Problem 1: Gerrymandering Suppose that among a population, some fraction p of the voters support Party A, and 1 -p support Party B. In an election of N voters, let X be the number of votes cast for Party A. . What is the distribution of X? . What is the probability that Party A wins power in the election, or ties? . Suppose an election is held with 303 votes cast, where the probability of Party A support is 51%. What is the probability that Party A wins power? Consider separating the population into three regions or districts of equal population, District 1, 2, 3, with relative fractions of Party A support pi; p2, P3. . What is the relationship between p1, p2, p3 and p? . If n votes are cast in each district, what is the probability that Party A wins at least two districts and takes power? Suppose that Party A is responsible for redistricting, and in doing so determines the concentration of Party A support in each district. . For a given value of p, how should the districts be structured (values of p1, p2, p3) to maximize the probability that Party A wins at least two districts? . If 101 votes are cast in each district (structured as above) for 303 total votes, where p = 0.51 as before, what is the probability that Party A wins and takes power? Generate a graph, as a function of p, of a) the probability that Party A takes power in the undistricted election, and b) the probability that Party A takes power in the maximally gerrymandered election. Assume 303 total votes cast, with 101 votes in each district