Variations on the Gambler s Ruin Problem


 Marcia Gilmore
 2 years ago
 Views:
Transcription
1 Variations on the Gambler s Ruin Problem Mat Willmott December 6, 2002 Abstract. This aer covers the history and solution to the Gambler s Ruin Problem, and then exlores the odds for each layer to win in three variations on the roblem: the attrition variation, a variation in which one layer has an infinite number of oints, and a three layer variation, in which two layers lay against a house layer. 1. Introduction. Throughout the years, the Gambler s Ruin Problem has been rominent in alied mathematics. With differing levels of comlexity, variations on the roblem arise in all tyes of games, from a child s board games to comlicated casino games such as cras and blackjack. The theory behind the roblem is also used in horse racing and dog racing, generally by the track to be sure that it makes a rofit. There are many variations on the roblem, and as fast as older ones are being solved, newer ones are being formulated. Fourlayer and fivelayer variations, for examle, have already been roosed, and some have even been solved. The Gambler s Ruin Problem is clearly an imortant and growing toic in discrete alied mathematics; it was relevant in the 1600s and is still relevant today. Section 2 covers the history of the Gambler s Ruin Problem. Then Section 3 discusses and gives a solution to the most common form of the roblem, closely following the solution resented by DeGroot [2, ]. Sections 4 through 6 discuss some of the more common variations on the roblem. Section 4 resents and analyzes a game where the layers do not win oints on successful trials, but only lose them on unsuccessful ones. Section 5 discusses a variation in which one layer has an infinite money suly. Section 6 concludes the aer with a discussion of a three layer variation on the roblem. 2. History. The Gambler s Ruin Problem is one of the oldest roblems in robability theory. According to Edwards [3,. 73], Pascal osed a roblem similar to the Gambler s Ruin Problem in 1656 in a letter to Fermat. Caravaci later summarizes the letter: Let two men lay with three dice, the first layer scoring a oint whenever 11 is thrown, and the second whenever 14 is thrown. But instead of the oints accumulating in the ordinary way, let a oint 1
2 be added to a layer s score only if his oonent s score is nil, but otherwise let it be subtracted from his oonent s score. It is as if oosing oints form airs, and annihilate each other, so that the trailing layer always has zero oints. The winner is the first to reach twelve oints; what are the relative chances of each layer winning? This roblem, however, is not the Gambler s Ruin Problem in its common form. Edwards goes on to exlain that the common form comes from Huygens, who read and restated the roblem. Originally, Huygens looked at the above roblem as if oints were accumulated normally, and the winner were the first layer to lead by 12 oints. Later, he again restated the roblem as follows: Problem 21 Each layer starts with 12 oints, and a successful roll of the three dice for a layer getting an 11 for the first layer or a 14 for the second adds one to that layer s score and subtracts one from the other layer s score; the loser of the game is the first to reach zero oints. What is the robability of victory for each layer? The above roblem is known as the Gambler s Ruin Problem because one of the layers will run out of money be ruined at the end of the game. This form is the form most commonly used today. 3. The Basic Form of the Gambler s Ruin Problem. To solve Problem 21, we look at the game from the oint of view of one of the layers, say Player A. We name Player A s oonent Player B. Say Player A starts with i oints, and Player B starts with k i oints, so that the total number of oints in the game is k. For examle, in Section 2, we see that i = 12 and k = 24. Player A wins when he has k oints, and loses when he has zero oints. Also, say that the robability of A winning the next oint is, so the robability of B winning the next oint is q = 1. Finally, say that a i is the the robability of A reaching k oints before reaching 0 oints when starting with i oints; so a 0 = 0 and a k = 1. Let us refer to the event that A reaches k oints before B as W, the event of A winning the first oint as A 1, and the event of A losing the first oint B winning the first oint as B 1. Then P W = P A 1 P W A 1 + P B 1 P W B Substituting in the robability definitions made above, we get the following system of equations: 2
3 a 1 = a 2 + qa 0 = a 2 a 2 = a 3 + qa a k 2 = a k 1 + qa k 3 a k 1 = a k + qa k 2 = + qa k 2. Since + q = 1, we can relace each a i with a i + qa i. Then Equations 32 become a 2 a 1 = q a 1 a 3 a 2 = q q 2a1 a 2 a 1 = a 4 a 3 = q q 3a1 a 3 a 2 = 33 a k 1 a k 2 =. q k 2a1 a k 2 a k 3 = 1 a k 1 = q q k 1a1 a k 1 a k 2 =. And finally, we can sum Equations 33 to get k 1 q i. 1 a 1 = a For a fair game, = q = 1/2, and so Equation 34 becomes 1 a 1 = k 1a 1, that is, a 1 = 1/k. From Equations 33, we can see that for = q, we have a 2 = 2a 1 and a 3 = 3a 1 and so on, so that i=1 a i = i/k. 35 For an unfair game, q, and so Equation 34 becomes This exression can be simlified to 1 a 1 = a 1 q k q q 1. a 1 = q 1 q k 1. 3
4 Using Equations 32, this formula can be generalized to a i = q i 1 q k We can use Equation 36 to solve Problem 21. Assume Player A is rolling for an 11. Since the chances of rolling a 14 on three dice is 15/216 and the chances of rolling an 11 is 27/216, we have q/ = 5/9. Plugging this ratio into Equation 36 and using i = 12 and k = 24, we get a i = , which is about 1156/1157, the same solution that Pascal came u with. The Gambler s Ruin Problem can be modified and generalized to aly to many different tyes of games with different numbers of layers, different tyes of layers, and different rules. 4. The Attrition Variation. One of the most common variations on the Gambler s Ruin Problem is called attrition. In attrition, one layer does not win a oint from the other layer on a successful lay, such as a roll of an 11 for Player A in Problem 21. Instead, the losing layer simly discards a oint. As W. D. Kaigh describes it in [4,.22], this variation alies to many situations from oular board games like Risk to bestofseven sorts series like the World Series or the Stanley Cu finals. The robability that Player A wins the game, and B is ruined, can be found by examining A s total score at the end of the game. We denote the number of oints that A has lost at the end of the game by L A. We define the event W, the robabilities a i,, and q, and the variables i and k as in Section 3. We now see that 0 L A i 1. As a result, we see that i 1 P W = P W L A = x. 41 x=0 For simlicity s sake, let us refer to Player B s starting score as b, where b = k i. Then for A to win while losing x oints in the rocess, we want Player A to win b oints in the same amount of time that it takes for Player B to win x oints. Using basic binomial robability [2,.84 85], we rewrite this condition as b + x 1 P W L A = x = b q x. 42 x Combining Equation 41 with Equation 42, we obtain the robability that A wins the game, or that B is ruined, as i 1 b + x 1 a i = b q x. 43 x x=0 4
5 5. One Player with Limited Points vs. One Player with Infinite Points. In another common variation on the Gambler s Ruin Problem, one layer, B say, has an infinite oint suly. Obviously, this hyothesis eliminates the game art of the roblem because B can never lose. However, we can still examine the numerical consequences of having a layer with an infinite oint suly. For < 1/2, we have q/ > 1, and as k goes to infinity in Equation 36, we see that a i always aroaches zero. For > 1/2, we have q/ < 1, and as k goes to infinity in Equation 36, we see that the chance that A is not ruined is q i. a i = 1 Player A can never win because B has an infinite number of oints and therefore can not be ruined, so this equation only gives the robability that A will continue to lay forever. For = 1/2, we use Equation 35 to find the chance that A is not ruined. Clearly, as k aroaches infinity, i/k aroaches zero, so Player A loses all of his oints. We can conclude that A is always ruined eventually for 1/2, and has a robability of a i = 1 q/ i of laying forever for > 1/2. 6. A Generalization of the Problem to Three Players. We can generalize the Gambler s Ruin Problem to three layers. Player A and Player B lay games against each other, but they also lay a combined game against a searate layer, called C. In this variation, however, A and B lace two halfoint bets on every lay, rather than one fulloint bet. In the game between A and B, say A wins the halfoint bet with robability 1, and B wins with robability q 1 = 1 1. Also, A and B both contribute a halfoint to a combined fulloint bet against C. Say A and B win this bet with robability 2, and C wins with robability q 2 = 1 2. Now suose that A, B, and C start with i, j, and l oints resectively. Set k = i + j + l. Then k is the total number of oints in the game; it is the number of oints the winning layer has at the end of the game when the other two layers are ruined. We denote the event that Player X gains a oint by G X, and the event that Player X loses a oint by L X, where X = A, B, or C. Then we define the four ossibilities on each lay as := P G A and L C = 1 2, q := P L A and G C = q 1 q 2, 61 r := P G B and L C = q 1 2, s := P L B and G C = 1 q 2. 5
6 Let x and y be the total scores for A and B resectively. Player A is ruined when x = 0; Player B is ruined when y = 0; Player C is ruined when x + y = k. Therefore, the first ortion of the game, that is, the three layer ortion before one layer is ruined, can be described as a two dimensional random walk within the triangle bounded by x = 0, y = 0, and x+y = k. Once one of the boundaries of the triangle is reached, the three layer ortion of the game ends and the game becomes a standard two layer Gambler s Ruin Problem. This two dimensional walk is illustrated in Figure ,k B wins C ruined A ruined r q * s 0,0 C wins B ruined k,0 A wins Figure 61. Reresentation of the two dimensional random walk created by the three layer game. The oint may move u, down, left, or right according to the robabilities given. As Barnett describes in [1, ], there are six ossible ways for the game to lay out: any one of the three layers may be ruined first, and then either of the two remaining layers may be ruined, leaving a single victorious layer. Thus, each layer has two ways of winning. For examle, A may win by first ruining B in the three layer game, and then ruining C in the two layer game or by ruining C in the three layer game, and then ruining B in the two layer game. Informally, this organization of the game is described as: 6
7 P A wins = P B is ruined first P A goes on to ruin C + P C is ruined first P A goes on to ruin B, P B wins = P A is ruined first P B goes on to ruin C P C is ruined first P B goes on to ruin A, P C wins = P A is ruined first P C goes on to ruin B + P B is ruined first P C goes on to ruin A. Converting these equations into more useful ones, however, is more difficult than it first sounds. The robability of a layer winning the two layer ortion of the game deends on the starting number of oints of the two layer ortion, that is, the ending number of oints of the three layer ortion. Therefore, we must sum over all the ossible scores for the start of the two layer ortion of the game. Define the function ux, y to be the robability that the three layer ortion of the game ends at the oint x, y. Thus, we combine Equation 36 with Equations 62 and simlify to get the robabilites that A, B, or C resectively win the game: P A = P B = P C = k 2 q 2 2 n 1 k 1 q 1 q 2 n=1 2 k un, n 1 1 q 1 n=1 1 k un, k n, 1 k 2 q 2 2 n 1 k 1 1 q q 2 n=1 2 k u0, n + 1 k n n=1 q 1 k un, k n, k 2 2 q 2 k n 1 un, u0, n. q 2 k 1 n=1 Barnett continues to show that we can break down the robability of ultimate victory for each layer by defining a better ux, y. Define E i,j x, y to be the average number of times the two dimensional walk starting at i, j leaves the oint x, y. Then we see that the ux, y that we used in defining the robabilities is written as follows: un, 0 = se i,j n, 1 for n = 1, 2,..., k 2; u0, n = qe i,j 1, n for n = 1, 2,..., k 2; 64 un, k n = re i,j n, k n 1 + E i,j n 1, k n for n = 1, 2,..., k 1. Finally, we can ut Equations 64 into Equations 63 to obtain formulas for P A, P B, and P C. 7
8 References [1] Barnett, V. D., A threelayer extension of the gambler s ruin roblem, Journal of Alied Probability , [2] DeGroot, Morris H., Probability and Statistics, AddisonWesley Publishing Co., [3] Edwards, A. W. F., Pascal s roblem: the gambler s ruin, International Statistical Review , [4] Kaigh, W. D., An attrition roblem of gambler s ruin, Mathematics Magazine ,
6.042/18.062J Mathematics for Computer Science December 12, 2006 Tom Leighton and Ronitt Rubinfeld. Random Walks
6.042/8.062J Mathematics for Comuter Science December 2, 2006 Tom Leighton and Ronitt Rubinfeld Lecture Notes Random Walks Gambler s Ruin Today we re going to talk about onedimensional random walks. In
More informationStat 134 Fall 2011: Gambler s ruin
Stat 134 Fall 2011: Gambler s ruin Michael Lugo Setember 12, 2011 In class today I talked about the roblem of gambler s ruin but there wasn t enough time to do it roerly. I fear I may have confused some
More information1 Gambler s Ruin Problem
Coyright c 2009 by Karl Sigman 1 Gambler s Ruin Problem Let N 2 be an integer and let 1 i N 1. Consider a gambler who starts with an initial fortune of $i and then on each successive gamble either wins
More informationPythagorean Triples and Rational Points on the Unit Circle
Pythagorean Triles and Rational Points on the Unit Circle Solutions Below are samle solutions to the roblems osed. You may find that your solutions are different in form and you may have found atterns
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 information3.2 Roulette and Markov Chains
238 CHAPTER 3. DISCRETE DYNAMICAL SYSTEMS WITH MANY VARIABLES 3.2 Roulette and Markov Chains In this section we will be discussing an application of systems of recursion equations called Markov Chains.
More informationOPTIMAL EXCHANGE BETTING STRATEGY FOR WINDRAWLOSS MARKETS
OPTIA EXCHANGE ETTING STRATEGY FOR WINDRAWOSS ARKETS Darren O Shaughnessy a,b a Ranking Software, elbourne b Corresonding author: darren@rankingsoftware.com Abstract Since the etfair betting exchange
More informationThe fast Fourier transform method for the valuation of European style options inthemoney (ITM), atthemoney (ATM) and outofthemoney (OTM)
Comutational and Alied Mathematics Journal 15; 1(1: 16 Published online January, 15 (htt://www.aascit.org/ournal/cam he fast Fourier transform method for the valuation of Euroean style otions inthemoney
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 informationCombinatorics: The Fine Art of Counting
Combinatorics: The Fine Art of Counting Week 7 Lecture Notes Discrete Probability Continued Note Binomial coefficients are written horizontally. The symbol ~ is used to mean approximately equal. The Bernoulli
More informationPRIME NUMBERS AND THE RIEMANN HYPOTHESIS
PRIME NUMBERS AND THE RIEMANN HYPOTHESIS CARL ERICKSON This minicourse has two main goals. The first is to carefully define the Riemann zeta function and exlain how it is connected with the rime numbers.
More informationComplex Conjugation and Polynomial Factorization
Comlex Conjugation and Polynomial Factorization Dave L. Renfro Summer 2004 Central Michigan University I. The Remainder Theorem Let P (x) be a olynomial with comlex coe cients 1 and r be a comlex number.
More informationThe Cubic Formula. The quadratic formula tells us the roots of a quadratic polynomial, a polynomial of the form ax 2 + bx + c. The roots (if b 2 b+
The Cubic Formula The quadratic formula tells us the roots of a quadratic olynomial, a olynomial of the form ax + bx + c. The roots (if b b+ 4ac 0) are b 4ac a and b b 4ac a. The cubic formula tells us
More informationCBus Voltage Calculation
D E S I G N E R N O T E S CBus Voltage Calculation Designer note number: 3121256 Designer: Darren Snodgrass Contact Person: Darren Snodgrass Aroved: Date: Synosis: The guidelines used by installers
More informationCoordinate Transformation
Coordinate Transformation Coordinate Transformations In this chater, we exlore maings where a maing is a function that "mas" one set to another, usually in a way that reserves at least some of the underlyign
More informationDouble Integrals in Polar Coordinates
Double Integrals in Polar Coorinates Part : The Area Di erential in Polar Coorinates We can also aly the change of variable formula to the olar coorinate transformation x = r cos () ; y = r sin () However,
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 informationRisk and Return. Sample chapter. e r t u i o p a s d f CHAPTER CONTENTS LEARNING OBJECTIVES. Chapter 7
Chater 7 Risk and Return LEARNING OBJECTIVES After studying this chater you should be able to: e r t u i o a s d f understand how return and risk are defined and measured understand the concet of risk
More informationStat 20: Intro to Probability and Statistics
Stat 20: Intro to Probability and Statistics Lecture 16: More Box Models Tessa L. ChildersDay UC Berkeley 22 July 2014 By the end of this lecture... You will be able to: Determine what we expect the sum
More informationHOMEWORK (due Fri, Nov 19): Chapter 12: #62, 83, 101
Today: Section 2.2, Lesson 3: What can go wrong with hyothesis testing Section 2.4: Hyothesis tests for difference in two roortions ANNOUNCEMENTS: No discussion today. Check your grades on eee and notify
More informationStatistics and Random Variables. Math 425 Introduction to Probability Lecture 14. Finite valued Random Variables. Expectation defined
Expectation Statistics and Random Variables Math 425 Introduction to Probability Lecture 4 Kenneth Harris kaharri@umich.edu Department of Mathematics University of Michigan February 9, 2009 When a large
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 informationEE302 Division 1 Homework 2 Solutions.
EE302 Division Homework 2 Solutions. Problem. Prof. Pollak is flying from L to Paris with two lane changes, in New York and London. The robability to lose a iece of luggage is the same,, in L, NY, and
More informationExamples of infinite sample spaces. Math 425 Introduction to Probability Lecture 12. Example of coin tosses. Axiom 3 Strong form
Infinite Discrete Sample Spaces s of infinite sample spaces Math 425 Introduction to Probability Lecture 2 Kenneth Harris kaharri@umich.edu Department of Mathematics University of Michigan February 4,
More informationElementary Statistics and Inference. Elementary Statistics and Inference. 16 The Law of Averages (cont.) 22S:025 or 7P:025.
Elementary Statistics and Inference 22S:025 or 7P:025 Lecture 20 1 Elementary Statistics and Inference 22S:025 or 7P:025 Chapter 16 (cont.) 2 D. Making a Box Model Key Questions regarding box What numbers
More informationRISKy Business: An InDepth Look at the Game RISK
RISKy Business: An InDepth Look at the Game RISK Sharon Blatt Advisor: Dr. Crista Coles Elon University Elon, NC 744 slblatt@hotmail.com Introduction Games have been of interest to mathematicians for
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 informationSQUARE GRID POINTS COVERAGED BY CONNECTED SOURCES WITH COVERAGE RADIUS OF ONE ON A TWODIMENSIONAL GRID
International Journal of Comuter Science & Information Technology (IJCSIT) Vol 6, No 4, August 014 SQUARE GRID POINTS COVERAGED BY CONNECTED SOURCES WITH COVERAGE RADIUS OF ONE ON A TWODIMENSIONAL GRID
More informationlecture 25: Gaussian quadrature: nodes, weights; examples; extensions
38 lecture 25: Gaussian quadrature: nodes, weights; examles; extensions 3.5 Comuting Gaussian quadrature nodes and weights When first aroaching Gaussian quadrature, the comlicated characterization of the
More informationP (A) = lim P (A) = N(A)/N,
1.1 Probability, Relative Frequency and Classical Definition. Probability is the study of random or nondeterministic experiments. Suppose an experiment can be repeated any number of times, so that we
More informationFind Probabilities Using Permutations. Count permutations. Choices for 1st letter
13.2 Find Probabilities Using Permutations Before You used the counting rincile. Now You will use the formula for the number of ermutations. Why? So you can find the number of ossible arrangements, as
More informationA Simple Model of Pricing, Markups and Market. Power Under Demand Fluctuations
A Simle Model of Pricing, Markus and Market Power Under Demand Fluctuations Stanley S. Reynolds Deartment of Economics; University of Arizona; Tucson, AZ 85721 Bart J. Wilson Economic Science Laboratory;
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 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 informationFrequentist vs. Bayesian Statistics
Bayes Theorem Frequentist vs. Bayesian Statistics Common situation in science: We have some data and we want to know the true hysical law describing it. We want to come u with a model that fits the data.
More informationConfidence Intervals for CaptureRecapture Data With Matching
Confidence Intervals for CatureRecature Data With Matching Executive summary Caturerecature data is often used to estimate oulations The classical alication for animal oulations is to take two samles
More informationPrice Elasticity of Demand MATH 104 and MATH 184 Mark Mac Lean (with assistance from Patrick Chan) 2011W
Price Elasticity of Demand MATH 104 and MATH 184 Mark Mac Lean (with assistance from Patrick Chan) 2011W The rice elasticity of demand (which is often shortened to demand elasticity) is defined to be 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 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 informationChapter 2: Systems of Linear Equations and Matrices:
At the end of the lesson, you should be able to: Chapter 2: Systems of Linear Equations and Matrices: 2.1: Solutions of Linear Systems by the Echelon Method Define linear systems, unique solution, inconsistent,
More information36 Odds, Expected Value, and Conditional Probability
36 Odds, Expected Value, and Conditional Probability What s the difference between probabilities and odds? To answer this question, let s consider a game that involves rolling a die. If one gets the face
More informationComputational Finance The Martingale Measure and Pricing of Derivatives
1 The Martingale Measure 1 Comutational Finance The Martingale Measure and Pricing of Derivatives 1 The Martingale Measure The Martingale measure or the Risk Neutral robabilities are a fundamental concet
More informationBinomial Random Variables. Binomial Distribution. Examples of Binomial Random Variables. Binomial Random Variables
Binomial Random Variables Binomial Distribution Dr. Tom Ilvento FREC 8 In many cases the resonses to an exeriment are dichotomous Yes/No Alive/Dead Suort/Don t Suort Binomial Random Variables When our
More informationQuestion 1 Formatted: Formatted: Formatted: Formatted:
In many situations in life, we are presented with opportunities to evaluate probabilities of events occurring and make judgments and decisions from this information. In this paper, we will explore four
More informationMonitoring Frequency of Change By Li Qin
Monitoring Frequency of Change By Li Qin Abstract Control charts are widely used in rocess monitoring roblems. This aer gives a brief review of control charts for monitoring a roortion and some initial
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 informationModule 5: Multiple Random Variables. Lecture 1: Joint Probability Distribution
Module 5: Multile Random Variables Lecture 1: Joint Probabilit Distribution 1. Introduction irst lecture o this module resents the joint distribution unctions o multile (both discrete and continuous) random
More informationThe Graphical Method. Lecture 1
References: Anderson, Sweeney, Williams: An Introduction to Management Science  quantitative aroaches to decision maing 7 th ed Hamdy A Taha: Oerations Research, An Introduction 5 th ed Daellenbach, George,
More informationSynopsys RURAL ELECTRICATION PLANNING SOFTWARE (LAPER) Rainer Fronius Marc Gratton Electricité de France Research and Development FRANCE
RURAL ELECTRICATION PLANNING SOFTWARE (LAPER) Rainer Fronius Marc Gratton Electricité de France Research and Develoment FRANCE Synosys There is no doubt left about the benefit of electrication and subsequently
More informationThe Mathematics of Gambling
The Mathematics of Gambling with Related Applications Madhu Advani Stanford University April 12, 2014 Madhu Advani (Stanford University) Mathematics of Gambling April 12, 2014 1 / 23 Gambling Gambling:
More informationIndex Numbers OPTIONAL  II Mathematics for Commerce, Economics and Business INDEX NUMBERS
Index Numbers OPTIONAL  II 38 INDEX NUMBERS Of the imortant statistical devices and techniques, Index Numbers have today become one of the most widely used for judging the ulse of economy, although in
More informationV. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPECTED VALUE
V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPETED VALUE A game of chance featured at an amusement park is played as follows: You pay $ to play. A penny and a nickel are flipped. You win $ if either
More informationSession 8 Probability
Key Terms for This Session Session 8 Probability Previously Introduced frequency New in This Session binomial experiment binomial probability model experimental probability mathematical probability outcome
More informationThe Online Freezetag Problem
The Online Freezetag Problem Mikael Hammar, Bengt J. Nilsson, and Mia Persson Atus Technologies AB, IDEON, SE3 70 Lund, Sweden mikael.hammar@atus.com School of Technology and Society, Malmö University,
More informationFREQUENCIES OF SUCCESSIVE PAIRS OF PRIME RESIDUES
FREQUENCIES OF SUCCESSIVE PAIRS OF PRIME RESIDUES AVNER ASH, LAURA BELTIS, ROBERT GROSS, AND WARREN SINNOTT Abstract. We consider statistical roerties of the sequence of ordered airs obtained by taking
More informationStochastic Derivation of an Integral Equation for Probability Generating Functions
Journal of Informatics and Mathematical Sciences Volume 5 (2013), Number 3,. 157 163 RGN Publications htt://www.rgnublications.com Stochastic Derivation of an Integral Equation for Probability Generating
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 informationGaming the Law of Large Numbers
Gaming the Law of Large Numbers Thomas Hoffman and Bart Snapp July 3, 2012 Many of us view mathematics as a rich and wonderfully elaborate game. In turn, games can be used to illustrate mathematical ideas.
More informationAddition and Subtraction Games
Addition and Subtraction Games Odd or Even (Grades 13) Skills: addition to ten, odd and even or more Materials: each player has cards (Ace=1)10, 2 dice 1) Each player arranges their cards as follows. 1
More informationThe Impact of a Finite Bankroll on an EvenMoney Game
The Impact of a Finite Bankroll on an EvenMoney Game Kelvin Morin Manitoba Lotteries Corporation kelmorin@hotmail.com / morink@mlc.mb.ca 2003 Calculating the average cost of playing a table game is usually
More informationLoglikelihood and Confidence Intervals
Stat 504, Lecture 3 Stat 504, Lecture 3 2 Review (contd.): Loglikelihood and Confidence Intervals The likelihood of the samle is the joint PDF (or PMF) L(θ) = f(x,.., x n; θ) = ny f(x i; θ) i= Review:
More informationWorldwide Casino Consulting Inc.
Card Count Exercises George Joseph The first step in the study of card counting is the recognition of those groups of cards known as Plus, Minus & Zero. It is important to understand that the House has
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 informationCard Games. ***All card games require the teacher to take out the face cards except aces. The aces have a value of one.
Card Games ***All card games require the teacher to take out the face cards except aces. The aces have a value of one. Salute: (Multiplication or Addition) Three students are needed to play this game.
More information(This result should be familiar, since if the probability to remain in a state is 1 p, then the average number of steps to leave the state is
How many coin flis on average does it take to get n consecutive heads? 1 The rocess of fliing n consecutive heads can be described by a Markov chain in which the states corresond to the number of consecutive
More informationChapter 4 Lecture Notes
Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a realvalued function defined on the sample space of some experiment. For instance,
More informationAn important observation in supply chain management, known as the bullwhip effect,
Quantifying the Bullwhi Effect in a Simle Suly Chain: The Imact of Forecasting, Lead Times, and Information Frank Chen Zvi Drezner Jennifer K. Ryan David SimchiLevi Decision Sciences Deartment, National
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 informationLarge firms and heterogeneity: the structure of trade and industry under oligopoly
Large firms and heterogeneity: the structure of trade and industry under oligooly Eddy Bekkers University of Linz Joseh Francois University of Linz & CEPR (London) ABSTRACT: We develo a model of trade
More informationMathematical goals. Starting points. Materials required. Time needed
Level S2 of challenge: B/C S2 Mathematical goals Starting points Materials required Time needed Evaluating probability statements To help learners to: discuss and clarify some common misconceptions about
More informationSituation Based Strategic Positioning for Coordinating a Team of Homogeneous Agents
Situation Based Strategic Positioning for Coordinating a Team of Homogeneous Agents Luís Paulo Reis, Nuno Lau 2 and Eugénio Costa Oliveira LIACC Artificial Intelligence and Comuter Science Lab., University
More information(SEE IF YOU KNOW THE TRUTH ABOUT GAMBLING)
(SEE IF YOU KNOW THE TRUTH ABOUT GAMBLING) Casinos loosen the slot machines at the entrance to attract players. FACT: This is an urban myth. All modern slot machines are stateoftheart and controlled
More information$2 4 40 + ( $1) = 40
THE EXPECTED VALUE FOR THE SUM OF THE DRAWS In the game of Keno there are 80 balls, numbered 1 through 80. On each play, the casino chooses 20 balls at random without replacement. Suppose you bet on the
More informationThe MagnusDerek Game
The MagnusDerek Game Z. Nedev S. Muthukrishnan Abstract We introduce a new combinatorial game between two layers: Magnus and Derek. Initially, a token is laced at osition 0 on a round table with n ositions.
More informationPOISSON PROCESSES. Chapter 2. 2.1 Introduction. 2.1.1 Arrival processes
Chater 2 POISSON PROCESSES 2.1 Introduction A Poisson rocess is a simle and widely used stochastic rocess for modeling the times at which arrivals enter a system. It is in many ways the continuoustime
More informationExpectations. Expectations. (See also Hays, Appendix B; Harnett, ch. 3).
Expectations Expectations. (See also Hays, Appendix B; Harnett, ch. 3). A. The expected value of a random variable is the arithmetic mean of that variable, i.e. E() = µ. As Hays notes, the idea of the
More informationKINDERGARTEN Practice addition facts to 5 and related subtraction facts. 1 ST GRADE Practice addition facts to 10 and related subtraction facts.
MATH There are many activities parents can involve their children in that are math related. Children of all ages can always practice their math facts (addition, subtraction, multiplication, division) in
More informationCABRS CELLULAR AUTOMATON BASED MRI BRAIN SEGMENTATION
XI Conference "Medical Informatics & Technologies"  2006 Rafał Henryk KARTASZYŃSKI *, Paweł MIKOŁAJCZAK ** MRI brain segmentation, CT tissue segmentation, Cellular Automaton, image rocessing, medical
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 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 informationFlying Things. Preparation and Materials. Planning chart. Using This Activity. 174 Activity 20 Flying Things The Math Explorer.
Leader Overview ACTIVITY 20 Flying Things Making and testing aer airlanes is great fun and will burn off some energy on a rainy day. In this activity, math is used to make a aer airlane contest fair and
More informationUniversiteitUtrecht. Department. of Mathematics. Optimal a priori error bounds for the. RayleighRitz method
UniversiteitUtrecht * Deartment of Mathematics Otimal a riori error bounds for the RayleighRitz method by Gerard L.G. Sleijen, Jaser van den Eshof, and Paul Smit Prerint nr. 1160 Setember, 2000 OPTIMAL
More informationHey, That s Not Fair! (Or is it?)
Concept Probability and statistics Number sense Activity 9 Hey, That s Not Fair! (Or is it?) Students will use the calculator to simulate dice rolls to play two different games. They will decide if the
More informationIEEM 101: Inventory control
IEEM 101: Inventory control Outline of this series of lectures: 1. Definition of inventory. Examles of where inventory can imrove things in a system 3. Deterministic Inventory Models 3.1. Continuous review:
More informationSOME PROPERTIES OF EXTENSIONS OF SMALL DEGREE OVER Q. 1. Quadratic Extensions
SOME PROPERTIES OF EXTENSIONS OF SMALL DEGREE OVER Q TREVOR ARNOLD Abstract This aer demonstrates a few characteristics of finite extensions of small degree over the rational numbers Q It comrises attemts
More informationChapter 3. Special Techniques for Calculating Potentials. r ( r ' )dt ' ( ) 2
Chater 3. Secial Techniues for Calculating Potentials Given a stationary charge distribution r( r ) we can, in rincile, calculate the electric field: E ( r ) Ú Dˆ r Dr r ( r ' )dt ' 2 where Dr r 'r. This
More informationProbability and Statistics
CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b  0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute  Systems and Modeling GIGA  Bioinformatics ULg kristel.vansteen@ulg.ac.be
More informationSoftmax Model as Generalization upon Logistic Discrimination Suffers from Overfitting
Journal of Data Science 12(2014),563574 Softmax Model as Generalization uon Logistic Discrimination Suffers from Overfitting F. Mohammadi Basatini 1 and Rahim Chiniardaz 2 1 Deartment of Statistics, Shoushtar
More informationEffect Sizes Based on Means
CHAPTER 4 Effect Sizes Based on Means Introduction Raw (unstardized) mean difference D Stardized mean difference, d g Resonse ratios INTRODUCTION When the studies reort means stard deviations, the referred
More informationStochastic Processes and Advanced Mathematical Finance. Ruin Probabilities
Steven R. Dunbar Department of Mathematics 203 Avery Hall University of NebraskaLincoln Lincoln, NE 685880130 http://www.math.unl.edu Voice: 4024723731 Fax: 4024728466 Stochastic Processes and Advanced
More informationDiscrete Math I Practice Problems for Exam I
Discrete Math I Practice Problems for Exam I The ucoming exam on Thursday, January 12 will cover the material in Sections 1 through 6 of Chater 1. There may also be one question from Section 7. If there
More informationStability Improvements of Robot Control by Periodic Variation of the Gain Parameters
Proceedings of the th World Congress in Mechanism and Machine Science ril ~4, 4, ianin, China China Machinery Press, edited by ian Huang. 868 Stability Imrovements of Robot Control by Periodic Variation
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 informationMath 728 Lesson Plan
Math 728 Lesson Plan Tatsiana Maskalevich January 27, 2011 Topic: Probability involving sampling without replacement and dependent trials. Grade Level: 812 Objective: Compute the probability of winning
More informationIntroduction to NPCompleteness Written and copyright c by Jie Wang 1
91.502 Foundations of Comuter Science 1 Introduction to Written and coyright c by Jie Wang 1 We use timebounded (deterministic and nondeterministic) Turing machines to study comutational comlexity of
More informationFirst Law, Heat Capacity, Latent Heat and Enthalpy
First Law, Heat Caacity, Latent Heat and Enthaly Stehen R. Addison January 29, 2003 Introduction In this section, we introduce the first law of thermodynamics and examine sign conentions. Heat and Work
More informationJoint Distributions. Lecture 5. Probability & Statistics in Engineering. 0909.400.01 / 0909.400.02 Dr. P. s Clinic Consultant Module in.
3σ σ σ +σ +σ +3σ Joint Distributions Lecture 5 0909.400.01 / 0909.400.0 Dr. P. s Clinic Consultant Module in Probabilit & Statistics in Engineering Toda in P&S 3σ σ σ +σ +σ +3σ Dealing with multile
More informationMath Board Games. For School or Home Education. by Teresa Evans. Copyright 2005 Teresa Evans. All rights reserved.
Math Board Games For School or Home Education by Teresa Evans Copyright 2005 Teresa Evans. All rights reserved. Permission is given for the making of copies for use in the home or classroom of the purchaser
More informationStochastic Processes and Advanced Mathematical Finance. Duration of the Gambler s Ruin
Steven R. Dunbar Department of Mathematics 203 Avery Hall University of NebraskaLincoln Lincoln, NE 685880130 http://www.math.unl.edu Voice: 4024723731 Fax: 4024728466 Stochastic Processes and Advanced
More information6.3 Conditional Probability and Independence
222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted
More information