Problem 72 - A First Course in Probability

Question

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?

Step-by-Step Solution

Answer

In conclusion, the probability that a given steering committee member's vote is decisive and reverses the final decision is \(P(R) = (p^2 + (1-p)^2)({4\choose2}p^2(1-p)^2 + {4\choose3}p^3(1-p))\), while the probability that a given council member not on the steering committee has a decisive vote is \(P(F) = (2p^2 - p^3)({3\choose0}(1-p)^3 + {3\choose1}p(1-p)^2 + {3\choose3}p^3) * p\).

Step 1: Probability that a given steering committee member gives a decisive vote

To find the probability that a given steering committee member's vote is decisive, we need to consider the following events: A1: The other two steering committee members approve the legislation unanimously. A2: The other two steering committee members reject the legislation unanimously. The probability for A1 and A2 is: \(P(A1) = p^2\) \(P(A2) = (1-p)^2\) Now, consider that the given member's vote is decisive. If the given steering committee member votes "no" in the case of A1, then the legislation will fail to pass the committee. Similarly, if the given steering committee member votes "yes" in the case of A2, the legislation will pass the committee. The probability that the given steering committee member's vote is decisive is: \(P(D) = P(A1) + P(A2) = p^2 + (1-p)^2\)

Step 2: Probability that the given steering committee member's vote reverses the final decision

We will now find the probability that by reversing the given steering committee member's vote, the legislation will either be accepted or rejected by the full council. Since there are 4 council members not on the steering committee, each of them can vote either "yes" or "no", resulting in 16 possible outcomes. We only need to consider the cases when 2 or 3 council members approve the legislation. If all 4 council members approve it, reversing the given steering committee member's vote will not change the result; similarly, if all 4 council members reject, it will not change the result either. Let B1 represent the event that 2 of the 4 council members approve the legislation and B2 represent the event that 3 of the 4 council members approve the legislation. The probability of B1 and B2 is: \(P(B1) = {4\choose2}p^2(1-p)^2\) \(P(B2) = {4\choose3}p^3(1-p)\) Now, the probability that the final decision is reversed by the steering committee member's vote is given by: \(P(R) = P(D) * (P(B1) + P(B2) = (p^2 + (1-p)^2)({4\choose2}p^2(1-p)^2 + {4\choose3}p^3(1-p))\)

Step 3: Probability that a given council member not on the steering committee has a decisive vote

For a council member not on the steering committee to be decisive, the following events must occur: C1: The legislation passes the steering committee with at least 2 votes. C2: The given council member must vote in such a way that the overall result changes. We know that the legislation passes the steering committee with probability \(p^2 + 2p^2(1-p)= 2p^2 - p^3\). The probability of passing the full council is given by: \(P(C1) = 2p^2 - p^3\) Now, consider the eight possible outcomes from the other three council members: 1. 0 out of 3 members approve 2. 1 out of 3 members approve 3. 2 out of 3 members approve 4. 3 out of 3 members approve The probability that the given council member has a decisive vote is: \(P(C2) = P(0) * p + P(1) * p + P(3) * (1-p) = ({3\choose0}(1-p)^3 + {3\choose1}p(1-p)^2 + {3\choose3}p^3) * p\) The overall probability that a given council member not on the steering committee has a decisive vote is: \(P(F) = P(C1) * P(C2) = (2p^2 - p^3)({3\choose0}(1-p)^3 + {3\choose1}p(1-p)^2 + {3\choose3}p^3) * p\) Therefore, the probability that a given steering committee member's vote is decisive and reverses the final decision is P(R), and the probability that a given council member not on the steering committee has a decisive vote is P(F).