Why does the phrase colored in red below ($\color{red}{\text{the people who move have a percentage of Democrats which is between these two values}}$) matter? How does it assist with intuiting the [peaceful coexistence](https://en.wikipedia.org/wiki/Peaceful_coexistence) of $P_{new}(D|B) > P_{old}(D|B)$ and
$P_{new}(D|B^C) > P_{old}(D|B^C)$?
>59. The book _Red State, Blue State, Rich State, Poor State_ by Andrew Gelman [12]
discusses the following election phenomenon: within any U.S. state, a wealthy voter is
more likely to vote for a Republican than a poor voter, yet the wealthier states tend to
favor Democratic candidates!
>
>(a) Assume for simplicity that there are only 2 states (called Red and Blue), each
of which has 100 people, and that each person is either rich or poor, and either a
Democrat or a Republican. Make up numbers consistent with the above, showing how
this phenomenon is possible, by giving a 2 2 table for each state (listing how many
people in each state are rich Democrats, etc.). So within each state, a rich voter is more
likely to vote for a Republican than a poor voter, but the percentage of Democrats is
higher in the state with the higher percentage of rich people than in the state with the
lower percentage of rich people.
>
>(b) In the setup of (a) (not necessarily with the numbers you made up there), let
D be the event that a randomly chosen person is a Democrat (with all 200 people
equally likely), and B be the event that the person lives in the Blue State. Suppose
that 10 people move from the Blue State to the Red State. Write $P_{old}$ and $P_{new}$ for
probabilities before and after they move. Assume that people do not change parties,
so we have $P_{new}(D) = P_{old}(D)$. Is it possible that _both_ $P_{new}(D|B) > P_{old}(D|B)$ and
$P_{new}(D|B^C) > P_{old}(D|B^C)$ are true? If so, explain how it is possible and why it does not contradict the law of total probability $P(D) = P(D|B)P(B) + P(D|B^C)P(B^C)$; if not,
show that it is impossible.
>
>## Solution:
>(b) Yes, it is possible. Suppose with the numbers from (a) that 10 people move from the
Blue State to the Red State, of whom 5 are Democrats and 5 are Republicans. Then
$P_{new}(D|B) = 75/90 > 80/100 = P_{old}(D|B)$ and $P_{new}(D|B^C) = 30/110 > 25/100 = P_{old}(D|B^C)$. Intuitively, this makes sense since the Blue State has a higher percentage
of Democrats initially than the Red State, and $\color{red}{\text{the people who move have a percentage
of Democrats which is between these two values}}$.
>
>This result does not contradict the law of total probability since the weights $P(B), P(B^C)$ also change: $P_{new}(B) = 90/200$, while $P_{old}(B) = 1/2$. The phenomenon could not occur if an equal number of people also move from the Red State to the Blue State (so that
P(B) is kept constant).
Blitzstein, *Introduction to Probability* (2019 2 edn), Chapter 2, Exercise 59, p 96.
p 18 in the publicly downloadable PDF of curbed solutions.
DNB
·
2021-12-31T08:55:07Z (almost 3 years ago)