Day 25: Heart of Mathematics

Your exam will cover the material since the previous exam:

• infinity,
• dimension,
• geometry, and
• statistics.

• Why do all the polls differ?
• What's this Bradley Effect? Or the Reverse Bradley Effect?
• What is the Electoral College, and why can one candidate win more peoples' votes but lose the election?

• Samples versus Populations
• What is statistical bias?
• Cicada population data
• Avoiding bias. Mindscape #5, p. 581.
• When is enough enough? Data, that is.

• accurate
• unbiased
• "sufficient in number to reflect the population's variability"

• Are you rich?
• Do you cheat on your spouse?
• Do you have AIDS?

• Let's calculate the true rates of maleness and femaleness in class...
• Each person will flip a coin:
  • If it's heads, answer "Male" (the embarrassing answer).
  • If it's tails, answer truthfully.
• How do we now estimate the true rate?
  • Guess that half the "Male" answers are from heads; the other half are truthful answers.
• An example: Mindscape #2, p. 581.
• Do you see that we're losing information? We're losing power to predict the true rate of embarrassing maleness....
• How much information can we afford to lose?

• with a die?
• with two tosses of a coin? (this will be the subject of Friday's class, and will be important for doing problem #12 of your homework!)

Three voting schemes, three different winners.

1. Plurality voting -- Chad wins
2. Vote for Two -- Courtney wins
3. Borda Count Method -- Gwyn wins

4. Here's my favorite system: Instant run-off, until we have a majority. Who wins?

Look, it seems like this should be easy. We want a system that has the following properties, given each voter's list of ordered rankings for the candidates:

1. Better is better: If a voter changes his or her vote in favor of Candidate X, then that shouldn't hurt Candidate X.
2. Ignore the irrelevant: throwing out a loser will not change the outcome.
3. Go along with the consensus: if everyone agrees, then we have a winner.

Here's the main, astonishing result:

That is, one person makes the decision.

It was not a "fluke" that different methods gave different results. As mentioned in the text, mathematicians can "rig" a seemingly benign system that will hand the election over to a preferred candidate.

Example:

Look at the beautiful symmetry in this table.... It's responsible for the problem, in many ways.