Investigations into the Realm of Math Psych: Mental Capacity and the Mind Game of Poker

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Investigations into the Realm of Math Psych
Mental Capacity and the Mind Game of Poker

• By Laura Williams

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• Outline
• Section 1 Intro
• Section 2 Probability of a Full House
• Section 3 Maximum Intellectual Capacity For
  Remembering
• Music
• Section 4 Conclusion
• Key words probability, combinations,
  feasibility, binary
• numbers, channel capacity

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• Section 1 Introduction
• Applications of math psych probability in
  poker, experiments in the limit of human judgment
• Problem A Mathematically model how to determine
  the likelihood poker players might obtain a Full
  House
• Problem B Calculate the channel capacity, the
  maximum amount of information a person can absorb
  and recall without error (Miller 136), for
  musical tones.
• Purpose Use probability and graph orientation
  to link math psych to discrete mathematics.
• Focus How does one calculate the probability of
  a Full House, how to find the maximum channel
  capacity for a person in terms of musical memory.

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• Definitions (To be discussed aloud in detail-for
• clarification)
• -Probability (Class Text)
• -Combination (Class Text)
• -Feasibility (My own Definition based on class
  notes)
• -Binary Numbers (http//en.wikipedia.org/wiki/Bina
  ry_numeral_system)
• -Channel Capacity (Previous slide)

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• Section 2 Probability of a Full House
• Problem A The goal of Poker is to come up with
  the best possible hand, and knowing the rules of
  probability helps (along with strategies for
  outwitting opponents). However, a layperson
  without any prior knowledge of game may have
  difficulty deciding which hand to try for.
• -Suppose a player is dealt a hand of five cards
  in favor of a Full House.
• Q How can we determine his odds for successfully
  obtaining that particular type of hand by the end
  of the game?
• A Analyze the problem in terms of probability.
  The likelihood of a Full House depends on the
  number of outcomes in favor of exactly one
  3-of-a-Kind and one Pair, as well as the total
  number of outcomes in the game.

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• Model of Problem A
• 1) Pick 4C3 combinations of a particular card
  type and multiply that by13 ranks. 4C3 indicates
  that out of four cards of the same rank (e.g. 7
  of clubs, diamonds, hearts, and spades), one has
  to pick 3 of those cards to obtain 3-of-a-kind.
  Also, there are 13 ranks of cards in a poker
  game, hence multiply by 13.
• 2) Take 4C2 combinations of a particular card
  type and multiply that by 12. By similar logic as
  1, 4C2 indicates that out of 4 cards of the same
  rank, one must pick 2 of those cards to obtain a
  pair. Likewise, since the 3-of-a-kind accounts
  for one rank, the pair must come from one of the
  other 12, so multiply 4C2 by that number.

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• Model (Contd)
• The first two steps comprise the favorable
  outcome for a Full House. Now we seek to find the
  total number of outcomes in a poker game. There
  are 52 cards in a standard deck, and every poker
  hand consists of 5 cards, so the total outcome
  space is 52C5.
• Divide the number of favorable outcomes by the
  total outcome space to calculate the probability
  of a Full House. The result is P(Full House)
  P(Three-of-kind and pair) (134C3)(124C2)
  / 52C5 .0014405762.
• Diagram is from (http//www.indepthinfo.com/
• probability-poker/full-house.shtml)

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• Mathematical Application and difficulty
• -The probability of a Full House may not seem
  challenging if one understands logic and has a
  keen intuition, but for a layperson randomly
  playing, the solving process becomes tedious.
  Finding the odds of both a 3-of-a-kind and a pair
  simultaneously involves the concept of
  combinations discussed in the first week of
  class we must take n4 cards of the same rank
  and group them by r3 for a 3-of-kind and r2 for
  a pair. But we are not done there. Take n52
  cards in a deck and group them by n5 for the
  total possible arrangements. And dont forget to
  multiply by 13 ranks and then 12 remaining ranks,
  respectively. And divide as well! This particular
  approach of math psych relates to discrete math
  because it uses combinations, and it poses a
  challenge due to the overwhelming number of
  possible card combinations, and just because of
  the complexity of Poker itself!!!

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• Section 3 Maximum Intellectual Capacity For
  Remembering Music
• Problem B According to a review of the limits of
  human capacity, an individual can only absorb a
  certain of what he or she perceives.
• -The author discusses the concept of absolute
  judgment in a 1-D space, referring to the
  observer as a communication channel which
  relays info based on the stimuli presented to him
  (Miller 136).
• Q A layperson might ask Well, how would I go
  about finding this maximum human capacity that
  the author alludes to?
• A Maintain a record of his mental and physical
  responses and note how often he correctly
  identifies info w/out error. Over time, notice
  how data interpretation changes and use the
  average to calculate a value for maximum channel
  capacity. This numerical data comes from comes
  the idea that for every 2 bits of information
  presented to the test subject, the difficulty of
  accurately recognizing an item increases
  exponentially.

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• Model of Problem B
• Record the average number of musical notes a
  person can recollect given a set time period.
• Set up a rectangular array of 0s and 1s (e.g. a
  matrix), and allow each digit to represent one
  bit of computer data, where bits are relayed as
  increments of 2n.
• 3) Find the maximum value for the absorption
  rate, which corresponds to the upper bound that
  makes finding all orientations for a set time
  period feasible.
• 4) For every 2 bits of information presented, the
  difficulty of accurately recognizing an item
  increases exponentially.
• 5) Based on the authors conclusion, while an
  exceptional genius may recall 50 tones at a
  time, a normal person only remembers and
  identifies six songs, on average. Therefore, the
  maximum channel capacity for an average person is
  around 6 songs in a set time period.

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• Here are two graphs (Miller 137-140) which
  illustrate limits on human capacity

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• Here are two more graphs representing channel
  capacity
• (http//webscripts.softpedia.com/screenshots/MIMO-
  Rayleigh-fading-Channel--Capacity-17585.png)
• (http//www.yobology.info/harbin2005/part1/img2.gi
  f)

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• Mathematical Application and Difficulty
• -Graph orientation and feasibility, covered in
  lecture, help when modeling the problem of
  maximizing musical memory. I presented two graphs
  depicting pitches and auditory loudness (from
  Miller) as portals for showing the upper bound on
  the max. channel capacity. Both graphs indicate
  that the average ratio of input info to
  transmitted info is between 2 and 3bits, which
  means that the average person can handle about
  2.5 bits of info at once. From what Miller
  concluded, the max. channel capacity in terms of
  musical tones is 6 songs.
• -This model does not use a whole lot of formulas,
  but rather the author theoretically determines
  the max. absorption rate for music from
  experimental results of others and knowledge of
  binary numbers. From both graphs, 6 is an u.b.,
  and because this is a small of songs, it is
  feasible to conclude that the max. channel
  capacity for recalling musical tones w/out error
  is about 6.

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• However, if a layperson wanted to solve the
  problem on his own, he could do so by utilizing
  calculus, matrix theory, and graph theory.
• -Calculus -find max. absorption rate by setting
  f(memory) 0 and solving for X, where X total
  of musical tones correctly identified.
• -Computer science and graph orientation- derive a
  function and a matrix for a realistic estimate of
  the max. rate. The max. value obtained by finding
  the 1st derivative and setting it to 0 yields the
  average channel capacity for a human being. By
  the tool of mathematics, it is possible to
  calculate, on average, how much an individual can
  reasonably take in w/out making errors when
  responding.
• -Binary encoding and decoding- determine bits if
  a graph is not available.

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• Section 4 Conclusion
• In conclusion, the layperson should understand
• -That math psych is one tool for applying
  discrete math
• -Poker is mentally challenging, not just from a
  psychological standpoint, but also in terms of
  math.
• -Create a discrete math model by calculating the
  probability of a Full House and explaining the
  procedure step-by-step.
• -How to apply graph theory to a real-world
  problem.
• -The max. rate for learning music can be obtained
  from analyzing graphs.
• -By building a model using knowledge of adjacency
  matrices and computer science, one can find the
  upper bound of such a rate and determine whether
  or not it is feasible to increase bits of info a
  test subject must retain.
• -In general, discrete topics such as probability
  and graph orientation can be used in the
  formulation of math models.

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• References
• Miller, George A. The Magical Number Seven, Plus
  or Minus Two Some Limits on Our Capacity for
  Processing Information. Readings in
  Mathematical Psychology. John Wiley and Sons,
  Inc. New York, 1963, pg. 135-140.
• Roberts, Fred. Applied Combinatorics, 2nd ed.
  Prentice Hall.
• http//www.indepthinfo.com/probability-poker/full-
  house.shtml
• http//webscripts.softpedia.com/screenshots/MIMO-R
  ayleigh-fading-Channel--Capacity-17585.png
• http//en.wikipedia.org/wiki/Binary_numeral_system
  
• http//www.yobology.info/harbin2005/part1/img2.gif