Consider a political discussion group consisting of 8 Democrats, 8 Republicans, and 3 Independents. Suppose that two group members are randomly selected, in succession, to attend a political convention. Find the probability of selecting two Democrats.
Consider a political discussion group consisting of 8 Democrats, 8 Republicans, and 3 Independents. Suppose that two group members are randomly selected, in succession, to attend a political convention. Find the probability of selecting two Democrats.
MATLAB: An Introduction with Applications
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ISBN:9781119256830
Author:Amos Gilat
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![### Probability in Political Groups: Selecting Two Democrats
#### Problem Statement
Consider a political discussion group consisting of:
- 8 Democrats
- 8 Republicans
- 3 Independents
Suppose that two group members are randomly selected, in succession, to attend a political convention. Find the probability of selecting two Democrats.
#### Solution Explanation
To solve this problem, we need to determine the probability of selecting two Democrats in succession from the political discussion group.
1. **Total Members in the Group**:
The group has a total of \( 8 + 8 + 3 = 19 \) members.
2. **Probability of Selecting the First Democrat**:
Since there are 8 Democrats in the group, the probability of selecting a Democrat first is:
\[
\frac{8}{19}
\]
3. **Probability of Selecting the Second Democrat**:
Assuming the first member selected was a Democrat (without replacement), there are now:
- 7 Democrats left
- Total group members: \( 19 - 1 = 18 \)
The probability of selecting another Democrat in the second selection is:
\[
\frac{7}{18}
\]
4. **Overall Probability**:
The overall probability of selecting two Democrats in succession is given by the product of the two individual probabilities:
\[
\frac{8}{19} \times \frac{7}{18} = \frac{56}{342}
\]
Simplify the fraction:
\[
\frac{56}{342} = \frac{28}{171}
\]
Thus, the probability of randomly selecting two Democrats, in succession, from the group is \( \frac{28}{171} \).
### Conclusion
Understanding the probability of selecting members of specific political affiliations helps in addressing various questions related to group dynamics and representation. This particular method can be applied to different scenarios where random selection from a group is involved.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F78db42f9-6d62-48c9-a117-b436e469106b%2F38ec03fa-be0c-4331-8808-1a117f45d7b1%2Fhe57hnp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Probability in Political Groups: Selecting Two Democrats
#### Problem Statement
Consider a political discussion group consisting of:
- 8 Democrats
- 8 Republicans
- 3 Independents
Suppose that two group members are randomly selected, in succession, to attend a political convention. Find the probability of selecting two Democrats.
#### Solution Explanation
To solve this problem, we need to determine the probability of selecting two Democrats in succession from the political discussion group.
1. **Total Members in the Group**:
The group has a total of \( 8 + 8 + 3 = 19 \) members.
2. **Probability of Selecting the First Democrat**:
Since there are 8 Democrats in the group, the probability of selecting a Democrat first is:
\[
\frac{8}{19}
\]
3. **Probability of Selecting the Second Democrat**:
Assuming the first member selected was a Democrat (without replacement), there are now:
- 7 Democrats left
- Total group members: \( 19 - 1 = 18 \)
The probability of selecting another Democrat in the second selection is:
\[
\frac{7}{18}
\]
4. **Overall Probability**:
The overall probability of selecting two Democrats in succession is given by the product of the two individual probabilities:
\[
\frac{8}{19} \times \frac{7}{18} = \frac{56}{342}
\]
Simplify the fraction:
\[
\frac{56}{342} = \frac{28}{171}
\]
Thus, the probability of randomly selecting two Democrats, in succession, from the group is \( \frac{28}{171} \).
### Conclusion
Understanding the probability of selecting members of specific political affiliations helps in addressing various questions related to group dynamics and representation. This particular method can be applied to different scenarios where random selection from a group is involved.
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