The beta-binomial distribution plays a prominent role in the Bayesian estimation of a Bernoulli success probability p which we wish to estimate based on data. Let \mathbf{X}=\{X_1, X_2, \cdots X_{n_1}\} be a sample of independent and identically distributed Bernoulli random variables X_i \sim \text{Bernoulli}(p). Suppose, our knowledge of p - in Bayesian fashion - is uncertain and is modeled by the prior distribution p \sim \text{Beta}(\alpha,\beta). If Y_1=\sum_{i=1}^{n_1} X_i then through compounding, the prior predictive distribution of
