Free-throw shooting rate with a Beta prior

Question

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 $p$ ?

Accepted Answer

Intuition first. A Beta prior stands in for "pretend I already saw some makes and misses." 	ext{Beta}(3,1) is like 3 pretend makes and 1 pretend miss, an optimistic starting point. Updating with real makes and misses is bookkeeping: add real makes to the first number, real misses to the second. The rule. Beta is conjugate to make/miss data, so the posterior is still a Beta: 	ext{Beta}(a,b)\ +\ k 	ext{ makes and } (n-k) 	ext{ misses}\ ightarrow\ 	ext{Beta}(a+k,\ b+n-k). Plug in. Here a=3, b=1, with 6 makes and 2 misses: 	ext{posterior} = 	ext{Beta}(3+6,\ 1+2) = 	ext{Beta}(9, 3). Post…

Free-throw shooting rate with a Beta prior

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