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?

Short answer

If the specific member is in the steering committee $2p^3(1-p)\left(5p^2-12p+8\right)$.

Otherwise$10p^3(1-p{)}^3$.

Step-by-step solution

Given Information

$7$ council members

$3$ of them are in the steering committee

Majority vote to pass the steering committee, then the council

One member is specified

Event:

$E$- if one specific council member changed his vote, the outcome of the legislation would be changed.

$A$- that specific council member voted "FOR",$P\left(A\right)=p$

$S_i$- the legislation passed the steering committee with $i=0,1,2,3$votes.

Explanation

The specified member is a steering committee member Condition upon $A$, using the Bayes formula

$P\left(E\right)=P(E\mid A)P\left(A\right)+P\left(E\mid A^c\right)P\left(A^c\right)$

$P(E\mid A)=P\left(E\mid A^c\right)$, because the same distributions of votes make the specified vote not interchangeable.

$\Rightarrow P\left(E\right)=P(E\mid A)$

$P(E\mid A)$

If$A$ happened, $E$ means that the bill had passed, but if special member changed opinion it wouldn't have. Then condition upon whether it passed with $S_2$ or $S_3$, using conditional probability$P(·\mid A)$.

$P(E\mid A)=P\left(E\mid S_{2}A\right)P\left(S_2\mid A\right)+P\left(E\mid S_{3}A\right)P\left(S_3\mid A\right)$

$=\left[1-(1-p{)}^4-4p(1-p{)}^3\right]·P\left(S_2\mid A\right)+4p(1-p{)}^{3}P\left(S_3\mid A\right)$

$=\left[1-(1-p{)}^4-4p(1-p{)}^3\right]2p(1-p)+4p(1-p{)}^3·p^2$

$=2p^3(1-p)\left(5p^2-12p+8\right)$

Explanation

The specified member is NOT a steering committee member

Use the Bayes formula to condition upon events $S_0,S_1,S_2,S_3$, one and only one of which occurs:

$P\left(E\right)=P\left(E\mid S_0\right)P\left(S_0\right)+P\left(E\mid S_1\right)P\left(S_1\right)+P\left(E\mid S_2\right)P\left(S_2\right)+P\left(E\mid S_3\right)P\left(S_3\right)$

The members of the council who are not steering committee members can influence the decision only if the legislation arrives at council vote

$\Rightarrow P\left(E\mid S_0\right)=P\left(E\mid S_1\right)=0$

$P\left(E\mid S_2\right)$

This is the probability of event that precisely one of the three not specified committee members votes "FOR", other two against, with two votes from the steering committee, the result is either $4:3$ or $3:4$

$P\left(E\mid S_2\right)=3p(1-p{)}^2$

Step 4: 

$P\left(E\mid S_3\right)$

This is the probability of event that no other committee member votes "FOR", the result is either $4:3$or $3:4$

$P\left(E\mid S_3\right)=(1-p{)}^3$$P(E\mid A)=P\left(E\mid S_{2}A\right)P\left(S_2\mid A\right)+P\left(E\mid S_{3}A\right)P\left(S_3\mid A\right)$$=3p(1-p{)}^2·3p^2(1-p)+(1-p{)}^3{p}^3$$=10p^3(1-p{)}^3$

Final Answer

If the specific member is in the steering committee $2p^3(1-p)\left(5p^2-12p+8\right)$.

Otherwise $10p^3(1-p{)}^3$.