Physical Sciences and Mathematics Commons^™

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 …

A New Approximation Scheme For Monte Carlo Applications, Bo Jones

CMC Senior Theses

Approximation algorithms employing Monte Carlo methods, across application domains, often require as a subroutine the estimation of the mean of a random variable with support on [0,1]. One wishes to estimate this mean to within a user-specified error, using as few samples from the simulated distribution as possible. In the case that the mean being estimated is small, one is then interested in controlling the relative error of the estimate. We introduce a new (epsilon, delta) relative error approximation scheme for [0,1] random variables and provide a comparison of this algorithm's performance to that of an existing approximation scheme, both …

Cycle Lengths Of Θ-Biased Random Permutations, Tongjia Shi

HMC Senior Theses

Consider a probability distribution on the permutations of n elements. If the probability of each permutation is proportional to θ^K, where K is the number of cycles in the permutation, then we say that the distribution generates a θ-biased random permutation. A random permutation is a special θ-biased random permutation with θ = 1. The m^th moment of the r^th longest cycle of a random permutation is Θ(n^m), regardless of r and θ. The joint moments are derived, and it is shown that the longest cycles of a permutation can either be positively or …

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.

Constructing Phylogenetic Trees Using Maximum Likelihood, Anna Cho

Scripps Senior Theses

Maximum likelihood methods are used to estimate the phylogenetic trees for a set of species. The probabilities of DNA base substitutions are modeled by continuous-time Markov chains. We use these probabilities to estimate which DNA bases would produce the data that we observe. The topology of the tree is also determined using base substitution probabilities and conditional likelihoods. Felsenstein [2] introduced this method of finding an estimate for the maximum likelihood phylogenetic tree. We will explore this method in detail in this paper.

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.

Recounting The Odds Of An Even Derangement, Arthur T. Benjamin, Curtis D. Bennet, Florence Newberger

All HMC Faculty Publications and Research

No abstract provided in this article.

What's Best?, Arthur T. Benjamin, Matthew T. Fluet '99

All HMC Faculty Publications and Research

No abstract provided in this article.

A Rational Solution To Cootie, Arthur T. Benjamin, Matthew T. Fluet '99

All HMC Faculty Publications and Research

No abstract provided in this article.

Optimization In Chemical Kinetics, Arthur T. Benjamin, Gordon J. Hogenson '92

All HMC Faculty Publications and Research

No abstract provided in this article.

The Bisection Method: Which Root?, Arthur T. Benjamin

All HMC Faculty Publications and Research

No abstract provided in this article.

Reliable Computation In The Presence Of Noise, Nicholas Pippenger

All HMC Faculty Publications and Research

This talk concerns computation by systems whose components exhibit noise (that is, errors committed at random according to certain probabilistic laws). If we aspire to construct a theory of computation in the presence of noise, we must possess at the outset a satisfactory theory of computation in the absence of noise.

A theory that has received considerable attention in this context is that of the computation of Boolean functions by networks (with perhaps the strongest competition coming from the theory of cellular automata; see [G] and [GR]). The theory of computation by networks associates with any two sets Q and …