In the two-player board game Go, black and white take turns putting a stone on the board, with black having the first move. Sondre Glimsdal, the \(O g\) Go club chairman, wonders what percentage of the games is won by each color. He believes it is fairly even, so his prior for the proportion of games won by white is \(\beta_{(7,7)}\).

(a) Sondre looks at 20 randomly chosen games, and sees that 13 were won by white, and 7 were won by black. What is Sondre's posterior probability distribution for the proportion of games won by white?

(b) Sondre looks at another 20 random games, and finds that 11 of these were won by white, and 9 were won by black. What is Sondre's new posterior probability distribution for the proportion of games won by white?

(c) What is Sondre's probability that white wins at least half of the games?