(Voting Systems and Communication Diversity) A large group of voters is to decide between two candidates. Each voter is to vote for either candidate A or candidate B. There are 50 million people who have decided to vote for candidate A and 49 million who have decided to vote for candidate B. An additional 100 million people have no preference what so ever and will each vote by making an independent random equiprobable choice. (All numbers are large enough so that the central limit theorem applies.)

a. What is the probability that candidate A will win in the popular vote?

b. An adversary is able to cast an additional 0.9 million false votes for candidate B by fraud. Can the adversary change the outcome? What now is the probability that candidate A will win? How does this change if the adversary can record one million (or more) false votes?

Suppose now that an electoral system for voting is used as follows: the voters are divided into fifty regions of equal size. Assume that each region has the three types of voters in the same proportion: one million voters for candidate A, 0.98 million voters for candidate B, and 2 million undecided. The majority popular vote in each region determines the single electoral vote from that region. The candidate with the largest number of electoral votes wins.

c. What is the probability that candidate A will win in the electoral vote?

d. Can the adversary with 0.9 million false votes change the outcome by fraud? With what strategy? Must the false voters be distributed over more than one region? How many? What now is the probability that candidate A will win? How does this change if the adversary can cast one million (or more) false votes?