A previous study indicates that the percentage of chain smokers, p, who have lung cancer follows a beta distribution with mean 70% and standard deviation 10%. Suppose a new data set collected shows that 81 out of 120 chain smokers have lung cancer.
a) Determine the posterior distribution of the percentage of chain smokers who have lung cancer by combining the new data and the prior information.
b) What is the posterior probability that p is larger than 50%?