You are given the prior hyperparameters of a binomial process, observation data, and a value \(p\). Find the posterior distribution for \(\pi\) and its Normal approximation. Further, calculate the probability \(P(\pi \leq p)\) both by exact calculation on \(\beta\), and by using the Normal approximation.

(a) Prior hyperparameters: \(a_{0}=2, b_{0}=2\). Observations: 17 positive, 29 negative. \(p=0.4\).

(b) Prior: \(\beta_{(1,7)}\). Observed: \(k=4\) positive and \(l=89\) negative. \(p=0.07\).

(c) Prior hyperparameters: \(a_{0}=0, b_{0}=0\). Observed: \(k=42\) positive and \(l=13\) negative. \(p=0.7\).

(d) Prior: \(\beta_{(0.5,0.5)}\). Observed: \(k=434\) positive and \(l=177\) negative. \(p=0.7\).