3.75. 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? What is the corresponding probability for a given council member not on the steering committee?
Please be fast.
Do the problem in the image, but considering only the case p = 1/2. Answer only the first question:
What is the
if that person's vote were reversed, then the final fate of the legislation would be reversed? (Note:
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 27 = 128 possible outcomes. A key
question obviously is: For how many of these 128 outcomes is voter I '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?
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