Probability the next pitch is a strike

Question

A pitcher has thrown 12 strikes out of 20 pitches. You start from a flat prior $\text{Beta}(1, 1)$ on the true strike probability $p$ , treating every value as equally plausible beforehand.

Using Bayesian updating, what is the probability that the next pitch is a strike?

Accepted Answer

Intuition first. You want the chance the next single pitch is a strike, which the Bayesian approach gets by averaging the strike probability over all still-plausible values of p, weighted by belief. That average equals the posterior mean. The update rule. Beta is conjugate to strike/ball data, so add strikes to the first number and balls to the second: 	ext{Beta}(a,b)\ +\ k 	ext{ strikes and } (n-k) 	ext{ balls}\ ightarrow\ 	ext{Beta}(a+k,\ b+n-k). Here a=1, b=1, with 12 strikes and 8 balls: 	ext{posterior} = 	ext{Beta}(1+12,\ 1+8) = 	ext{Beta}(13, 9). Probability the next pitch is a…

Probability the next pitch is a strike

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