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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 reversed,then the final fate of the legislation would bereversed?

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(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?

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