Sondre Glimsdal wants to estimate \(\pi\), the proportion of \(G o^{1}\) games won by white. His prior is \(\pi \sim \beta_{(7,7)}\). Sondre updates his estimate by observing new games. Find the \(80 \%\) symmetric credible interval for \(\pi\) after each update.

(a) 1st observation: white wins 13 and black wins 7 .

(b) 2nd observation: white wins 11 and black wins 9.

(c) How many new games does he need to observe to ensure that the width of \(I_{0.2}^{\pi}\) is less than 0.05 ?

(d) 3rd observation: white wins 348 and black wins 255 .