Suppose a state has 10 representatives and a population of 12 million; party affiliations are $50 \%$ Republican and $50 \%$ 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?
a. What would be the most likely distribution of House seats?
5 Republicans, 5 Democrats
b. The maximum number of Republicans representatives could be The minimum number of Republicans representatives could be
Answer
For part b, if the districts could be drawn without restriction, the party in power could potentially gerrymander the districts to their advantage. In the extreme case, they could draw the districts such that all of the representatives sent to Congress are from their party. Therefore, the maximum number of Republican representatives could be \(\boxed{10}\), and the minimum could be \(\boxed{0}\).
Step 1 ：Suppose a state has 10 representatives and a population of 12 million; party affiliations are 50% Republican and 50% Democrat.
Step 2 ：For part a, if the districts were drawn randomly and the population is evenly split between Republicans and Democrats, the most likely distribution of House seats would be an even split as well. This is because, on average, each district would have the same number of Republicans and Democrats. So, the most likely distribution of House seats would be \(\boxed{5}\) Republicans and \(\boxed{5}\) Democrats.
Step 3 ：For part b, if the districts could be drawn without restriction, the party in power could potentially gerrymander the districts to their advantage. In the extreme case, they could draw the districts such that all of the representatives sent to Congress are from their party. Therefore, the maximum number of Republican representatives could be \(\boxed{10}\), and the minimum could be \(\boxed{0}\).
