\begin{tabular}{|l|l|l|l|}
\hline
& No Degree & College Degree & Total \\
\hline
Liberal & 42 & 18 & 60 \\
\hline
Mixed & 24 & 64 & 88 \\
\hline
Conservative & 14 & 88 & 102 \\
\hline
Total & 80 & 170 & 250 \\
\hline
\end{tabular}

Match the statement with the correct probability.

A. Probability of a randomly selected person identifying as Liberal.
[tex]\[
\frac{60}{250}
\][/tex]

B. Probability of a randomly selected person identifying as Liberal and holds a College Degree.
[tex]\[
\frac{18}{250}
\][/tex]

C. Probability of a randomly selected person identifying as Liberal or holds a College Degree.
[tex]\[
\frac{42}{250}
\][/tex]



Answer :

To solve the problem of finding the probabilities associated with the given data table, let's carefully analyze the information provided. Here's a structured approach to the solution:

1. Total number of respondents: According to the table, the total number of respondents in each category is given in the last row.
- [tex]$Total = 80 + 170 + 250 = 500$[/tex]

2. Probability of a randomly selected person identifying as Liberal:
- The number of respondents identifying as Liberal from the given data:
- [tex]$Liberal = 80$[/tex]
- Therefore, the probability of a randomly selected person identifying as Liberal is:
[tex]\[ P(\text{Liberal}) = \frac{\text{Number of Liberals}}{\text{Total number of respondents}} = \frac{80}{500} \][/tex]
- Simplifying the fraction:
[tex]\[ P(\text{Liberal}) = \frac{80}{500} = \frac{8}{50} = \frac{4}{25} \][/tex]

3. Probability of a randomly selected person identifying as Liberal and holding a College Degree:
- The number of respondents identifying as Liberal and holding a College Degree should be provided directly or computed by dividing:
- [tex]$Liberal \land \text{College Degree} = 42$[/tex] (as per problem statement, not from the given data)
- Therefore, the probability is:
[tex]\[ P(\text{Liberal} \land \text{College Degree}) = \frac{\text{Number of Liberals with College Degree}}{\text{Total number of respondents}} = \frac{42}{500} \][/tex]
- Simplifying the fraction:
[tex]\[ P(\text{Liberal} \land \text{College Degree}) = \frac{42}{500} = \frac{21}{250} \][/tex]

4. Probability of a randomly selected person identifying as Liberal, or identifying as Mixed, or identifying as Conservative:
- Since these are all the categories provided, and they cover the complete set of respondents, we can infer:
[tex]\[ P(\text{Liberal} \lor \text{Mixed} \lor \text{Conservative}) = 1 \][/tex]

With this detailed explanation, we now have matching probabilities.

- Probability of a randomly selected person identifying as Liberal: [tex]$\frac{4}{25}$[/tex]
- Probability of a randomly selected person identifying as Liberal and holding a College Degree: [tex]$\frac{42}{250}$[/tex]