The recent Democratic and Republican presidential state primary elections were highlighted by the difference in the way winning candidates were awarded delegates. In Republican states, the winner is awarded all the state's delegates; conversely, the Democratic state winner is awarded delegates in proportion to the percentage of votes. This led to a Chance (Fall 2007) article on making your vote count. Consider the following scenario where you are one of five voters (for example, on a county commission where you are one of the five commissioners voting on an issue).
a. Determine the number of ways the five commissioners can vote, where each commissioner votes either for or against. (These outcomes represent the sample points for the experiment.)
b. Assume each commissioner is equally likely to vote for or against. Assign reasonable probabilities to the sample points, part a.
c. Your vote counts (i.e., is the decisive vote) only if the other four voters split, two in favor and two against. Assuming you are commissioner number 5, how many of the sample points in part a result in a 2-2 split for the other four commissioners?
d. Use your answers to parts a-c to find the probability that your vote counts.
e. Now suppose you convince two other commissioners to "vote in bloc," i.e., you all agree to vote among yourselves first, and whatever the majority decides is the way all three will vote. With only five total voters, this guarantees that the bloc vote will determine the outcome. In this scenario, your vote counts only if the other two commissioners in the bloc split votes, one in favor and one against. Find the probability that your vote counts.