Question

Making your vote count. Refer to the Chance (Fall 2007) study on making your vote count, Exercise 3.29 (p. 171). Recall the scenario where you are one of five county commissioners voting on an issue, and each commissioner is equally likely to vote for or against.

1. Your vote counts (i.e., is the decisive vote) only if the other four voters split, two in favour and two against. Use the binomial distribution to find the probability that your vote counts.
2. If you convince two other commissioners to “vote in bloc” (i.e., you all agree to vote among yourselves first, and whatever the majority decides is the way all three will vote, guaranteeing that the issue is decided by the bloc), your vote counts only if these two commissioners split their bloc votes, one in favour and one against. Again, use the binomial distribution to find the probability that your vote counts.

Short answer

1. The probability that your vote count is 0.375 if the other four voters split.

2. The probability that your vote count is 0.5 if the other two votes split.

Step-by-step solution

(a) Given the information

It is given that five county commissioners are voting on an issue, and each commissioner is equally likely to vote for or against it. Given that the four voters split, so $\mathrm{n}=4$.

To find the probability that your vote counts is if the other four voters split, two in favour and two against, so$\mathrm{x}=2$

Hence, the probability that in favour and against is

• p(favor)=p(against)
• p(favor)=0.5
• p(against)=0.5

localid="1664387128029" $$\begin{gathered} \mathrm{q}=1-\mathrm{p} \\ \mathrm{Therefore},\mathrm{q}=1-0.5 \\ =0.5 \end{gathered}$$

The formula for the probability function of the binomial distribution is,

$\mathrm{P}\left(\mathrm{x}\right){=}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{x}}{\mathrm{p}}^{\mathrm{x}}{\mathrm{q}}^{\mathrm{n}-\mathrm{x}}$

To determine the probability that your vote is p(x=2) as

$$" width="9">P(x=2)=4C2(0.5)2(0.5)4-2

$$\begin{gathered} =\frac{4!}{2!2!}{(0.5)}^2{(0.5)}^2 \\ =0.375 \end{gathered}$$

(b) Probability of the vote count

If only two commissioners split their bloc votes, one in favour and one against,

$\mathrm{n}=2\mathrm{and}\mathrm{x}=1$

Here, the probability that in favour and the probability that against is equal,

So, p (favour)=p(against)

Substituting the values in the above formula,

$$\begin{gathered} p(x=1){=}^2{C}_1{\left(0.5\right)}^1{\left(0.5\right)}^{2-1} \\ P(x=1)=0.5 \end{gathered}$$

Therefore, the probability that your vote count is 0.5.