Suppose a population of n% Democrats and (100-n)% Republicans was randomly distributed into 10 equally sized districts. Suppose also that n is greater than 50.
What is the most likely outcome?
Is it that about n% of the 10 districts will have a majority of Democratic voters?
Or, is it that each district will most likely have a majority of Democratic voters, resulting in 10 Democratic leaning districts?
Or, is it a 3rd possibility?
Also, does anyone have a quick way to write a Monte-Carlo simulation for this in Python, say?
Thanks in advance.
The answer depends on the size of the population. If the population is $10$---so, one district per person, then the fraction of Democratic-majority districts is exactly the fraction of Democrats.
If Democrats account for the proportion $p$ of the population $m$, then the proportion of Democrats in any given district has $$\textrm{mean }p \qquad \textrm{ and} \qquad \textrm{standard deviation }O(m^{-1 / 2}). $$ So, if $p > \frac{1}{2}$, then as $m \to \infty$ the probability that any given district has a Democratic majority approaches $1$, and thus so does the probability that all $10$ districts have a Democratic majority, too.