Comparing p Across Groups

The table below shows the probability (as a percentage) that each group’s p is the highest or lowest among all groups (approximated via Monte Carlo simulation).

Group	Prob(Highest p)	Prob(Lowest p)

The Binomial Distribution Across Multiple Groups

\[ P(X = k) = \binom{n}{k} p^{k} (1-p)^{n-k}, \quad p \sim \mathrm{Beta}(\alpha, \beta). \]

When you have multiple binomial processes (groups), each may have a different underlying success probability \(p_i\). By treating each group’s unknown \(p_i\) as a random variable with a Beta posterior, you can visualize the PDFs together and compare their likelihoods of being the highest or lowest among the groups.