Beta-Binomial Distribution Calculator

About this calculator

The beta-binomial distribution is a compound distribution: instead of assuming a fixed success probability like the ordinary binomial does, it treats that probability as itself uncertain and drawn from a Beta(α, β) distribution. The result models overdispersed count data — situations where outcomes vary more than a plain binomial would predict because the underlying rate differs from trial to trial or group to group.

This calculator computes the probability mass P(k), mean and variance for given trials n, successes k, and shape parameters α and β. It’s a core tool in Bayesian statistics for updating beliefs about a success rate, in A/B testing and marketing analytics where conversion rates vary across users or segments, and in ecology and genetics for modeling count data with natural between-group variability.