Certainly! Let's go through each part of the question step-by-step.
### a. Probability that the person is an Independent and said "Yes"
We need to determine the probability that a randomly chosen person from the group is an Independent and said "Yes" to the legalization of marijuana.
From the table:
- The number of Independents who said "Yes" is 298.
- The total number of people surveyed is 1037.
The probability [tex]\( P(\text{Independent and Yes}) \)[/tex] is calculated as follows:
[tex]\[ P(\text{Independent and Yes}) = \frac{\text{Number of Independents who said "Yes"}}{\text{Total number of people surveyed}} \][/tex]
[tex]\[ P(\text{Independent and Yes}) = \frac{298}{1037} \][/tex]
Calculating this percentage:
[tex]\[ P(\text{Independent and Yes}) = \frac{298}{1037} \approx 0.2873 \][/tex]
Converting this to a percentage:
[tex]\[ 0.2873 \times 100 = 28.7\% \][/tex]
So, the probability that a randomly chosen person is an Independent and said "Yes" is [tex]\(28.7\%\)[/tex].
### b. Probability that the person is a Republican who said "No"
We need to determine the probability that a randomly chosen person from the group is a Republican who said "No" to the legalization of marijuana.
From the table:
- The number of Republicans who said "No" is 135.
- The total number of people surveyed is 1037.
The probability [tex]\( P(\text{Republican and No}) \)[/tex] is calculated as follows:
[tex]\[ P(\text{Republican and No}) = \frac{\text{Number of Republicans who said "No"}}{\text{Total number of people surveyed}} \][/tex]
[tex]\[ P(\text{Republican and No}) = \frac{135}{1037} \][/tex]
Calculating this percentage:
[tex]\[ P(\text{Republican and No}) = \frac{135}{1037} \approx 0.1301 \][/tex]
Converting this to a percentage:
[tex]\[ 0.1301 \times 100 = 13.0\% \][/tex]
So, the probability that a randomly chosen person is a Republican who said "No" is [tex]\(13.0\%\)[/tex].
### c. Are the events "Yes" and "Democrat" mutually exclusive?
To determine whether the events "Yes" and "Democrat" are mutually exclusive, we need to establish if it is possible for a voter to be both a Democrat and have voted "Yes."
From the table:
- The number of Democrats who said "Yes" is 231.
Mutually exclusive events are two events that cannot happen at the same time. Since there are 231 voters who are Democrats and who voted "Yes," it is clear that a voter can indeed be both a Democrat and vote "Yes."
Therefore, the events "Yes" and "Democrat" are not mutually exclusive.
#### The correct answer is:
B. "Yes" and "Democrat" are not mutually exclusive because a voter can vote "Yes" and be a "Democrat."
Thus, we have answered the question step-by-step confirming the probabilities and the nature of the events provided.
