Stat

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
you start setting up the computation. Would your answer be different if you consider voter 2?