The Probability Lifesaver
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URSA Scholars Week
Undergraduate Research and Scholarly Achievement Baylor University presents URSA Scholars Week March 22 - March 26, 2010 Dear Colleagues: Welcome to Baylor’s Third Annual URSA Scholars’ Week! One of Baylor’s documented core strengths is a vital, transformative undergraduate classroom experience. Undergraduate research is a wonderful extension of that experience, providing students the opportunity to explore real world research / scholarship in partnership with their teachers. The presentations before you are the fruit of another core Baylor value – relationship. They represent a rare glimpse into the investment routinely made by Baylor faculty in their students, centered on individual mentoring and the sharing of research skills that can inform a lifetime of academic endeavor. I am grateful for your presence at URSA Scholars’ Day 2010. What has been said in previous years continues to be so: the best is yet to come. Sincerely, Truell W. Hyde, Ph.D. Vice Provost for Research Baylor Sciences Building Atrium March 22 and 25, 2010 • 12:00 – 2:00 Board 1 Sarah Garza, Senior, Environmental Science Jessi Carrothers, Senior, University Scholar/Pre-Medical Austin Cook-Lindsay, Senior, Environmental Science Faculty Mentor: Susan Bratton Title: Participation in Undergraduate Research a Key to Successful Applications for Fellowships, Research Assistantships and Graduate School Admissions Undergraduate research participation is a key component of medical school, graduate school and job applications. Senior authorship of publications is especially valuable for obtaining competitive graduate admissions, fellowships, and research assistantships. Research participation demonstrates competence in problem solving, ability to follow protocols and in professional writing and presentation. Faculty research mentors provide superior letters of reference as they are able to verify applicant characteristics, such as imagination, dependability, work ethic and academic integrity. -
How Superman Developed Into a Jesus Figure
HOW SUPERMAN DEVELOPED INTO A JESUS FIGURE CRISIS ON INFINITE TEXTS: HOW SUPERMAN DEVELOPED INTO A JESUS FIGURE By ROBERT REVINGTON, B.A., M.A. A Thesis Submitted to the School of Graduate Studies in Partial Fulfillment of the Requirements for the Degree of Master of Arts McMaster University © Copyright by Robert Revington, September 2018 MA Thesis—Robert Revington; McMaster University, Religious Studies McMaster University MASTER OF ARTS (2018) Hamilton, Ontario, Religious Studies TITLE: Crisis on Infinite Texts: How Superman Developed into a Jesus Figure AUTHOR: Robert Revington, B.A., M.A (McMaster University) SUPERVISOR: Professor Travis Kroeker NUMBER OF PAGES: vi, 143 ii MA Thesis—Robert Revington; McMaster University, Religious Studies LAY ABSTRACT This thesis examines the historical trajectory of how the comic book character of Superman came to be identified as a Christ figure in popular consciousness. It argues that this connection was not integral to the character as he was originally created, but was imposed by later writers over time and mainly for cinematic adaptations. This thesis also tracks the history of how Christians and churches viewed Superman, as the film studios began to exploit marketing opportunities by comparing Superman and Jesus. This thesis uses the methodological framework of intertextuality to ground its treatment of the sources, but does not follow all of the assumptions of intertextual theorists. iii MA Thesis—Robert Revington; McMaster University, Religious Studies ABSTRACT This thesis examines the historical trajectory of how the comic book character of Superman came to be identified as a Christ figure in popular consciousness. Superman was created in 1938, but the character developed significantly from his earliest incarnations. -
The Birthday Problem (2.7)
Combinatorics (2.6) The Birthday Problem (2.7) Prof. Tesler Math 186 Winter 2020 Prof. Tesler Combinatorics & Birthday Problem Math 186 / Winter 2020 1 / 29 Multiplication rule Combinatorics is a branch of Mathematics that deals with systematic methods of counting things. Example How many outcomes (x, y, z) are possible, where x = roll of a 6-sided die; y = value of a coin flip; z = card drawn from a 52 card deck? (6 choices of x) × (2 choices of y) × (52 choices of z) = 624 Multiplication rule The number of sequences (x1, x2,..., xk) where there are n1 choices of x1, n2 choices of x2,..., nk choices of xk is n1 · n2 ··· nk. This assumes the number of choices of xi is a constant ni that doesn’t depend on the other choices. Prof. Tesler Combinatorics & Birthday Problem Math 186 / Winter 2020 2 / 29 Addition rule Months and days How many pairs (m, d) are there where m = month 1,..., 12; d = day of the month? Assume it’s not a leap year. 12 choices of m, but the number of choices of d depends on m (and if it’s a leap year), so the total is not “12 × ” Split dates into Am = f (m, d): d is a valid day in month m g: A = A1 [···[ A12 = whole year jAj = jA1j + ··· + jA12j = 31 + 28 + ··· + 31 = 365 Addition rule If A ,..., A are mutually exclusive, then 1 n n n [ Ai = jAij i=1 i=1 X Prof. Tesler Combinatorics & Birthday Problem Math 186 / Winter 2020 3 / 29 Permutations of distinct objects Here are all the permutations of A, B, C: ABC ACB BAC BCA CAB CBA There are 3 items: A, B, C. -
A Note on the Exponentiation Approximation of the Birthday Paradox
A note on the exponentiation approximation of the birthday paradox Kaiji Motegi∗ and Sejun Wooy Kobe University Kobe University July 3, 2021 Abstract In this note, we shed new light on the exponentiation approximation of the probability that all K individuals have distinct birthdays across N calendar days. The exponen- tiation approximation imposes a pairwise independence assumption, which does not hold in general. We sidestep it by deriving the conditional probability for each pair of individuals to have distinct birthdays given that previous pairs do. An interesting im- plication is that the conditional probability decreases in a step-function form |not in a strictly monotonical form| as the more pairs are restricted to have distinct birthdays. The source of the step-function structure is identified and illustrated. The equivalence between the proposed pairwise approach and the existing permutation approach is also established. MSC 2010 Classification Codes: 97K50, 60-01, 65C50. Keywords: birthday problem, pairwise approach, permutation approach, probability. ∗Corresponding author. Graduate School of Economics, Kobe University. 2-1 Rokkodai-cho, Nada, Kobe, Hyogo 657-8501 Japan. E-mail: [email protected] yDepartment of Economics, Kobe University. 2-1 Rokkodai-cho, Nada, Kobe, Hyogo 657-8501 Japan. E-mail: [email protected] 1 1 Introduction Suppose that each of K individuals has his/her birthday on one of N calendar days consid- ered. Let K¯ = f1;:::;Kg be the set of individuals, and let N¯ = f1;:::;Ng be the set of ¯ calendar days, where 2 ≤ K ≤ N. Let Ak;n be an event that individual k 2 K has his/her birthday on day n 2 N¯ . -
Probability Theory
Probability Theory Course Notes — Harvard University — 2011 C. McMullen March 29, 2021 Contents I TheSampleSpace ........................ 2 II Elements of Combinatorial Analysis . 5 III RandomWalks .......................... 15 IV CombinationsofEvents . 24 V ConditionalProbability . 29 VI The Binomial and Poisson Distributions . 37 VII NormalApproximation. 44 VIII Unlimited Sequences of Bernoulli Trials . 55 IX Random Variables and Expectation . 60 X LawofLargeNumbers...................... 68 XI Integral–Valued Variables. Generating Functions . 70 XIV RandomWalkandRuinProblems . 70 I The Exponential and the Uniform Density . 75 II Special Densities. Randomization . 94 These course notes accompany Feller, An Introduction to Probability Theory and Its Applications, Wiley, 1950. I The Sample Space Some sources and uses of randomness, and philosophical conundrums. 1. Flipped coin. 2. The interrupted game of chance (Fermat). 3. The last roll of the game in backgammon (splitting the stakes at Monte Carlo). 4. Large numbers: elections, gases, lottery. 5. True randomness? Quantum theory. 6. Randomness as a model (in reality only one thing happens). Paradox: what if a coin keeps coming up heads? 7. Statistics: testing a drug. When is an event good evidence rather than a random artifact? 8. Significance: among 1000 coins, if one comes up heads 10 times in a row, is it likely to be a 2-headed coin? Applications to economics, investment and hiring. 9. Randomness as a tool: graph theory; scheduling; internet routing. We begin with some previews. Coin flips. What are the chances of 10 heads in a row? The probability is 1/1024, less than 0.1%. Implicit assumptions: no biases and independence. 10 What are the chance of heads 5 out of ten times? ( 5 = 252, so 252/1024 = 25%). -
A Non-Uniform Birthday Problem with Applications to Discrete Logarithms
A NON-UNIFORM BIRTHDAY PROBLEM WITH APPLICATIONS TO DISCRETE LOGARITHMS STEVEN D. GALBRAITH AND MARK HOLMES Abstract. We consider a generalisation of the birthday problem that arises in the analysis of algorithms for certain variants of the discrete logarithm problem in groups. More precisely, we consider sampling coloured balls and placing them in urns, such that the distribution of assigning balls to urns depends on the colour of the ball. We determine the expected number of trials until two balls of different colours are placed in the same urn. As an aside we present an amusing “paradox” about birthdays. Keywords: birthday paradox, discrete logarithm problem (DLP), probabilistic analysis of randomised algorithms 1. Introduction In the classical birthday problem one samples uniformly with replacement from a set of size N until the same value is sampled twice. It is known that the expected time at which this match first occurs grows as πN/2. The word “birthday” arises from a common application of this result: the expected value of the minimum number of people in a room before two of them have the same birthday is approximately 23.94 (assuming birthdays are uniformly distributed over the year). The birthday problem can be generalised in a number of ways. For example, the assumption that births are uniformly distributed over the days in the year is often false. Hence, researchers have studied the expected time until a match occurs for general distributions. One can also generalise the problem to multi-collisions (e.g., 3 people having the same birthday) or coincidences among individuals of different “types” (e.g., in a room with equal numbers of boys and girls, when can one expect a boy and girl to share the same birthday). -
The Matching, Birthday and the Strong Birthday Problem
Journal of Statistical Planning and Inference 130 (2005) 377–389 www.elsevier.com/locate/jspi The matching, birthday and the strong birthday problem: a contemporary review Anirban DasGupta Department of Statistics, Purdue University, 1399 Mathematical Science Building, West Lafayette, IN 47907, USA Received 7 October 2003; received in revised form 1 November 2003; accepted 10 November 2003 Dedicated to Herman Chernoff with appreciation and affection on his 80th birthday Available online 29 July 2004 Abstract This article provides a contemporary exposition at a moderately quantitative level of the distribution theory associated with the matching and the birthday problems. A large number of examples, many not well known, are provided to help a reader have a feeling for these questions at an intuitive level. © 2004 Elsevier B.V. All rights reserved. Keywords: Birthday problem; Coincidences; Matching problem; Poisson; Random permutation; Strong birthday problem 1. Introduction My first exposure to Professor Chernoff’s work was in an asymptotic theory class at the ISI. Later I had the opportunity to read and teach a spectrum of his work on design of experiments, goodness of fit, multivariate analysis and variance inequalities. My own modest work on look-alikes in Section 2.8 here was largely influenced by the now famous Chernoff faces. It is a distinct pleasure to write this article for the special issue in his honor. This article provides an exposition of some of the major questions related to the matching and the birthday problems. The article is partially historical, and partially forward looking. For example, we address a new problem that we call the strong birthday problem. -
Approximations to the Birthday Problem with Unequal Occurrence Probabilities and Their Application to the Surname Problem in Japan*
Ann. Inst. Statist. Math. Vol. 44, No. 3, 479-499 (1992) APPROXIMATIONS TO THE BIRTHDAY PROBLEM WITH UNEQUAL OCCURRENCE PROBABILITIES AND THEIR APPLICATION TO THE SURNAME PROBLEM IN JAPAN* SHIGERU MASE Faculty of Integrated Arts and Sciences, Hiroshima University, Naka-ku, Hiroshima 730, Japan (Received July 4, 1990; revised September 2, 1991) Abstract. Let X1,X2,...,X~ be iid random variables with a discrete dis- tribution {Pi}~=l. We will discuss the coincidence probability Rn, i.e., the probability that there are members of {Xi} having the same value. If m = 365 and p~ ~ 1/365, this is the famous birthday problem. Also we will give two kinds of approximation to this probability. Finally we will give two applica- tions. The first is the estimation of the coincidence probability of surnames in Japan. For this purpose, we will fit a generalized zeta distribution to a fre- quency data of surnames in Japan. The second is the true birthday problem, that is, we will evaluate the birthday probability in Japan using the actual (non-uniform) distribution of birthdays in Japan. Key words and phrases: Birthday problem, coincidence probability, non-uni- formness, Bell polynomial, approximation, surname. I. Introduction It is frequently observed that, even within a small group, there are people with the same surname. It may not be considered so curious to find persons with the same surname in a group compared to those with the same birthday. But if we consider the variety of surnames and the relatively small portions of each surname in some countries, this fact becomes less trivial than is first seen. -
A FIRST COURSE in PROBABILITY This Page Intentionally Left Blank a FIRST COURSE in PROBABILITY
A FIRST COURSE IN PROBABILITY This page intentionally left blank A FIRST COURSE IN PROBABILITY Eighth Edition Sheldon Ross University of Southern California Upper Saddle River, New Jersey 07458 Library of Congress Cataloging-in-Publication Data Ross, Sheldon M. A first course in probability / Sheldon Ross. — 8th ed. p. cm. Includes bibliographical references and index. ISBN-13: 978-0-13-603313-4 ISBN-10: 0-13-603313-X 1. Probabilities—Textbooks. I. Title. QA273.R83 2010 519.2—dc22 2008033720 Editor in Chief, Mathematics and Statistics: Deirdre Lynch Senior Project Editor: Rachel S. Reeve Assistant Editor: Christina Lepre Editorial Assistant: Dana Jones Project Manager: Robert S. Merenoff Associate Managing Editor: Bayani Mendoza de Leon Senior Managing Editor: Linda Mihatov Behrens Senior Operations Supervisor: Diane Peirano Marketing Assistant: Kathleen DeChavez Creative Director: Jayne Conte Art Director/Designer: Bruce Kenselaar AV Project Manager: Thomas Benfatti Compositor: Integra Software Services Pvt. Ltd, Pondicherry, India Cover Image Credit: Getty Images, Inc. © 2010, 2006, 2002, 1998, 1994, 1988, 1984, 1976 by Pearson Education, Inc., Pearson Prentice Hall Pearson Education, Inc. Upper Saddle River, NJ 07458 All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher. Pearson Prentice Hall™ is a trademark of Pearson Education, Inc. Printed in the United States of America 10987654321 ISBN-13: 978-0-13-603313-4 ISBN-10: 0-13-603313-X Pearson Education, Ltd., London Pearson Education Australia PTY. Limited, Sydney Pearson Education Singapore, Pte. Ltd Pearson Education North Asia Ltd, Hong Kong Pearson Education Canada, Ltd., Toronto Pearson Educacion´ de Mexico, S.A. -
Superman Ii the Richard Donner Cut Mp4 Download Please Enable Your VPN When Downloading Torrents
superman ii the richard donner cut mp4 download Please enable your VPN when downloading torrents. If you torrent without a VPN, your ISP can see that you're torrenting and may throttle your connection and get fined by legal action! Loading, please wait. Synopsis. Picking up where "Superman: The Movie" left off, three criminals, General Zod (Terence Stamp), Ursa, (Sarah Douglas), and Non (Jack O'Halloran) from the planet Krypton are released from the Phantom Zone by a nuclear explosion in space. They descend upon Earth where they could finally rule. Superman, meanwhile, is in love with Lois Lane (Margot Kidder), who finds out who he really is. Lex Luthor (Gene Hackman) escapes from prison and is determined to destroy Superman by joining forces with the three criminals. —Keith Howley. Picking up where "Superman: The Movie" left off, three criminals, General Zod (Terence Stamp), Ursa, (Sarah Douglas), and Non (Jack O'Halloran) from the planet Krypton are released from the Phantom Zone by a nuclear explosion in space. They descend upon Earth where they could finally rule. Superman, meanwhile, is in love with Lois Lane (Margot Kidder), who finds out who he really is. Lex Luthor (Gene Hackman) escapes from prison and is determined to destroy Superman by joining forces with the three criminals. —Keith Howley. Uploaded By: OTTO April 04, 2019 at 04:21 AM. Director. Tech specs. Movie Reviews. Great film. Simply put, Superman II is one of the best action/comic-oriented films of all time. I'd rank it second only to the original Superman: The Movie and only X-Men is right up there. -
A (Probably) Exact Solution to the Birthday Problem
A (Probably) Exact Solution to the Birthday Problem David Brink July 2011 Abstract. Given a year with n ≥ 1 days, the Birthday Problem asks for the minimal number X (n) such that in a class of X (n) students, the probability of finding two students with the same birthday is at least 50 percent. We derive heuristically an exact formula for X (n) and argue that the probability that a counter-example to this formula exists is less than one in 45 billion. We then give a new derivation of the asymptotic expansion of Ramanujan's Q-function and note its curious resemblance to the formula for X (n). 1 Introduction It is a surprising fact, apparently first noticed by Davenport around 1927, that in a class of 23 students, the probability of finding two students with the same birthday is more than 50 percent. This observation, and the multitude of related problems that it raises, have been discussed by many distinguished authors, including von Mises [19], Gamow [9, pp. 204{206], Littlewood [17, p. 18], Halmos [11, pp. 103{104], and Davenport [6, pp. 174{175]. The interested reader is referred to [22] for a historical account and to [12] to experience the problem experimentally. Given a year with n ≥ 1 days, the variant of the Birthday Problem studied here asks for the min- imal number X (n) such that in a class of X (n) students, the probability of a birthday coincidence is at least 50 percent. In other words, X (n) is the minimal integer x such that 1 2 x − 1 P (x) := 1 − 1 − ··· 1 − (1) n n n 1 is less than or equal to 2 . -
A Monte Carlo Simulation of the Birthday Problem
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-2007 A Monte Carlo Simulation of the Birthday Problem Stacey Aldag University of Nebraska-Lincoln Follow this and additional works at: https://digitalcommons.unl.edu/mathmidexppap Part of the Science and Mathematics Education Commons Aldag, Stacey, "A Monte Carlo Simulation of the Birthday Problem" (2007). MAT Exam Expository Papers. 29. https://digitalcommons.unl.edu/mathmidexppap/29 This Article is brought to you for free and open access by the Math in the Middle Institute Partnership at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in MAT Exam Expository Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Master of Arts in Teaching (MAT) Masters Exam Stacey Aldag In partial fulfillment of the requirements for the Master of Arts in Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics. David Fowler, Advisor July 2007 Aldag-MAT Expository Paper-1 A Monte Carlo Simulation of the Birthday Problem Stacey Aldag July 2007 Aldag-MAT Expository Paper-2 A Monte Carlo Simulation of the Birthday Problem Question, how many people would you need in a group in order for there to be a 50-50 chance that at least two people will share a birthday? Answer, 23 people. But how can this be? There are 365 days in a year and half of that would be 182, so why wouldn’t you need at least 182 people to have a 50-50 chance? Strangely enough the answer to this question is only 23 people are necessary to have a 50% chance at least two people in the group will share a birthday.