Physical Sciences and Mathematics Commons^™

Big Two And N-Card Poker Probabilities, Brian Wu, Chai Wah Wu

Communications on Number Theory and Combinatorial Theory

Between the poker hands of straight, flush, and full house, which hand is more common? In standard 5-card poker, the order from most common to least common is straight, flush, full house. The same order is true for 7-card poker such as Texas hold'em. However, is the same true for n-card poker for larger n? We study the probability of obtaining these various hands for n-card poker for various values of n≥5. In particular, we derive closed expressions for the probabilities of flush, straight and full house and show that the probability of a flush is less than a straight …

Divisibility Probabilities For Products Of Randomly Chosen Integers, Noah Y. Fine

Rose-Hulman Undergraduate Mathematics Journal

We find a formula for the probability that the product of n positive integers, chosen at random, is divisible by some integer d. We do this via an inductive application of the Chinese Remainder Theorem, generating functions, and several other combinatorial arguments. Additionally, we apply this formula to find a unique, but slow, probabilistic primality test.

(R1971) Analysis Of Feedback Queueing Model With Differentiated Vacations Under Classical Retrial Policy, Poonam Gupta, Naveen Kumar, Rajni Gupta

Applications and Applied Mathematics: An International Journal (AAM)

This paper analyzes an M/M/1 retrial queue under differentiated vacations and Bernoulli feedback policy. On receiving the service, if the customer is not satisfied, then he may join the retrial group again with some probability and demand for service or may leave the system with the complementary probability. Using the probability generating functions technique, the steady-state solutions of the system are obtained. Furthermore, we have obtained some of the important performance measures such as expected orbit length, expected length of the system, sojourn times and probability of server being in different states. Using MATLAB software, we have represented the graphical …

Application Of Probabilistic Ranking Systems On Women’S Junior Division Beach Volleyball, Cameron Stewart, Michael Mazel, Bivin Sadler

SMU Data Science Review

Women’s beach volleyball is one of the fastest growing collegiate sports today. The increase in popularity has come with an increase in valuable scholarship opportunities across the country. With thousands of athletes to sort through, college scouts depend on websites that aggregate tournament results and rank players nationally. This project partnered with the company Volleyball Life, who is the current market leader in the ranking space of junior beach volleyball players. Utilizing the tournament information provided by Volleyball Life, this study explored replacements to the current ranking systems, which are designed to aggregate player points from recent tournament placements. Three …

Irreducibility And Galois Groups Of Random Polynomials, Hanson Hao, Eli Navarro, Henri Stern

Rose-Hulman Undergraduate Mathematics Journal

In 2015, I. Rivin introduced an effective method to bound the number of irreducible integral polynomials with fixed degree d and height at most N. In this paper, we give a brief summary of this result and discuss the precision of Rivin's arguments for special classes of polynomials. We also give elementary proofs of classic results on Galois groups of cubic trinomials.

Applying The Data: Predictive Analytics In Sport, Anthony Teeter, Margo Bergman

Access*: Interdisciplinary Journal of Student Research and Scholarship

The history of wagering predictions and their impact on wide reaching disciplines such as statistics and economics dates to at least the 1700’s, if not before. Predicting the outcomes of sports is a multibillion-dollar business that capitalizes on these tools but is in constant development with the addition of big data analytics methods. Sportsline.com, a popular website for fantasy sports leagues, provides odds predictions in multiple sports, produces proprietary computer models of both winning and losing teams, and provides specific point estimates. To test likely candidates for inclusion in these prediction algorithms, the authors developed a computer model, and test …

Covid-19 And Quantitative Literacy: Focusing On Probability, Michael A. Lewis

Numeracy

The COVID-19 pandemic is arguably the worst crisis the world has faced, so far, in this new century. We haven’t seen a pandemic like this since the 1918 Flu at the beginning of the last century, and, as of this writing, there appears to be no end in sight. What those of us who’re focused on quantitative methods have noticed, in addition to the many people dying, becoming ill, and losing their livelihoods, is the importance of quantitative literacy to an understanding of what’s going on. That’s what this article is about. Specifically, it’s about how the COVID-19 pandemic is …

Pair-A-Dice Lost: Experiments In Dice Control, Robert H. Scott Iii, Donald R. Smith

UNLV Gaming Research & Review Journal

This paper presents our findings from experiments designed to test whether we could use a custom-made dice throwing machine applying common dice control methods to produce dice rolls that differ from random. In earlier research we calculated the percentages of control a craps player needs to break even or beat the house (Smith and Scott, 2018). Using the most common practices of dice control in craps, we established how dice should be configured (i.e., set) and thrown to achieve certain outcomes such as not rolling a seven in the point cycle. We decided to run experiments to see if a …

Probabilities Involving Standard Trirectangular Tetrahedral Dice Rolls, Rulon Olmstead, Doneliezer Baize

Rose-Hulman Undergraduate Mathematics Journal

The goal is to be able to calculate probabilities involving irregular shaped dice rolls. Here it is attempted to model the probabilities of rolling standard tri-rectangular tetrahedral dice on a hard surface, such as a table top. The vertices and edges of a tetrahedron were projected onto the surface of a sphere centered at the center of mass of the tetrahedron. By calculating the surface areas bounded by the resultant geodesics, baseline probabilities were achieved. Using a 3D printer, dice were constructed of uniform density and the results of rolling them were recorded. After calculating the corresponding confidence intervals, the …

Calculus Of The Impossible: Review Of The Improbability Principle (2014) By David Hand And The Logic Of Miracles (2018) By Lásló Mérő, Samuel L. Tunstall

Numeracy

David J. Hand. 2014. The Improbability Principle: Why Coincidences, Miracles, and Rare Events Happen Every Day (New York, NY: Scientific American/Farrar, Straus and Giroux) 288 pp. ISBN: 978-0374175344.

Lásló Mérő. 2018. The Logic of Miracles: Making Sense of Rare, Really Rare, and Impossibly Rare Events (New Haven, CT: Yale University Press) 288 pp. ISBN: 978-0300224153.

David Hand and Lásló Mérő both grapple with the occurrence of seemingly impossible events in these two popular science books. In this comparative review, I describe the two books, and explain why I prefer Hand's treatment of the impossible.

Golden Arm: A Probabilistic Study Of Dice Control In Craps, Donald R. Smith, Robert Scott Iii

UNLV Gaming Research & Review Journal

This paper calculates how much control a craps shooter must possess on dice outcomes to eliminate the house advantage. A golden arm is someone who has dice control (or a rhythm roller or dice influencer). There are various strategies for dice control in craps. We discuss several possibilities of dice control that would result in several different mathematical models of control. We do not assert whether dice control is possible or not (there is a lack of published evidence). However, after studying casino-legal methods described by dice-control advocates, we can see only one realistic mathematical model that describes the resulting …

Ladies' Night, Robert Dawson

Journal of Humanistic Mathematics

"Lady" Jane is an expert at her racket. The Joint Statistical Meetings are in Vegas, and she reckons it's payday. But she's taking on the professionals.

The Battle Against Malaria: A Teachable Moment, Randy K. Schwartz

Journal of Humanistic Mathematics

Malaria has been humanity’s worst public health problem throughout recorded history. Mathematical methods are needed to understand which factors are relevant to the disease and to develop counter-measures against it. This article and the accompanying exercises provide examples of those methods for use in lower- or upper-level courses dealing with probability, statistics, or population modeling. These can be used to illustrate such concepts as correlation, causation, conditional probability, and independence. The article explains how the apparent link between sickle cell trait and resistance to malaria was first verified in Uganda using the chi-squared probability distribution. It goes on to explain …

Probabilistic Earthquake Hazard Assessment For Ankara And Its Environs, Bülent Özmen, Berna Burçak Başbuğ Erkan

Turkish Journal of Earth Sciences

The capital and the second largest city of Turkey, Ankara, is generally considered to be safe in terms of seismic activities and earthquake hazard. However, recent studies and earthquakes experienced in the region showed that Ankara is not indeed seismically safe. As the number of studies on Ankara’s seismic hazard increases, the number of scientists who claim that the earthquake hazard in Ankara is higher than expected also increases. However, to date no detailed analysis has been undertaken as to the earthquake hazard facing Ankara. This study has compiled data from the earthquake catalogues available in Turkey and employed the …

Confidence Interval, Ursula Whitcher

Journal of Humanistic Mathematics

A poem about estimating probabilities.

Mathematics And The Hunger Games, Michael A. Lewis

Journal of Humanistic Mathematics

The Hunger Games plot features a dystopian future in which twelve outer districts are oppressed by a centralized capital. The story focuses on the heroism of a sixteen-year-old girl named Katniss and how she tries to rise above the oppression that she experiences. It also features a special lottery and other twists that are sources of mathematical interest. This essay focuses on some of the mathematical issues raised by The Hunger Games in an effort to show that this story can be used to teach students (as well as other interested parties) some important concepts from mathematics.

On Lattice Structure Of The Probability Functions On L*, Mashaallah Mashinchi, Ghader Khaledi

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, the set of all probability functions on L* is studied, where L* is the lattice of bothvalued fuzzy sets or intuitionistic fuzzy sets. It is shown that the set of all probability functions on L* endowed with two appropriate operations has a monoid structure which is also a distributive complete lattice. Also the lattice structure of the set of all probability functions on L* induced by an appropriate function on [0, 1] to itself is studied. Some lattice (dual) isomorphisms are discussed that suggests probabilities on L* could be considered in the framework of theories modeling imprecision.

Book Review: What’S Luck Got To Do With It? The History, Mathematics, And Psychology Of The Gambler’S Illusion By Joseph Mazur, Michael Lugo

Journal of Humanistic Mathematics

This review of Joseph Mazur's book on the history of gambling, for a general audience, is in three parts, paralleling the structure of the book. The first part briefly outlines Mazur's coverage of the history of probability from prehistory to the present day, with a focus on gambling. The second part examines the relationship between the mathematics of gambling and probability theory, and summarizes classical problems in probability arising from gambling such as Galileo's dice and the Pascal-Fermat problem of points. The third part, on psychology, discusses the gambler's illusion and psychological motivations for gambling.

Constructive Criticism, Ronald C. Serlin

Journal of Modern Applied Statistical Methods

Attempts to attain knowledge as certified true belief have failed to circumvent Hume’s injunction against induction. Theories must be viewed as unprovable, improbable, and undisprovable. The empirical basis is fallible, and yet the method of conjectures and refutations is untouched by Hume’s insights. The implications for statistical methodology is that the requisite severity of testing is achieved through the use of robust procedures, whose assumptions have not been shown to be substantially violated, to test predesignated range null hypotheses. Nonparametric range null hypothesis tests need to be developed to examine whether or not effect sizes or measures of association, as …

The Moral Significance Of Indetectable Effects, Sven Ove Hansson

RISK: Health, Safety & Environment (1990-2002)

A reassessment of Parfit's fifth "mistake in moral mathematics."

Book Review Of Deborah G. Mayo, Error And The Growth Of Experimental Knowledge (University Of Chicago Press 1996), Kristin S. Shrader-Frechette

RISK: Health, Safety & Environment (1990-2002)

Review of the book: Deborah G. Mayo, Error and the Growth of Experimental Knowledge (University of Chicago Press 1996). Figures, index, preface, references. ISBN 0-226-51197-9 [493 pp. $74.00 Cloth; $29.95 Paper. 5801 S. Ellis Ave., Chicago, IL 60637.]

Reply To Valverde, Paul B. Thompson

RISK: Health, Safety & Environment (1990-2002)

Professor Thompson responds to Valverde's argument, in the last issue, that his approach to Risk puts too much emphasis on the distinction between Risk subjectivism and Risk objectivism. In doing so, he asserts, inter alia, that anchoring Risk judgments in a probabilistic framework does not go far enough in rejecting reigning Risk-analysis notions of "real Risk."

The Cognitive Status Of Risk: A Response To Thompson, L. James Valverde

RISK: Health, Safety & Environment (1990-2002)

Discussing the role that probability theory should play in Risk analysis and management, Dr. Valverde argues that Thompson's approach puts too much emphasis on the distinction between Risk subjectivism and Risk objectivism in addressing the question, "When are Risks real?"

Predictive Probabilities In Employee Drug-Testing, John M. Gleason, Darold T. Barnum

RISK: Health, Safety & Environment (1990-2002)

Substance abuse in the U.S. has been estimated to cost $99 billion annually through lower productivity. Yet the authors urge caution in attempting to reduce these costs and health and safety Risks. In doing so, they cite commonly high frequencies of false negatives and false positives in employee drug tests - the latter having the potential to do great injustice to many drug-free employees.

Risk Objectivism And Risk Subjectivism: When Are Risks Real, Paul B. Thompson

RISK: Health, Safety & Environment (1990-2002)

Typically, those who discuss Risk management envision a two-step process wherein, first, Risk is more or less objectively appraised and, second, the acceptability of those Risks is subjectively evaluated. This paper questions the philosophical foundations of that approach.