# A town council of 7 members contains a steering committee of size 3. New ideas for legislation go first to the steering committee and then

A town council of 7 members contains a steering committee of size 3. New ideas for legislation go first to the steering committee and then on to the council as a whole if at least 2 of the 3 committee members approve the legislation. Once at the full council, the legislation requires a majority vote (of at least 4) to pass. Consider a new piece of legislation, and suppose that each town council member will approve it, independently, with probability p. What is the probability that a given steering committee member’s vote is decisive in the sense that if that person’s vote were reversed, then the final fate of the legislation would be reversed? (Warning: There are some answers on the web for this book problem which are incorrect.) To set up the problem: note there are 7 voters. Set up voters 1-3 on the steering committee, and voters 4-7 as not on the steering committee. Focus on voter 1. Thus there are 2^7 = 128 possible outcomes. A key question obviously is: For how many of these 128 outcomes is voter 1’s vote decisive? To make sure you understand the problem, it’s best to answer this question for a few example outcomes, before