In a jury, there are 12 selected jurors. We are looking to see how many prospective people should go to jury duty. Suppose whenever a pool of n prospective jurors is called into voir dire, the number of rejected jurors is X, where X|theta is B(n,theta). Suppose also that historically, the proportion of rejected jurors has fluctuated in the 30%-60% range, inspiring the assignment of the Uniform (.3,.6) prior on theta. Suppose also that two recent school-system cases are randomly selected from among all school system cases and that in the first case, 21 out of 27 prospective jurors were unable to serve, and in the second case, 30 out of 40 were unable to serve.
a. What is the distribution of theta given these observations?
b. What is the mean of the prior distribution? To 4 decimals, what is the mean of the posterior distribution? Why does the relationship between these two numbers make qualitative sense?
c. What is the conditional probability that 1 randomly selected prospective juror will be unable to serve?
d. What is the unconditional probability that of 5 randomly selected prospective jurors, 3 will be unable to serve?
e. What are the name and parameters of the posterior distribution of theta if the prior had been Uniform {0,1}?