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]
