R Simulations: Monty Hall problem


 Kelley Hawkins
 2 years ago
 Views:
Transcription
1 R Simulations: Monty Hall problem Monte Carlo Simulations Monty Hall Problem Statistical Analysis Simulation in R Exercise 1: A Gift Giving Puzzle Exercise 2: Gambling Problem R Simulations: Monty Hall problem Ying Sun SAMSI Undergraduate Workshop February 24, 2011 Ying Sun R Simulations: Monty Hall problem
2 R Simulations: Monty Hall problem Monte Carlo Simulations Monty Hall Problem Statistical Analysis Simulation in R Exercise 1: A Gift Giving Puzzle Exercise 2: Gambling Problem R Simulations: Monty Hall problem 1 Monte Carlo Simulations 2 Monty Hall Problem 3 Statistical Analysis 4 Simulation in R 5 Exercise 1: A Gift Giving Puzzle 6 Exercise 2: Gambling Problem Ying Sun R Simulations: Monty Hall problem
3 R Simulations: Monty Hall problem Monte Carlo Simulations Monty Hall Problem Statistical Analysis Simulation in R Exercise 1: A Gift Giving Puzzle Exercise 2: Gambling Problem Monte Carlo Simulations What is Monte Carlo simulation: A problem solving technique used to approximate the probability of certain outcomes by running multiple trial runs, called simulations, using random variables. Why use simulations: Some situations do not lend to precise mathematical treatment. Others may be difficult, timeconsuming to analyze. Simulations may approximate realworld results, yet require less time and effort. Ying Sun R Simulations: Monty Hall problem
4 R Simulations: Monty Hall problem Monte Carlo Simulations Monty Hall Problem Statistical Analysis Simulation in R Exercise 1: A Gift Giving Puzzle Exercise 2: Gambling Problem Conduct A Simulation 1 Describe the possible outcomes. 2 Link each outcome to one or more random numbers. 3 Choose a source of random numbers. 4 Choose a random number. 5 Based on the random number, note the simulated outcome. Repeat steps 4 and 5 multiple times; preferably, until the outcomes show a stable pattern. 6 Analyze the simulated outcomes and report results. Ying Sun R Simulations: Monty Hall problem
5 Switch or Not Switch In September of 1991 a reader of Marilyn Vos Savant s Sunday Parade column wrote in and asked the following question: Suppose you re on a game show, and you re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what s behind the other doors, opens another door, say No. 3, which has a goat. He then says to you, Do you want to pick door No. 2? Is it to your advantage to take the switch?
6 Monty Hall Problem This problem was given the name The Monty Hall Paradox in honor of the long time host of the television game show Let s Make a Deal. Articles about the controversy appeared in the New York Times and other papers around the country. Marilyn s answer was that the contestant should switch doors and she received nearly 10,000 responses from readers, most of them disagreeing with her. Several were from mathematicians and scientists whose responses ranged from hostility to disappointment at the nation s lack of mathematical skills. They assumed that each door has an equal probability and concluded that switching does not matter.
7 Conditional Probability Suppose the player chooses door No.1. Let A 1 : Door No.1 has the car, A 2 : Door No.2 has the car, A 3 : Door No.3 has the car, O: Host opens door No.3. P(A 1 ) = P(A 2 ) = P(A 3 ) = 1 3 If door No. 1 has the car, the host could open door No.2 or 3, P(O A 1 ) = 1 2 If door No. 2 has the car, the host must open door No.3, P(O A 2 ) = 1 If door No. 3 has the car, the host can not open door No.3, P(O A 3 ) = 0
8 Bayes Theorem Bayes theorem: relates the conditional and marginal probabilities of events. P(A 1 O) = P(O A 1 1)P(A 1 ) 3 i=1 P(O A i)p(a i ) = = 1 3 P(A 2 O) = P(O A 1 2)P(A 2 ) 3 i=1 P(O A i)p(a i ) = = 2 3 The player chooses door No.1, p 1 = P(switch and win) = 2 3, p 2 = P(not switch and win) = 1 3.
9 Simulation in R Suppose the player plays this game n = 10 times. Which door has the car: > car=sample(3,10,replace=t) > car [1] Which door is chosen: > door=sample(3,10,replace=t) > door [1] Switch and win: the car is not behind the chosen door. > switchwin=(door!=car) > switchwin [1] TRUE FALSE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE > sum(switchwin)/10 [1] 0.7 Not switch and win: the car is behind the chosen door. > noswitchwin=(door==car) > noswitchwin [1] FALSE TRUE TRUE FALSE FALSE FALSE TRUE FALSE FALSE FALSE > sum(noswitchwin)/10 [1] 0.3
10 R Simulations: Monty Hall problem Monte Carlo Simulations Monty Hall Problem Statistical Analysis Simulation in R Exercise 1: A Gift Giving Puzzle Exercise 2: Gambling Problem The Law of Large Numbers The law of large numbers: It describes the result of performing the same experiment a large number of times. The average of the results obtained from a large number of trials should be close to the expected value. It will tend to become closer as more trials are performed. Increase the number of trials n. # of switch and win ˆp 1 = n p 1 = 2 3. # of not switch and win ˆp 2 = n p 2 = 1 3. Ying Sun R Simulations: Monty Hall problem
11 Uncertainties Bootstrap Method: It allows one to estimate the sampling distribution of the estimators, ˆp 1 and ˆp 2 (statistics). We can construct confidence intervals for p 1 and p 2 (parameters). R function gameshow(n) returns ˆp 1 and ˆp 2 when the player plays the game n times. 95% bootstrap confidence intervals: B=1000 prob=null n=1000 for (i in 1:B){ prob=rbind(prob,gameshow(n)) } p1hat=prob[,1] p2hat=prob[,2] quantile(p1hat,c(0.025,0.975)) quantile(p2hat,c(0.025,0.975))
12 Normal Approximation Sampling distribution: if min{np, n(1 p)} 10 and n is large, ( ) p(1 p) ˆp N p,. n Check the histgrams and boxplots of ˆp 1 for different n. boxplot(p1hat,ylim=c(0.4,1)) hist(p1hat,freq=f,xlim=c(0.4,1)) If we know the sampling distribution in theory, we can only estimate p once and use the normal approximation to construct a confidence interval. 95% confidence interval for p 1 : ( ) ˆp1 (1 ˆp 1 ) ˆp1 (1 ˆp 1 ) ˆp , ˆp n n
13 A Gift Giving Puzzle A probability problem: n people put their names into a hat, then they all draw a name. The draw is successful if no one draws their own name. How likely is that? Theoretical solution: The idea is to count the total number of permutations and then subtract out any permutation that fixes one or more points. The trick is to make sure there are no double counts. The formula specifies how to add and subtract various subsets (fixing one point, two points, three points, etc). ( n n! 1 = n! ) ( n (n 1)! + 2 n ( 1) k k=0 k! p = n ( 1) k k=0. k! ) (n 2)!... + ( 1) n ( n n ) (n n)!
14 Simulation in R Suppose n.p = 10 people put their names into a hat: > name=1:10 > name [1] They all draw a name: draw=sample(name) > draw [1] Check if no one draws their own name: > check=sum(draw==name) > check [1] 0 Success? > check==0 [1] TRUE The gift(n.p,n.sim) function does this simulation n.sim times.
15 Gambling Problem A gambler starts with $100. She plays a game in which she is allowed to bet any amount of money (up to the amount that she has). If she wins, she receives twice her original stake back, while if she loses, she loses the amount she bet. The probability she wins the game is p = She will stop playing if she reaches $0 or if she reaches $200. She wishes to maximize the probability that she stops at $200. Is it better to bet in small increments (i.e., bet $5 at a time until she reaches $0 or $200) or bet it all at once (i.e. $100 on the first bet)? The gambling(start,bet,n.sim) function does this simulation n.sim times with the starting money and betting money as input arguments.
LET S MAKE A DEAL! ACTIVITY
LET S MAKE A DEAL! ACTIVITY NAME: DATE: SCENARIO: Suppose you are on the game show Let s Make A Deal where Monty Hall (the host) gives you a choice of three doors. Behind one door is a valuable prize.
More informationBook Review of Rosenhouse, The Monty Hall Problem. Leslie Burkholder 1
Book Review of Rosenhouse, The Monty Hall Problem Leslie Burkholder 1 The Monty Hall Problem, Jason Rosenhouse, New York, Oxford University Press, 2009, xii, 195 pp, US $24.95, ISBN 9780195#67898 (Source
More informationWeek 2: Conditional Probability and Bayes formula
Week 2: Conditional Probability and Bayes formula We ask the following question: suppose we know that a certain event B has occurred. How does this impact the probability of some other A. This question
More informationProbability. a number between 0 and 1 that indicates how likely it is that a specific event or set of events will occur.
Probability Probability Simple experiment Sample space Sample point, or elementary event Event, or event class Mutually exclusive outcomes Independent events a number between 0 and 1 that indicates how
More informationDiscrete Math in Computer Science Homework 7 Solutions (Max Points: 80)
Discrete Math in Computer Science Homework 7 Solutions (Max Points: 80) CS 30, Winter 2016 by Prasad Jayanti 1. (10 points) Here is the famous Monty Hall Puzzle. Suppose you are on a game show, and you
More informationMonty Hall, Monty Fall, Monty Crawl
Monty Hall, Monty Fall, Monty Crawl Jeffrey S. Rosenthal (June, 2005; appeared in Math Horizons, September 2008, pages 5 7.) (Dr. Rosenthal is a professor in the Department of Statistics at the University
More informationSimulation Exercises to Reinforce the Foundations of Statistical Thinking in Online Classes
Simulation Exercises to Reinforce the Foundations of Statistical Thinking in Online Classes Simcha Pollack, Ph.D. St. John s University Tobin College of Business Queens, NY, 11439 pollacks@stjohns.edu
More informationLecture 13. Understanding Probability and LongTerm Expectations
Lecture 13 Understanding Probability and LongTerm Expectations Thinking Challenge What s the probability of getting a head on the toss of a single fair coin? Use a scale from 0 (no way) to 1 (sure thing).
More informationIntroduction and Overview
Introduction and Overview Probability and Statistics is a topic that is quickly growing, has become a major part of our educational program, and has a substantial role in the NCTM Standards. While covering
More informationThe Casino Lab STATION 1: CRAPS
The Casino Lab Casinos rely on the laws of probability and expected values of random variables to guarantee them profits on a daily basis. Some individuals will walk away very wealthy, while others will
More informationAppendix D: The Monty Hall Controversy
Appendix D: The Monty Hall Controversy Appendix D: The Monty Hall Controversy  Page 1 Let's Make a Deal Prepared by Rich Williams, Spring 1991 Last Modified Fall, 2004 You are playing Let's Make a Deal
More informationPractical Probability:
Practical Probability: Casino Odds and Sucker Bets Tom Davis tomrdavis@earthlink.net April 2, 2011 Abstract Gambling casinos are there to make money, so in almost every instance, the games you can bet
More informationUnit 19: Probability Models
Unit 19: Probability Models Summary of Video Probability is the language of uncertainty. Using statistics, we can better predict the outcomes of random phenomena over the long term from the very complex,
More informationConditional Probability
Chapter 4 Conditional Probability 4. Discrete Conditional Probability Conditional Probability In this section we ask and answer the following question. Suppose we assign a distribution function to a sample
More informationDiscrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22
CS 70 Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice, roulette
More informationWe rst consider the game from the player's point of view: Suppose you have picked a number and placed your bet. The probability of winning is
Roulette: On an American roulette wheel here are 38 compartments where the ball can land. They are numbered 136, and there are two compartments labeled 0 and 00. Half of the compartments numbered 136
More informationTerm Project: Roulette
Term Project: Roulette DCY Student January 13, 2006 1. Introduction The roulette is a popular gambling game found in all major casinos. In contrast to many other gambling games such as black jack, poker,
More informationProbabilities. Probability of a event. From Random Variables to Events. From Random Variables to Events. Probability Theory I
Victor Adamchi Danny Sleator Great Theoretical Ideas In Computer Science Probability Theory I CS 525 Spring 200 Lecture Feb. 6, 200 Carnegie Mellon University We will consider chance experiments with
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice,
More informationConditional Probability, Hypothesis Testing, and the Monty Hall Problem
Conditional Probability, Hypothesis Testing, and the Monty Hall Problem Ernie Croot September 17, 2008 On more than one occasion I have heard the comment Probability does not exist in the real world, and
More informationThe Math. P (x) = 5! = 1 2 3 4 5 = 120.
The Math Suppose there are n experiments, and the probability that someone gets the right answer on any given experiment is p. So in the first example above, n = 5 and p = 0.2. Let X be the number of correct
More informationSection 7C: The Law of Large Numbers
Section 7C: The Law of Large Numbers Example. You flip a coin 00 times. Suppose the coin is fair. How many times would you expect to get heads? tails? One would expect a fair coin to come up heads half
More informationSlide 1 Math 1520, Lecture 23. This lecture covers mean, median, mode, odds, and expected value.
Slide 1 Math 1520, Lecture 23 This lecture covers mean, median, mode, odds, and expected value. Slide 2 Mean, Median and Mode Mean, Median and mode are 3 concepts used to get a sense of the central tendencies
More informationMONT 107N Understanding Randomness Solutions For Final Examination May 11, 2010
MONT 07N Understanding Randomness Solutions For Final Examination May, 00 Short Answer (a) (0) How are the EV and SE for the sum of n draws with replacement from a box computed? Solution: The EV is n times
More informationPROBABILITY C A S I N O L A B
A P S T A T S A Fabulous PROBABILITY C A S I N O L A B AP Statistics Casino Lab 1 AP STATISTICS CASINO LAB: INSTRUCTIONS The purpose of this lab is to allow you to explore the rules of probability in the
More informationIntroduction to Discrete Probability. Terminology. Probability definition. 22c:19, section 6.x Hantao Zhang
Introduction to Discrete Probability 22c:19, section 6.x Hantao Zhang 1 Terminology Experiment A repeatable procedure that yields one of a given set of outcomes Rolling a die, for example Sample space
More informationWeek 4: Gambler s ruin and bold play
Week 4: Gambler s ruin and bold play Random walk and Gambler s ruin. Imagine a walker moving along a line. At every unit of time, he makes a step left or right of exactly one of unit. So we can think that
More information6.3 Probabilities with Large Numbers
6.3 Probabilities with Large Numbers In general, we can t perfectly predict any single outcome when there are numerous things that could happen. But, when we repeatedly observe many observations, we expect
More informationThe Kelly criterion for spread bets
IMA Journal of Applied Mathematics 2007 72,43 51 doi:10.1093/imamat/hxl027 Advance Access publication on December 5, 2006 The Kelly criterion for spread bets S. J. CHAPMAN Oxford Centre for Industrial
More informationStatistical Fallacies: Lying to Ourselves and Others
Statistical Fallacies: Lying to Ourselves and Others "There are three kinds of lies: lies, damned lies, and statistics. Benjamin Disraeli +/ Benjamin Disraeli Introduction Statistics, assuming they ve
More informationMidterm Exam #1 Instructions:
Public Affairs 818 Professor: Geoffrey L. Wallace October 9 th, 008 Midterm Exam #1 Instructions: You have 10 minutes to complete the examination and there are 6 questions worth a total of 10 points. The
More informationMidterm Exam #1 Instructions:
Public Affairs 818 Professor: Geoffrey L. Wallace October 9 th, 008 Midterm Exam #1 Instructions: You have 10 minutes to complete the examination and there are 6 questions worth a total of 10 points. The
More informationExpected Value and the Game of Craps
Expected Value and the Game of Craps Blake Thornton Craps is a gambling game found in most casinos based on rolling two six sided dice. Most players who walk into a casino and try to play craps for the
More informationProbability: The Study of Randomness Randomness and Probability Models. IPS Chapters 4 Sections 4.1 4.2
Probability: The Study of Randomness Randomness and Probability Models IPS Chapters 4 Sections 4.1 4.2 Chapter 4 Overview Key Concepts Random Experiment/Process Sample Space Events Probability Models Probability
More informationBasic Probability. Probability: The part of Mathematics devoted to quantify uncertainty
AMS 5 PROBABILITY Basic Probability Probability: The part of Mathematics devoted to quantify uncertainty Frequency Theory Bayesian Theory Game: Playing Backgammon. The chance of getting (6,6) is 1/36.
More informationMath 141. Lecture 2: More Probability! Albyn Jones 1. jones@reed.edu www.people.reed.edu/ jones/courses/141. 1 Library 304. Albyn Jones Math 141
Math 141 Lecture 2: More Probability! Albyn Jones 1 1 Library 304 jones@reed.edu www.people.reed.edu/ jones/courses/141 Outline Law of total probability Bayes Theorem the Multiplication Rule, again Recall
More informationElementary Statistics and Inference. Elementary Statistics and Inference. 17 Expected Value and Standard Error. 22S:025 or 7P:025.
Elementary Statistics and Inference S:05 or 7P:05 Lecture Elementary Statistics and Inference S:05 or 7P:05 Chapter 7 A. The Expected Value In a chance process (probability experiment) the outcomes of
More information+ Section 6.2 and 6.3
Section 6.2 and 6.3 Learning Objectives After this section, you should be able to DEFINE and APPLY basic rules of probability CONSTRUCT Venn diagrams and DETERMINE probabilities DETERMINE probabilities
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According
More informationChapter 2: Data quantifiers: sample mean, sample variance, sample standard deviation Quartiles, percentiles, median, interquartile range Dot diagrams
Review for Final Chapter 2: Data quantifiers: sample mean, sample variance, sample standard deviation Quartiles, percentiles, median, interquartile range Dot diagrams Histogram Boxplots Chapter 3: Set
More informationProbability Models.S1 Introduction to Probability
Probability Models.S1 Introduction to Probability Operations Research Models and Methods Paul A. Jensen and Jonathan F. Bard The stochastic chapters of this book involve random variability. Decisions are
More informationarxiv:1112.0829v1 [math.pr] 5 Dec 2011
How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly
More informationVideo Poker in South Carolina: A Mathematical Study
Video Poker in South Carolina: A Mathematical Study by Joel V. Brawley and Todd D. Mateer Since its debut in South Carolina in 1986, video poker has become a game of great popularity as well as a game
More informationAMS 5 CHANCE VARIABILITY
AMS 5 CHANCE VARIABILITY The Law of Averages When tossing a fair coin the chances of tails and heads are the same: 50% and 50%. So if the coin is tossed a large number of times, the number of heads and
More informationChicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011
Chicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011 Name: Section: I pledge my honor that I have not violated the Honor Code Signature: This exam has 34 pages. You have 3 hours to complete this
More informationThe Story. Probability  I. Plan. Example. A Probability Tree. Draw a probability tree.
Great Theoretical Ideas In Computer Science Victor Adamchi CS 55 Carnegie Mellon University Probability  I The Story The theory of probability was originally developed by a French mathematician Blaise
More informationAvoiding the Consolation Prize: The Mathematics of Game Shows
Avoiding the Consolation Prize: The Mathematics of Game Shows STUART GLUCK, Ph.D. CENTER FOR TALENTED YOUTH JOHNS HOPKINS UNIVERSITY STU@JHU.EDU CARLOS RODRIGUEZ CENTER FOR TALENTED YOUTH JOHNS HOPKINS
More informationPrediction Markets, Fair Games and Martingales
Chapter 3 Prediction Markets, Fair Games and Martingales Prediction markets...... are speculative markets created for the purpose of making predictions. The current market prices can then be interpreted
More informationTHE WINNING ROULETTE SYSTEM by http://www.webgoldminer.com/
THE WINNING ROULETTE SYSTEM by http://www.webgoldminer.com/ Is it possible to earn money from online gambling? Are there any 100% sure winning roulette systems? Are there actually people who make a living
More informationA Dutch Book for Group DecisionMaking?
A Dutch Book for Group DecisionMaking? Forthcoming in Benedikt Löwe, Eric Pacuit, JanWillem Romeijn (eds.) Foundations of the Formal Sciences VIReasoning about Probabilities and Probabilistic Reasoning.
More informationUsing gametheoretic probability for probability judgment. Glenn Shafer
Workshop on GameTheoretic Probability and Related Topics Tokyo University February 28, 2008 Using gametheoretic probability for probability judgment Glenn Shafer For 170 years, people have asked whether
More informationMonte Carlo Simulation (General Simulation Models)
Monte Carlo Simulation (General Simulation Models) STATGRAPHICS Rev. 9/16/2013 Summary... 1 Example #1... 1 Example #2... 8 Summary Monte Carlo simulation is used to estimate the distribution of variables
More informationLecture 4 : Bayesian inference
Lecture 4 : Bayesian inference The Lecture dark 4 energy : Bayesian puzzle inference What is the Bayesian approach to statistics? How does it differ from the frequentist approach? Conditional probabilities,
More informationBetting on Excel to enliven the teaching of probability
Betting on Excel to enliven the teaching of probability Stephen R. Clarke School of Mathematical Sciences Swinburne University of Technology Abstract The study of probability has its roots in gambling
More informationGoal Problems in Gambling and Game Theory. Bill Sudderth. School of Statistics University of Minnesota
Goal Problems in Gambling and Game Theory Bill Sudderth School of Statistics University of Minnesota 1 Three problems Maximizing the probability of reaching a goal. Maximizing the probability of reaching
More informationExperimental Uncertainty and Probability
02/04/07 PHY310: Statistical Data Analysis 1 PHY310: Lecture 03 Experimental Uncertainty and Probability Road Map The meaning of experimental uncertainty The fundamental concepts of probability 02/04/07
More informationPROBABILITY PROBABILITY
PROBABILITY PROBABILITY Documents prepared for use in course B0.305, New York University, Stern School of Business Types of probability page 3 The catchall term probability refers to several distinct ideas.
More informationMath 408, Actuarial Statistics I, Spring 2008. Solutions to combinatorial problems
, Spring 2008 Word counting problems 1. Find the number of possible character passwords under the following restrictions: Note there are 26 letters in the alphabet. a All characters must be lower case
More informationBetting systems: how not to lose your money gambling
Betting systems: how not to lose your money gambling G. Berkolaiko Department of Mathematics Texas A&M University 28 April 2007 / Mini Fair, Math Awareness Month 2007 Gambling and Games of Chance Simple
More informationIn the situations that we will encounter, we may generally calculate the probability of an event
What does it mean for something to be random? An event is called random if the process which produces the outcome is sufficiently complicated that we are unable to predict the precise result and are instead
More informationPROBABILITY SECOND EDITION
PROBABILITY SECOND EDITION Table of Contents How to Use This Series........................................... v Foreword..................................................... vi Basics 1. Probability All
More informationBetting interpretations of probability
Betting interpretations of probability Glenn Shafer June 21, 2010 Third Workshop on GameTheoretic Probability and Related Topics Royal Holloway, University of London 1 Outline 1. Probability began with
More informationChoice Under Uncertainty
Decision Making Under Uncertainty Choice Under Uncertainty Econ 422: Investment, Capital & Finance University of ashington Summer 2006 August 15, 2006 Course Chronology: 1. Intertemporal Choice: Exchange
More information6.042/18.062J Mathematics for Computer Science. Expected Value I
6.42/8.62J Mathematics for Computer Science Srini Devadas and Eric Lehman May 3, 25 Lecture otes Expected Value I The expectation or expected value of a random variable is a single number that tells you
More information13.0 Central Limit Theorem
13.0 Central Limit Theorem Discuss Midterm/Answer Questions Box Models Expected Value and Standard Error Central Limit Theorem 1 13.1 Box Models A Box Model describes a process in terms of making repeated
More informationCh. 13.2: Mathematical Expectation
Ch. 13.2: Mathematical Expectation Random Variables Very often, we are interested in sample spaces in which the outcomes are distinct real numbers. For example, in the experiment of rolling two dice, we
More informationProblem sets for BUEC 333 Part 1: Probability and Statistics
Problem sets for BUEC 333 Part 1: Probability and Statistics I will indicate the relevant exercises for each week at the end of the Wednesday lecture. Numbered exercises are backofchapter exercises from
More informationCurriculum Map Statistics and Probability Honors (348) Saugus High School Saugus Public Schools 20092010
Curriculum Map Statistics and Probability Honors (348) Saugus High School Saugus Public Schools 20092010 Week 1 Week 2 14.0 Students organize and describe distributions of data by using a number of different
More informationThe Normal Approximation to Probability Histograms. Dice: Throw a single die twice. The Probability Histogram: Area = Probability. Where are we going?
The Normal Approximation to Probability Histograms Where are we going? Probability histograms The normal approximation to binomial histograms The normal approximation to probability histograms of sums
More informationCh5: Discrete Probability Distributions Section 51: Probability Distribution
Recall: Ch5: Discrete Probability Distributions Section 51: Probability Distribution A variable is a characteristic or attribute that can assume different values. o Various letters of the alphabet (e.g.
More informationDefinition and Calculus of Probability
In experiments with multivariate outcome variable, knowledge of the value of one variable may help predict another. For now, the word prediction will mean update the probabilities of events regarding the
More informationSCRATCHING THE SURFACE OF PROBABILITY. Robert J. Russell, University of Paisley, UK
SCRATCHING THE SURFACE OF PROBABILITY Robert J. Russell, University of Paisley, UK Scratch cards are widely used to promote the interests of charities and commercial concerns. They provide a useful mechanism
More informationLOOKING FOR A GOOD TIME TO BET
LOOKING FOR A GOOD TIME TO BET LAURENT SERLET Abstract. Suppose that the cards of a well shuffled deck of cards are turned up one after another. At any timebut once only you may bet that the next card
More informationDecision Theory. 36.1 Rational prospecting
36 Decision Theory Decision theory is trivial, apart from computational details (just like playing chess!). You have a choice of various actions, a. The world may be in one of many states x; which one
More informationA UNIQUE COMBINATION OF CHANCE & SKILL
A UNIQUE COMBINATION OF CHANCE & SKILL The popularity of blackjack stems from its unique combination of chance and skill. The object of the game is to form a hand closer to 21 than the dealer without going
More informationפרויקט מסכם לתואר בוגר במדעים )B.Sc( במתמטיקה שימושית
המחלקה למתמטיקה Department of Mathematics פרויקט מסכם לתואר בוגר במדעים )B.Sc( במתמטיקה שימושית הימורים אופטימליים ע"י שימוש בקריטריון קלי אלון תושיה Optimal betting using the Kelly Criterion Alon Tushia
More informationDEVELOPING A MODEL THAT REFLECTS OUTCOMES OF TENNIS MATCHES
DEVELOPING A MODEL THAT REFLECTS OUTCOMES OF TENNIS MATCHES Barnett T., Brown A., and Clarke S. Faculty of Life and Social Sciences, Swinburne University, Melbourne, VIC, Australia ABSTRACT Many tennis
More informationLecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett
Lecture Note 1 Set and Probability Theory MIT 14.30 Spring 2006 Herman Bennett 1 Set Theory 1.1 Definitions and Theorems 1. Experiment: any action or process whose outcome is subject to uncertainty. 2.
More informationSOME ASPECTS OF GAMBLING WITH THE KELLY CRITERION. School of Mathematical Sciences. Monash University, Clayton, Victoria, Australia 3168
SOME ASPECTS OF GAMBLING WITH THE KELLY CRITERION Ravi PHATARFOD School of Mathematical Sciences Monash University, Clayton, Victoria, Australia 3168 In this paper we consider the problem of gambling with
More informationExpected Value and Variance
Chapter 6 Expected Value and Variance 6.1 Expected Value of Discrete Random Variables When a large collection of numbers is assembled, as in a census, we are usually interested not in the individual numbers,
More informationThe overall size of these chance errors is measured by their RMS HALF THE NUMBER OF TOSSES NUMBER OF HEADS MINUS 0 400 800 1200 1600 NUMBER OF TOSSES
INTRODUCTION TO CHANCE VARIABILITY WHAT DOES THE LAW OF AVERAGES SAY? 4 coins were tossed 1600 times each, and the chance error number of heads half the number of tosses was plotted against the number
More informationMA 1125 Lecture 14  Expected Values. Friday, February 28, 2014. Objectives: Introduce expected values.
MA 5 Lecture 4  Expected Values Friday, February 2, 24. Objectives: Introduce expected values.. Means, Variances, and Standard Deviations of Probability Distributions Two classes ago, we computed the
More information4. Continuous Random Variables, the Pareto and Normal Distributions
4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random
More informationNational Sun YatSen University CSE Course: Information Theory. Gambling And Entropy
Gambling And Entropy 1 Outline There is a strong relationship between the growth rate of investment in a horse race and the entropy of the horse race. The value of side information is related to the mutual
More informationCalculated Bets: Computers, Gambling, and Mathematical Modeling to Win Steven Skiena
Calculated Bets: Computers, Gambling, and Mathematical Modeling to Win Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena
More informationIntroduction to Probability
Massachusetts Institute of Technology Course Notes 0 6.04J/8.06J, Fall 0: Mathematics for Computer Science November 4 Professor Albert Meyer and Dr. Radhika Nagpal revised November 6, 00, 57 minutes Introduction
More informationUsing computer simulation to maximize profits and control risk.
Trading Strategies Using computer simulation to maximize profits and control risk. Copyright 2002 by Larry C. Sanders. All rights reserved. Published by LSS Limited. PO Box 981133 Park City, Utah 840981133
More information3. Data Analysis, Statistics, and Probability
3. Data Analysis, Statistics, and Probability Data and probability sense provides students with tools to understand information and uncertainty. Students ask questions and gather and use data to answer
More informationThe New Mexico Lottery
The New Mexico Lottery 26 February 2014 Lotteries 26 February 2014 1/27 Today we will discuss the various New Mexico Lottery games and look at odds of winning and the expected value of playing the various
More information2.5 Zeros of a Polynomial Functions
.5 Zeros of a Polynomial Functions Section.5 Notes Page 1 The first rule we will talk about is Descartes Rule of Signs, which can be used to determine the possible times a graph crosses the xaxis and
More informationA Simple Parrondo Paradox. Michael Stutzer, Professor of Finance. 419 UCB, University of Colorado, Boulder, CO 803090419
A Simple Parrondo Paradox Michael Stutzer, Professor of Finance 419 UCB, University of Colorado, Boulder, CO 803090419 michael.stutzer@colorado.edu Abstract The Parrondo Paradox is a counterintuitive
More informationMOVIES, GAMBLING, SECRET CODES, JUST MATRIX MAGIC
MOVIES, GAMBLING, SECRET CODES, JUST MATRIX MAGIC DR. LESZEK GAWARECKI 1. The Cartesian Coordinate System In the Cartesian system points are defined by giving their coordinates. Plot the following points:
More informationLecture 11 Uncertainty
Lecture 11 Uncertainty 1. Contingent Claims and the StatePreference Model 1) Contingent Commodities and Contingent Claims Using the simple twogood model we have developed throughout this course, think
More informationFourth Problem Assignment
EECS 401 Due on Feb 2, 2007 PROBLEM 1 (25 points) Joe and Helen each know that the a priori probability that her mother will be home on any given night is 0.6. However, Helen can determine her mother s
More information1 Portfolio Selection
COS 5: Theoretical Machine Learning Lecturer: Rob Schapire Lecture # Scribe: Nadia Heninger April 8, 008 Portfolio Selection Last time we discussed our model of the stock market N stocks start on day with
More informationQuestion: What is the probability that a fivecard poker hand contains a flush, that is, five cards of the same suit?
ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the
More informationLearn How to Use The Roulette Layout To Calculate Winning Payoffs For All Straightup Winning Bets
Learn How to Use The Roulette Layout To Calculate Winning Payoffs For All Straightup Winning Bets Understand that every square on every street on every roulette layout has a value depending on the bet
More informationLaw of Large Numbers. Alexandra Barbato and Craig O Connell. Honors 391A Mathematical Gems Jenia Tevelev
Law of Large Numbers Alexandra Barbato and Craig O Connell Honors 391A Mathematical Gems Jenia Tevelev Jacob Bernoulli Life of Jacob Bernoulli Born into a family of important citizens in Basel, Switzerland
More informationSummary of Formulas and Concepts. Descriptive Statistics (Ch. 14)
Summary of Formulas and Concepts Descriptive Statistics (Ch. 14) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume
More informationProbability and Expected Value
Probability and Expected Value This handout provides an introduction to probability and expected value. Some of you may already be familiar with some of these topics. Probability and expected value are
More information