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Investigations into the Realm of Math Psych: Mental Capacity and the Mind Game of Poker By Laura Williams 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 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. 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/Binary_numeral_system) -Channel Capacity (Previous slide) 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. 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. Model (Cont’d) 3) 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. 4) 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) = [(13*4C3)*(12*4C2)] / 52C5 = .0014405762. Diagram is from: (http://www.indepthinfo.com/ probability-poker/full-house.shtml) 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 n=4 cards of the same rank and group them by r=3 for a 3-of-kind and r=2 for a pair. But we are not done there. Take n=52 cards in a deck and group them by n=5 for the total possible arrangements. And don’t 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!!! 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. Model of Problem B 1) Record the average number of musical notes a person can recollect given a set time period. 2) Set up a rectangular array of 0’s and 1’s (e.g. a matrix), and allow each digit to represent one bit of computer data, where bits are relayed as increments of 2^n. 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 author’s 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. Here are two graphs (Miller 137-140) which illustrate limits on human capacity: Here are two more graphs representing channel capacity: (http://webscripts.softpedia.com/screenshots/MIMO-Rayleighfading-Channel--Capacity-17585.png) (http://www.yobology.info/harbin2005/part1/img2.gif) 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. 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. 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. 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-Rayleighfading-Channel--Capacity-17585.png http://en.wikipedia.org/wiki/Binary_numeral_system http://www.yobology.info/harbin2005/part1/img2.gif