The book Red State, Blue State, Rich State, Poor State by Andrew Gelman [13] discusses

Chapter 2, Problem 53

(choose chapter or problem)

The book Red State, Blue State, Rich State, Poor State by Andrew Gelman [13] discusses the following election phenomenon: within any U.S. state, a wealthy voter is more likely to vote for a Republican than a poor voter, yet the wealthier states tend to favor Democratic candidates! In short: rich individuals (in any state) tend to vote for Republicans, while states with a higher percentage of rich people tend to favor Democrats. (a) Assume for simplicity that there are only 2 states (called Red and Blue), each of which has 100 people, and that each person is either rich or poor, and either a Democrat or a Republican. Make up numbers consistent with the above, showing how this phenomenon is possible, by giving a 2 2 table for each state (listing how many people in each state are rich Democrats, etc.). (b) In the setup of (a) (not necessarily with the numbers you made up there), let D be the event that a randomly chosen person is a Democrat (with all 200 people equally likely), and B be the event that the person lives in the Blue State. Suppose that 10 people move from the Blue State to the Red State. Write Pold and Pnew for probabilities before and after they move. Assume that people do not change parties, so we have Pnew(D) = Pold(D). Is it possible that both Pnew(D|B) > Pold(D|B) and Pnew(D|Bc) > Pold(D|Bc) are true? If so, explain how it is possible and why it does not contradict the law of total probability P(D) = P(D|B)P(B) + P(D|Bc)P(Bc); if not, show that it is impossible.