69 Making your vote count. 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 a scenario where you are one of five county commissioners voting on an issue, where each commissioner is equally likely to vote for or against.

a. Your vote counts (i.e., is the decisive vote) only if the other four voters split, 2 in favor and 2 against. Use the binomial distribution to find the probability that your vote counts.

b. If 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, guaranteeing that the issue is decided by the bloc), your vote counts only if these 2 commissioners split their bloc votes, 1 in favor and 1 against. Again, use the binomial distribution to find the probability that your vote counts.