The posterior mean is not the observed rate
A beta–binomial update starts from a Beta(2, 2) prior and then observes y successes in n trials.
Why is the posterior mean not the observed rate y/n?
Worked answer
The posterior is \operatorname{Beta}(2+y, 2+n-y). Its mean is (2+y)/(4+n). That is a weighted average of the prior mean 1/2 and the observed rate y/n, with the four pseudo-counts from the prior pulling the mean toward 1/2. The observed rate is the maximum-likelihood estimate. It equals the posterior mean only for a prior that adds no pseudo-counts. The mean of draws from this posterior should agree with (2+y)/(4+n), up to Monte Carlo error. Agreement with y/n would mean the draws had ignored the prior.