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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

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

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.