Free-throw shooting rate with a Beta prior

A coach estimates the true free-throw make probability $p$ of a new player. Scouting reports make the coach optimistic, so the prior is $\text{Beta}(3, 1)$: like having already seen 3 makes and 1 miss, leaning high. In a short warmup the player makes 6 of 8 attempts.

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