Suppose a state has 

1010 

representatives and a population of 

88 

​million; party affiliations are 

4040​% 

Republican and 

6060​% 

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 this problem step by step:

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

Given:
- Total population: 8 million
- Number of representatives: 1010
- Party affiliations: 40% Republican, 60% 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: \( 40\% \) of 1010 seats = \( 0.4 \times 1010 = 404 \) representatives
- Democrats: \( 60\% \) of 1010 seats = \( 0.6 \times 1010 = 606 \) representatives

So, the most likely distribution of House seats would be:
- **Republicans:** 404 representatives
- **Democrats:** 606 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:**
- By concentrating Democrats into as few districts as possible, Republicans could potentially win a majority in more districts.
- In the extreme case, Republicans could win up to **1010 representatives**.

**Minimum number of Republican representatives:**
- Conversely, by spreading Republican voters thinly across all districts, Democrats could ensure that Republicans win as few districts as possible.
- In the extreme case, Republicans could end up with **0 representatives**.

### Summary:
- **Most likely distribution (Part 1a):**  
  - Republicans: 404 representatives  
  - Democrats: 606 representatives
- **Maximum number of Republican representatives (Part 1b):** 1010 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 number of representatives affect the stability of political representation?
2. What are the potential consequences of extreme gerrymandering on voter turnout?
3. How can mathematical models be used to propose fair district boundaries?
4. What are some historical examples of gerrymandering and their outcomes?
5. How does the distribution of population density influence gerrymandering strategies?

**Tip:** When analyzing political distributions, understanding both proportional representation and the effects of gerrymandering can provide insight into the fairness and effectiveness of electoral systems.

Explore the most likely distribution of House seats based on population and party affiliations, and understand the effects of unlimited gerrymandering on political representation. Calculate the expected number of Republican and Democrat representatives under random district drawing and explore the maximum and minimum outcomes with unrestricted gerrymandering.

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