March 28, 2006

From MathWiki

  • Amy did a `book report' on Algebra for Atheletes (http://www.algebraforathletes.com). This was an interesting way of bringing in applications of algebra for people who were not usually inclined to learn algebra.
  • We looked at news segment on an ABC News podcast about the final four. They stated that 'George Mason's odds of winning the tournament? 4 million to 1.' Of course, now that they have lost the tournament the odds of them winning are 0, but that is besides the point. Here is a team that was very low seated to begin with and made it to the final four and I couldn't believe that there was STILL a 4 million to 1 odds that they would win. So I tracked down the statistic on the USAToday website that they said was the source. This is what I found:

"Abolishment of slavery and adoption of a Bill of Rights might not have been a 4 million-to-1 shot — as USA TODAY's sports analyst Danny Sheridan put the Patriots' chances at the beginning of the NCAA tournament. But at the time, they probably weren't as good as Monday's 6-1 odds."

So that's what they meant. Danny Sheridan (http://www.dannysheridan.com/) computes the odds on sports all the time and this is what he was saying at the beginning of the tournament. NOT when George Mason was in the final four. I wonder how he figured out those odds in the first place. 4 million to one sounds like he is making it up. (http://www.amazon.com/gp/product/0393310728/002-6225368-6563247?v=glance&n=283155)

  • The next thing we did was three 'probability paradoxes.' I didn't tell you why I call these paradoxes, because a paradox (http://www.webster.com/dictionary/paradox) is a statement that is seemingly contradictory or opposed to common sense and yet is perhaps true. A probability paradox is a statement about probabilities which our intuition often leads us astray and is perhaps contradictory to what the mathematics says (my definition).
  • The first was the AIDS test paradox (http://garsia.math.yorku.ca/~zabrocki/math1590w06/paradoxes/diseaseparadox.pdf).
  • The second titled Three Sticks in a Bag (http://garsia.math.yorku.ca/~zabrocki/math1590w06/paradoxes/sticksparadox.pdf).
  • The third was called The Monty Hall Paradox (http://garsia.math.yorku.ca/~zabrocki/math1590w06/paradoxes/goatslides.pdf).
  • You notice that I explained these paradoxes with wheels to justify the probability. This is a really neat way of thinking about and explaining probability. Like any subject in this class, we don't get in any depth about any subject but if you would like a little introduction to probability (http://garsia.math.yorku.ca/~zabrocki/math1590w06/paradoxes/introprobabilty.pdf), look at these notes.
  • What also came up was this new game show 'Deal, or no deal!' with the game show host Howie Mandel. We listened to a story on NPR about economists (http://www.npr.org/templates/story/story.php?storyId=5243893) who would like to understand how we make decisions when given a choice between a definite $50 or a 50% chance at $100. This research is a mix of economics, mathematics and psychology. Lets go to commercial.
  • We also talked a little bit about the `triangular numbers which were also square numbers.' I couldn't come up with much so I went home and computed the first few. They are 1, 1 + ... + 8 = 62, 1 + ... + 49 = 352, 1 + ... + 288 = 2042, 1 + ... + 1681 = 11892, 1 + ... + 9800 = 69302, 1 + ... + 57121 = 403912, ...

What is the pattern? I entered 1, 6, 35, 204, 1681, 6930, 40391 in the online integer sequence database (http://www.research.att.com/~njas/sequences/index.html) and I found a(n)^2 is a triangular number: a(n) = 6*a(n-1) - a(n-2) (http://www.research.att.com/~njas/sequences/A001109). e.g. 35 = 6*6 - 1, 204 = 6*35 - 6, 1189 = 6*204 - 35, etc. Can we figure out why this is?

  • ANNOUNCEMENT: all work for 1590 is due on April 11.