A Brief Test of Your Bayesian Intuition

Here is a test of your Bayesian intuition: Suppose you assign a binomial chance parameter θ a beta(2,2) prior distribution. You anticipate collecting two observations. What is your expected posterior distribution?

NB. ChatGPT 3.5, Bard, and the majority of my fellow Bayesians get this wrong. The answer will be revealed in the next post.