Question

Exit poll candidates and voters.In an exit poll, 45% of voters said that the main issue affecting their choice of candidates was the economy, 35% said national security, and the remaining 20% were not sure. Suppose we select one of the voters who participated in the exit poll at random and ask for the main issue affecting his or her choice of candidates.

1. List the sample points for this experiment.
2. Assign reasonable probabilities to the sample points.
3. What is the probability that the main issue affecting randomly selected voters’ choice was either the economy or national security?

Short answer

1. There are three sample points which are as follows:$\left\{\mathbf{E}\mathbf{,}\mathbf{NS}\mathbf{,}\mathbf{N}\right\}$
2. The probability of respective sample points is-$\mathbf{P}\mathbf{\left(}\mathbf{E}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{45}\mathbf{,}\mathbf{P}\mathbf{\left(}\mathbf{NS}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{35}\mathbf{,}\mathbf{P}\mathbf{\left(}\mathbf{N}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{20}$
3. The probability that the main issue affecting randomly selected voter’s choice was either the economy or national security is 0.80

Step-by-step solution

Defining the events and listing sample points in mathematical format

As per the information given in the question, one notes down the events and assigns those initials. Let’s assign-

E = Economy

NS = National Security

N = Not Sure

The sample points are – $\left\{\mathbf{E}\mathbf{,}\mathbf{NS}\mathbf{,}\mathbf{N}\right\}$

Finding the probability of ‘economy’ and ‘national security as event

Given, one could ascertain that 45% of voters said that “economy” is the main issue affecting their choice. The probability of selecting a random voter whose choice affects the economy is $\mathbf{P}\mathbf{\left(}\mathbf{E}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{45}$

Given, one could ascertain that 35% of voters said that “national security” is the main issue affecting their choice. The probability of selecting a random voter whose choice affects national security is $\mathbf{P}\mathbf{\left(}\mathbf{NS}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{35}$

Locating probability of selecting a voter who is not sure

As per the information provided in the question, one could ascertain that 20% of the voters said that they are not sure about the main issue affecting their choice. The probability of selecting a random voter who is not sure is P(N) = 0.20

Assigning reasonable probabilities to sample points

The probabilities calculated in steps 1 to 3 can be assigned as $\mathbf{P}\mathbf{\left(}\mathbf{E}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{45}\mathbf{,}\mathbf{P}\mathbf{\left(}\mathbf{NS}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{35}\mathbf{,}\mathbf{P}\mathbf{\left(}\mathbf{N}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{20}$respectively.

Noting down the probability of ‘economy’ and ‘national security as event

We note down the result obtained in Step 2. The result obtained is $\mathbf{P}\mathbf{\left(}\mathbf{E}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{45}$ and $\mathbf{P}\mathbf{\left(}\mathbf{NS}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{35}$

Detecting probability that main issue affecting is either the economy or national security

Now, let ENS represent the event that the main issue affecting randomly selected voter’s choice is either the economy or national security. ENS will represent the combined probability of ‘economy’ and ‘national security.

$\mathbf{P}\mathbf{\left(}\mathbf{ENS}\mathbf{\right)}\mathbf{=}\mathbf{P}\mathbf{\left(}\mathbf{E}\mathbf{\right)}\mathbf{+}\mathbf{P}\mathbf{\left(}\mathbf{NS}\mathbf{\right)}$

$$\begin{gathered} \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{45}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{35} \\ \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{80} \end{gathered}$$