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Part 1
Suppose a state has 16 representatives and a population of 8 ​million; party affiliations are 30​% Republican and 70​% Democrat.
a.
If districts were drawn​ randomly, what would be the most likely distribution of House​ seats?
b.
If the districts could be drawn without restriction​ (unlimited gerrymandering), what would be the maximum and minimum number of Republican representatives who could be sent to​ Congress?
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Part 1
a. What would be the most likely distribution of House​ seats?

enter your response here ​Republicans,

enter your response here Democrats

Let's solve the problem step by step:

### Part 1(a): Most Likely Distribution of House Seats

Given:
- Total population: 8 million
- Number of representatives: 16
- Party affiliations: 30% Republican, 70% Democrat

If districts were drawn randomly and proportionally to the population, the most likely distribution of House seats would reflect the party affiliation percentages.

**Calculating the expected number of representatives:**
- Republicans: \( 30\% \) of 16 seats = \( 0.3 \times 16 = 4.8 \) ≈ 5 representatives
- Democrats: \( 70\% \) of 16 seats = \( 0.7 \times 16 = 11.2 \) ≈ 11 representatives

So, the most likely distribution of House seats would be:
- **Republicans:** 5 representatives
- **Democrats:** 11 representatives

### Part 1(b): Maximum and Minimum Number of Republican Representatives

If the districts could be drawn without restriction (unlimited gerrymandering), the distribution of seats could be manipulated to favor one party over the other.

**Maximum number of Republican representatives:**
- Gerrymandering could be used to concentrate Democrats into a few districts, allowing Republicans to win more districts with smaller majorities.
- In the extreme case, Republicans could win up to **16 representatives**.

**Minimum number of Republican representatives:**
- Conversely, districts could be drawn so that Republicans are spread thinly across all districts, allowing Democrats to win the majority in each district.
- In the extreme case, Republicans could end up with **0 representatives**.

### Summary:
- **Most likely distribution (Part 1a):**  
  - Republicans: 5 representatives  
  - Democrats: 11 representatives
- **Maximum number of Republican representatives (Part 1b):** 16 representatives
- **Minimum number of Republican representatives (Part 1b):** 0 representatives

Would you like more details on any part of this explanation?

### Related Questions:
1. How does the method of drawing districts affect the outcome of elections?
2. What is the significance of gerrymandering in political representation?
3. How can mathematical models be used to detect gerrymandering?
4. What are the legal and ethical implications of gerrymandering?
5. How might proportional representation differ from the current district-based system?

**Tip:** Gerrymandering can significantly impact the political landscape, often leading to a representation that does not accurately reflect the population's preferences.

Explore the most likely distribution of House seats based on party affiliations and the impact of gerrymandering on congressional representation. Understand the mathematical principles of probability and statistics applied to political science. Suitable for advanced high school students and beyond.

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