You are studying Bard's biased coin. Your prior hyperparameters for \(\pi\), the probability of heads, is \(a_{0}=9\) and \(b_{0}=12\). Find the \(90 \%\) symmetric credible interval for \(\pi\) after each update.

(a) You flip 23 heads and 18 tails.

(b) You flip again, and get 458 heads and 366 tails.

(c) You finally flip 5571 heads and 4429 tails.

(d) How many times do you need to flip the coin in total for the width of \(I_{0.1}^{\pi}\) to be less than 0.01 ?