A town council of 7 members contains a steeringcommittee of size 3. New ideas for legislation gofirst to the steering committee and then on to thecouncil as a whole if at least 2 of the 3 committeemembers approve the legislation. Once at thefull council the legislation requires a majority vote (of at least 4) to pass. Consider a new piece oflegislation and suppose that each town councilmember will approve it independently with probabilityp. What is the probability that a given steeringcommittee member s vote is decisive in thesense that if that person s vote were reversedthen the final fate of the legislation would bereversed?(Warning:There are some answers on the web for this book problem which are incorrect.) To set up theproblem: note there are 7 voters. Set up voters 1-3 on the steering committee and voters 4-7 as noton the steering committee. Focus on voter 1. Thus there are 2^7 = 128 possible outcomes. A keyquestion obviously is: For how many of these 128 outcomes is voter 1 s vote decisive? To make sureyou understand the problem it s best to answer this question for a few example outcomes beforeyou start setting up the computation. Would your answer be different if you consider voter 2?
