Bayesian fill rate from a Beta prior

A trading desk wants to estimate the chance $p$ that a resting limit order gets filled. Before seeing today's flow, your belief about $p$ is a $\text{Beta}(2, 2)$ distribution: centered at one-half and fairly wide, as if you had already watched 2 fills and 2 misses. Today you place 12 orders and 9 of them fill.

Using Bayesian updating, what is your new best estimate (the posterior mean) for the fill probability $p$?