February 7, 2006

From MathWiki

  • 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:
5 card poker hands and the number of each type
type of poker hand total hands of that type
royal flush 4
straight flush 36
four of a kind 624
full house 3,744
flush 5,108
straight 10,200
3 of a kind 54,912
two pair 123,552
one pair 1,098,240
high card 1,302,540
  • 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.

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