}$)

Mix a binomial distribution with a beta distribution for its success probability: $\Theta \sim \mathrm{Beta}(\alpha ,\beta )$ and $Y\mid \Theta \sim \mathrm{Binomial}(n,\Theta )$. Integrating their densities gives the displayed mass for $y=0,\dots ,n$. Its expectation is $n\alpha /(\alpha +\beta )$, and its variance is $n\alpha \beta (\alpha +\beta +n)/[(\alpha +\beta {)}^2(\alpha +\beta +1)]$. It is the posterior predictive distribution for repeated Bernoulli trials after a beta-prior update.

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.

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 $\left(\genfrac{}{}{0ex}{}{s}{y}\right)\left(\genfrac{}{}{0ex}{}{m+n-s}{n-y}\right)/\left(\genfrac{}{}{0ex}{}{m+n}{n}\right)$. Under the uniform-count predictive property, $s$ is uniform on $0,\dots ,m+n$. 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.