Defect rate from a skeptical Beta prior

A factory estimates the true defect probability $p$ of parts from a new machine. Engineers expect defects to be rare, so the prior is $\text{Beta}(1, 9)$: centered at $0.1$ and leaning low, like having already seen 1 defect and 9 good parts. In a batch of 30 parts, 2 are defective.

Using Bayesian updating, what is the posterior mean estimate of the defect probability $p$?