# Bayes' theorem and likelihood that an observation belongs to one of x clusters

I am trying to write a script is inspired by the following website. Here they examine mixture models, where they treat the distribution of batting averages as a mixture of two beta-binomial distributions, and need to guess which player belongs to which group.

My question: How would we assign players to clusters, and get a posterior probability that the player belongs to that cluster if we have more than two clusters?

If we have two clusters, we'd consider the likelihood each player would get the number of hits they did if they belong to cluster A or to cluster B. two clusters

By Bayes’ Theorem, we can simply use the ratio of one likelihood to the sum of the two likelihoods to get the posterior probability. posterior probability

How does this work in the scenario where there are more than two clusters?