91Ó°ÊÓ

$7$ 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.

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

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

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

$⇒P\left(E\right)=P(Eâˆ\pounds A)$

$P(Eâˆ\pounds A)$

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$P(·âˆ\pounds A)$.

$P(Eâˆ\pounds A)=P\left(Eâˆ\pounds S_{2}A\right)P\left(S_2âˆ\pounds A\right)+P\left(Eâˆ\pounds S_{3}A\right)P\left(S_3âˆ\pounds A\right)$

$=\left[1-(1-p{)}^4-4p(1-p{)}^3\right]·P\left(S_2âˆ\pounds A\right)+4p(1-p{)}^{3}P\left(S_3âˆ\pounds A\right)$

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

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

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:

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

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

$⇒P\left(Eâˆ\pounds S_0\right)=P\left(Eâˆ\pounds S_1\right)=0$

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 $3:4$

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

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

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