Day 27: Heart of Mathematics

Today:

• Final Exam: May 7, 3:10-5:10
  • It will be comprehensive, but emphasize recent topics.
  • All the sections we covered from chapters 5, 6, 7, and 8 are fair game. Expect problems that look like the homework or like activities that we did in class.

• The final paper/presentation:
  • Final project paper due today.

  • The Schedule of presentations. We start Monday! We must start right on time, and keep moving.

  • Here's the grading rubric for the presentations. This gives you an idea of how you'll be graded (and how you'll be grading your peers).

  • Important:
    • You need to be there for your classmates.
    • If you are not in class for the other presentations, you will lose points on your own.

• Something old, on chaos: Lorenz died just a few days ago (4/16/2008), at the age of 90. And I just saw a report, at the New York Times, that weather reporters are no good beyond three days out.

• Section 7.5 mindscapes returned.
  • #3: using a single question to grade the class
  • #12: Good surveys! Now push it all the way to the conclusion: make sure you calculate the percentages you'd expect in the "population".

• Section 7.6 mindscapes due.

• Section 8.3: Money Matters

  • Mindscapes: #1, 3, 4, 10, 12; Type up #9. Due Monday

  • How do the credit card companies make so much money?
    • Pretty straight forward: #6, p. 678
    • Here's a twist: #11, p. 679

  • How do mortgages work?
    • Other loans can be calculated the same way: for example, CitiFinancial's loan shark fishing attempt on me.
    • Here's another straight-forward mortgage problem: #16, p. 680

  • A "real-world" example: do you use those balance-transfer checks? I got this offer last week.
    • What are the hidden costs (buried in the fine print)?
    • What would $1000 cost me for a year, if I take it out May 1, and pay nothing back? (By the way, the balance transfer APR is 18.99%).

• Section 8.4: Peril at the Polls
  • A blast from the past: The uncola (7-up) versus the colas

  • Sixth grade vice-presidential vote

    	6 boys prefer	4 boys prefer	8 girls prefer	4 girls prefer
    First Choice	Chad	Chad	Gwyn	Courtney
    Second Choice	Courtney	Gwyn	Courtney	Gwyn
    Third Choice	Gwyn	Courtney	Chad	Chad

    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 a given candidate, then that shouldn't hurt the candidate.
    2. Ignore the irrelevant: throwing out a loser will not change the outcome.
    3. Go along with the concensus: if everyone agrees, then we have a winner.

    Here's the main, astonishing result:

    The only voting scheme that satisfies these three very reasonable requirements is a dictatorship!

    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.

  • "Transitivity" means that, if we prefer A to B, and B to C, then we'll prefer A to C. Condorcet's Paradox says there's no promise of transitivity!

    Example:

    	5 voters' rankins	5 voters' rankins	5 voters' rankins
    First Place	A	B	C
    Second Place	B	C	A
    Third Place	C	A	B

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

  • I want to share a story from my days in the Peace Corps in Togo.

• Evaluations

• The Butterfly Effect (for which Lorenz was famous).