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