A political analyst was interested in the proportion C of individuals who would vote for a controversial ballot measure. While he thought that it would be a close call, he was un sure of the precise value for C. He assessed a beta distribution for C with parameters r0 = 3, n0 = 6.

a. Plot this prior distribution for C.

b. The analyst is about to ask four individuals about their preferences. What is the probability that more than two of these individuals will express their support for the ballot measure?

c. Having questioned the four individuals, the analyst found that three would indeed vote for the ballot measure. Find the analyst's posterior distribution for C. Plot this posterior distribution and compare it with the prior distribution plotted in part a.

d. The analyst is now about to survey another 10 people. What is the probability that more than five of these people will support the ballot measure in this new poll?

e. Suppose that 6 of the 10 people surveyed said they would vote for the ballot measure. The analyst now must write up his results. What is his probability that the ballot measure will pass?