February 7, 2006
- I wanted to discuss a little about the debate in the newspapers this week about the changes to the Ontario mathematics curriculum. I brought in several articles from the Toronto Star
- Feb 1: List of university prep courses offered in Toronto high schools (http://garsia.math.yorku.ca/~zabrocki/math1590w06/geometrycalc/univprereqs.pdf)
- Feb 1: Subtract calculus from high school? (http://garsia.math.yorku.ca/~zabrocki/math1590w06/geometrycalc/subtractcalc.pdf)
- Feb 5: It's the geometry, stupid (http://garsia.math.yorku.ca/~zabrocki/math1590w06/geometrycalc/itsthegeomstupid.pdf)
- Feb 6: Editorial: Calculus counts in new economy (http://garsia.math.yorku.ca/~zabrocki/math1590w06/geometrycalc/cuthighschoolcalc.pdf)
- An additional letter explaining the rationale behind the Calculus cut (http://garsia.math.yorku.ca/~zabrocki/math1590w06/geometrycalc/ProCalcremovalMCB4U.pdf)
- Next Danielle brought in a 'show and tell' consiting of a tape of a Texas Hold 'Em game of poker. We watched it for about 10-15 minutes and then I asked the question 'Where is the math?' We listed at least four or five places we could spot math being used (I don't remember them precisely).
- with given cards showing it is possible to predict the probabilities of various hands for each of the players
- at certain points in the game there is cost associated with staying in and seeing the others cards, that cost needs to be weighed against the likelyhood of winning
- betting strategies were studied by John Nash in the ideas that contributed to game theory earned him a Nobel prize in economics. I have something to share on this idea but we will have to save it for another night.
- The order of each of the types of hands (pair, 2 pair, 3 of a kind, etc.) is determined by how many hands there are of each type. Then we counted that there were:
|type of poker hand||total hands of that type|
|four of a kind||624|
|3 of a kind||54,912|
- To determine the number of each of these hands this we came up with a list of things which determined the hand completely (e.g. for a 3 of a kind we need to know (which of the 13 types of cards appears 3 times, which of the 3 of the 4 suits appears, two different of the remaining 12 types of cards and a suit for each of these) we conclude from that description that there are 13*4*C(12,2)*4*4 = 832*C(12,2).
- Now in many of the descriptions that we had we determined the number of hands of each type in terms of this unknown quantity C(n,k) = the number of ways of choosing k elements from a set of size n.
- I argue that n! = the number of ways of ordering n things = the number of ways of picking k things from the n elements to go first * the number of ways of ordering those k things * the number of ways of ordering the remaining (n-k) things = C(n,k)*k!*(n-k)!. It follows that n! = C(n,k)*k!*(n-k)! and this implies that C(n,k) = n!/(k!*(n-k)!). In particular, C(12, 2) = 12!/(2!*10!) = 66 and hence the number of 3 of a kind hands is 832*66 = 54,912.
- I assigned 3 problems for homework due in two weeks. Faulty bricks, Jacobean Locks, and one more problem. The last problem is I want to ask what is the maximum number of regions n planes divide space into. If there is one plane it divides space into two regions, if there are two planes and these planes cross they divide space into 4 regions, if there are 3 planes we can divide space into 8 regions. How does this continue? To visualize this you may want to draw lines on a cube.
Previous class : January 31, 2006
Next class : February 21, 2006
Main class page : Mathematics 1590 Nature of Mathematics II