# Stat

**aprma_**

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?

- 9 years ago
- 3

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