1 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 problems, and the gaming industry remains a source of interesting applications of many of the methods taught in beginning probability courses. The use of examples from gambling can enhance the interest and usefulness of courses in probability, and equip students to participate intelligently in this form of entertainment. This paper explores how the many statistical functions of Microsoft Excel and utilities such as solver can be used to solve gambling problems. We use a simulation of roulette as a vehicle for experimentation; introduce means as a measure of house percentage; variance as a measure of volatility; Hypergeometric and binomial distributions to calculate chances of winning single bets and of being ahead after many bets; and use solver to balance a sports book. The use of Excel allows key ideas in probability to be explored through the investigation of realistic examples. Introduction Probability first arose as an area of study through gambling problems. Gamblers were interested in how to divide the stake when a game was interrupted. Gambling is now a huge industry, with total gambling losses in Victoria, Australia in reaching $3.19 billion. The gaming industry remains a source of interesting applications of many of the methods taught in beginning probability and statistics courses. The use of examples from gambling can enhance the interest and usefulness of statistics courses. Croucher (2000) discusses an introductory course in statistics based solely around gambling and sport. Gambling is becoming pervasive in our society, and an understanding of its principles will assist students to participate sensibly in this form of entertainment, either as providers or consumers. Interest can be heightened if the tedium of calculations can be reduced, and real examples used. Microsoft Excel is a powerful package with many built-in functions that can be used in statistics. This paper explores some statistical examples related to gambling easily implemented within Microsoft Excel. Gambling is an excellent vehicle to introduce the concepts of probability. The three usual definitions of probability rely on a 'mathematical' definition, based on equally likely outcomes, an 'experimental' definition based on relative frequency of occurrence in repeated trials, and a 'subjective' definition based on degree of belief. Most casino games can be approached via the first, long term profits to the casino or gambler via the second, and most sports betting via the third.
2 Simulating Roulette One of the simplest gambling games is Roulette. A roulette wheel has the numbers 1 to 36 with half coloured red and half black, in addition to a green zero (and sometimes a double zero). Various bets such as a straight, split, square, line, street, column, dozen, odd or even, red or black, high or low, can be laid on groups of numbers. Payouts are made as if there are 36 numbers, when in fact there are 37 (or 38 in a double zero wheel). So for example if $1 is bet on a single number and that comes up, the casino pays 35 to1. Thus the punter wins 35 dollars plus his own dollar back. If a $1 bet is made on the High numbers (19-36), and any of those numbers comes up, the casino pays 1 to1 (18 to 18) or even money. Thus the punter wins 1 dollar plus his own dollar back. The site has some information on roulette wheels and a roulette wheel you can operate for no cost. A roulette wheel is an excellent way to introduce the frequency definition of probability - the probability of an event is the number of outcomes in the event as a proportion of the total number of (equally likely) outcomes. The probability of all bets in roulette can be calculated by simple counting. Thus a bet on the High numbers (19-36) has 18 outcomes, so the probability of the High bet winning is 18/37. A square has four outcomes, so the probability is 4/37. Insights can be gained into many gaming situations by simulation. This is often the choice when a mathematical analysis is too difficult. However it also gives a vehicle whereby a student can experiment, and gain experience of the long run, without actually putting large sums of money at risk. Excel can be used to simulate a sequence of bets. Students can gain some practical experience of the behavior of random events before discussing issues that arise from that experience. We illustrate with roulette. This example is to create a simple simulation of a roulette wheel using Excel. It is too complicated to simulate allowing a full range of bets, but some simple ones can be easily implemented. In this case we will simulate betting the same size bet continually on a single number, and on a set of numbers. Students may be able to generalize depending on their Excel skills. Below is a set of instructions as issued to students for simulating roulette. 1. In Cell A1, type your favourite number between 0 and 36. This is the number you will bet on. 2. ColumnC: Head Bet number, and generate the sequence 1,2,3...in the column. 3. ColumnD: Head Winning Number, and use the formula RANDBETWEEN(0,36) to generate the number appearing on the roulette wheel. You may need to install the Analysis ToolPak to run this function. 4. Column E: Head Profit, and use an if statement to generate 1 or 35 depending on whether your number came up. 5. Column F: Head Cumulative Profit, and total profit up to now. 6. Copy Formulas down to allow for a betting sequence of 1000 bets. 7. Place in cell A4 your percentage profit/loss, in A6 the largest profit, and in A8 the largest loss. 8. Use the graph wizard to do a scatter plot of the cumulative profit against bet number. Place the graph at the top right of the spreadsheet. Comment on your graph. 9. Repeat several simulations by hitting F Add two more columns to allow betting on the even money bet High. Drag the profit column on to your graph. Place in column B the relevant statistics for this bet. Is the even money bet more or less volatile (risky) than the bet on a single number?
3 $ $ $ $ Figure 1. Four simulations of cumulative profit on 1000 bets on a single number and red in roulette. Figure 1 shows four examples of graphs produced by the above sheet. It is difficult to show 'Typical' graphs as they vary so widely. Students will quickly see the volatility of profit in the short term. Repeatedly hitting F9 gives a surprising variation in outcomes. Runs of good and bad luck characterize the graphs. There are large fluctuations, and even after 1000 bets, a player might be in front. It is obvious that the 'even money' bet is far less volatile than the bet on a single number, and there are many questions that arise naturally out of such simulations. Does a gambler generally win or lose? What is the chance he is ahead after a certain number of spins? What is the chance of winning/losing a given amount? How long does it take to win/lose a given amount? What is overall profit in the long run? Students could investigate these by repeating the above simulation many times, and recording the final profit as given in cell A4, or some other statistic each time. Again Excel has facilities for performing this automatically. For example, suppose we wish to estimate not only the final amount a player is ahead after 1000 bets, but the maximum amount a player is ahead and behind at any stage in the series of 1000 bets. The data table facility of Excel allows us to simulate the whole sequence of 1000 bets 100 times. The output from one of these sequences showed that betting on 'high' gave an average profit of - $24.70, with an average maximum ahead of $15.40 and average maximum behind of $ Betting on a single number produced -$27.60, $ and $ Such simulations clearly
4 demonstrate that while both bets lose about 2.5%, the number bet is the more volatile (exciting? risky? variable?). By experimenting with such a simulation, students may come up with their own problems, which might then be investigated via traditional probability calculations. House percentage and volatility A random variable is one whose value depends on a random experiment, and probability courses investigate random variables and their expected values. In gambling, as we saw in the above simulation, a random variable of particular interest is the amount returned to a punter or profit made on a bet. The expected profit made by the operator on a $1 bet is the house percentage, and can be considered as the long run price the gambler pays for playing the game. Just as different loans carry different interest rates, the games in a casino can vary markedly in their cost to play (from virtually zero to 40% or 50%). Yet many gamblers are unaware of the house percentages associated with different games or how to calculate them. Consider a bet of $1 on number 23 at Roulette, which pays 35 to 1 if it wins. With probability 1/37 a player will win and receive $36 (his original bet plus another $35) and with probability 36/37 the player loses and receives nothing. A simple table in Excel can be made with the various outcomes and their respective chances. Table 1: Return for betting on a single number (23) in roulette. Outcome 23 Not 23 Return 36 0 Total Probability 1/37 36/37 1 Probability x return 36/ /37 Of interest is the proportion of revenue that is returned to the punter via the various outcomes, which is obtained by calculating the probability of each return times the amount of the return. In this case, there is only one winning outcome, but in many other games, such as lotteries and poker machines, there are several winning outcomes and games are structured differently to appeal to the psychology of players. For example, a lottery may return a high proportion of the prize pool via its main or division 1 prize, whereas a poker machine may return a very low proportion of the amount bet via its biggest jackpot. Similarly, different poker machines may appeal to conservative or reckless gamblers by altering the proportion of stake returned in often occurring small prizes and rarely occurring large prizes. The sum of these proportions gives us the mean or expected value of the return. In the above case this is 36/37. Thus the expected return to a gambler betting on a single number in roulette is 36/37, or about 97.3%. Thus 2.7% is the house percentage - the casino makes on average nearly 3% of everything that is bet on a number. In fact this is the house percentage on virtually every bet at roulette.
5 There are various alternatives for setting up the table in Excel. It is usually easier to use columns rather than rows to list the various outcomes. If you install the Add-in Analysis VBA toolpack, the sumproduct function allows you to calculate the expected value directly, without calculating the individual values of return times probability. The table could also be calculated using the profit rather than return to the punter. In this case for a $1 bet the punter wins $35 with probability 1/37, and -$1 with probability 36/37, for an expected profit of -$1/37. So the gambler loses on average 2.7%. For a double zero wheel, as is common in the USA, the house percentage doubles to 5.4%. An alternative to the approach taken in Table 1, which is just as easy to implement in Excel and a bit more flexible, is to list all the individual outcomes from 0 to 36. These all have probability 1/37, and the returns for various bets could be listed. So for a bet on a square, for example, each of the four numbers in the square would return $9 if they came up. Another row could be introduced to allow bets of different sizes, and the total bet and total expected return calculated. This would allow various combinations of bets to be entered, and Excel would immediately give the expected return. For example, there are various systems promulgated, where people bet on several bets simultaneously. These could easily be tested, and shown to have the same expected return of 97.3%. Once such a sheet was constructed, it could easily be converted into a roulette wheel simulation. The randbetween function is used to generate a number between 0 and 36, and if statements are used to convert the probabilities into 1 or 0 depending on whether the outcome matches the generated number. The spreadsheets can also be modified to calculate the variance of random variables, as shown in Table 2. The SQRT function is used to calculate the standard deviation. Table 2: Mean and standard deviation of return X for a $1 bet on a single number x P(X=x) xp(x=x) (x-µ) 2 (x-µ) 2 P(X=x) Std Dev Variance Total Note that with careful planning, once a spreadsheet has been created, it can be reused for another problem. Consider the bet on 'High'. This has a return of $2 with probability 18/37, and 0 with probability 19/37. Entering these values in the first and second column of the above sheet, we automatically get the mean and standard deviation of the return for this bet. This is shown in Table 3, where we have used the alternative calculation formula for variance. Table 3: Mean and standard deviation of return X for a $1 bet on 'high' x P(X=x) xp(x=x) x 2 x 2 P(X=x) Std Dev Variance Total
6 Note that although this bet has the same mean return, the standard deviation is about one sixth of the single number bet. Thus the single number bet is 5.8 times more variable, or more volatile, or 5.8 times more exciting for the gambler and 5.8 times more risky for the operator. (That is why maximum bets on numbers are usually less than the even money bets). A gambler who each week had 100 bets on a number, and kept a record of his returns, would find they were about 6 times more variable than a similar gambler who had 100 bets on 'High' or Red each week. Spreadsheets similar in design to the above can be used to calculate the mean and variance of return for poker machines. Keno Excel has many functions which can be used to calculate probabilities. Two that are often useful in gambling applications are the Hypergeometric and binomial functions. As an example we can look at Keno. There are many versions of Keno, but typically players select from 1 up to 15 numbers from the numbers 1 to 80. Prizes depend on the number of hits in 20 numbers selected randomly from 80. The list of prizes for for tattersalls keno can be found by following the links on the website and they even give relevant formula and chance of winning (but not the house percentage). For example, for a Spot 8 Keno, the prize for matching all 8 numbers is $25000, 7 numbers is $1200 and 6 numbers is $90. The chance of winning these prizes is given as 1 in , 1 in 6,232 and 1 in 422. These Keno tables form a good example to apply the method described above, and the HYPERGEOMETIC function of Excel can be used to calculate the probabilities. Table 4 gives the table produced for the above game. The formula for the probability for matching 8 cell is HYPGEOMDIST(B5,8,20,80), where B5 is the address to the cell on the left. Taking the inverse of the probability figures confirms the chances given in the website. Note the return is a low 51% - in many cases, the returns in Keno are very low. If the game parameters (number of balls in population, number selected by operator, number of balls selected by player) are entered in separate cells, it is a very simple manner to set up a spreadsheet which will work for all Kenos. The payoffs for the various matches then have to be entered and Excel automatically calculates the chances and returns. Table 4. Payoff table and probabilities for Keno spot 8 Matches Probability Payoff Probability x Payoff 8 4.3E Total
7 Repeated trials While the house percentage gives the long term average cost of playing a game (or long term profit, from the viewpoint of the operator) we are also interested in the short term. Gamblers usually make a sequence of bets - what are the likely outcomes for the gambler? An operator needs to know the variation about the long term average. It is not much good having a profitable game in the long run, if they go broke in the short term. If the binomial distribution is covered, the BINOMDIST function of Excel can be used to calculate the chances of being in any position after any number of bets. For example betting on an even money bet, we will be level or behind after 1000 bets if we have won 500 or fewer bets. BINOMDIST(500,1000,18/37,1) gives our chance to be 0.81, so we only have a 19% chance of coming out in front after 1000 bets. Betting on a number, we will be behind if 27 bets or fewer win (since we have returned $36 for each winning bet). BINOMDIST(27,1000,1/37,1) gives.55, so using this strategy we have a 45% chance of being ahead. As discussed before, these two bets lose the same average amount in the long run - over many simulations they would each lose on average 1000/37 = $27, but the second is more volatile - when you win, you win more, when you lose, you lose more. This could lead to a discussion of variance. Since the profit after 1000 bets is the sum of the 1000 profits on each individual bets, it could also lead to a discussion of the central limit theorem and normality. In this case, parabolic error bands, within which the profit will lie with any degree of confidence, could be put on the graphs. Odds and bookmaking In many gaming situations, particularly sports betting, probabilities cannot be calculated but must be estimated using experience. In these cases, the estimated probabilities need to be calculated by working backwards from the prices or odds offered by the operator. A common way of expressing probabilities, particularly popular in gambling, is by odds. Odds of 2 to 1 against a particular event mean there are 2 chances of the event not occurring and 1 chance of it occurring. Thus the probability of the event is 1/3 or Thus odds of x:y against give a probability of y/(x+y). In a betting context, the first figure is the profit you make if you win (the bookmaker s contribution) and the second is the loss if you lose (the punter s contribution). If you place a bet of $1 at 3 to 1 against, and you win, you get back $4 your original bet of $1 plus the $3 you won. Events with greater than 50% chance are usually expressed as on. Thus odds of 5 to 2 on mean 2 to 5 against or to a probability of 5/7. Conversion of a bookmaker's odds Since a bookmaker hopes to make a profit, his odds do not correspond to probabilities they fail the important criteria of adding to one. In common with Haigh (1999) we will call y/(x+y) the relevant fraction. By adding these up, the amount the relevant fractions exceed one (the overround) can be used to estimate the bookmaker s margin, or percentage.
8 For example, in a sporting contest, odds of 6 to 4 on and 5 to 4 on might be offered on each competitor. This gives relevant fractions of.6 and.56, for a total of 1.16, and an overround of 16%. Thus the bookmaker s margin is approximately 16%, or more correctly 16/116 = 13.8%. In many totalisator systems, the price quoted will be the total amount returned to you for a single $1 bet. This is now common in sports betting and many situations where fixed odds are quoted. A price or payout of $2.50 (for a $1) bet means odds of 1.5 to 1 or 3 to 2, and a 'probability' of 1/2.5. In these cases the house percentage is even easier to calculate. Thus a horse race with 10 runners paying $7, $7, $6, $4, $9, $5, $26, $9, $15, and $21 gives a total relevant fraction of 1/7+1/7+1/6+1/4+1/9+1/5+1/26+1/9+1/15+1/21= 1.28 for a bookmaker's margin of 28/128 = 22% It is a simple exercise to create an Excel spreadsheet that allows the entry of the odds or prices of each possible result in one column, and calculates the relevant fraction (or probabilities in adjacent column) and thus gives the expected return to the bookmaker. Betting markets can be obtained from the Web or the newspaper, and prices entered into the table to find the house percentage The same method can be used on casino games, provided a set of exclusive and exhaustive outcomes are used. For example, consider the outcomes red, black and zero on a roulette wheel. These outcomes return $2, $2, and $36 for a total 1/2+1/2+1/36 = 37/36 = % and a house percentage of 2.8/102.8 = (you guessed it) 2.7%. Once the spreadsheet is constructed, any set of prices can be entered and the house percentage returned immediately. Such a table can also be used to investigate some betting scams. For example, the bookmaker s percentage if the favourite is nobbled (the probability of the favourite winning becomes zero) might become negative. So a punter who knows the favourite cannot win can make a profit in the long run (and with a sensible choice of bets on all outcomes, be certain of winning in the short term). Balancing a book In fact, a bookmaker constructing a book on an event, is not trying to predict the true probabilities of an event occurring, but the proportion of the punters who will bet on a particular outcome. The bookmaker is trying to balance his book, so he will not just make a profit in a long run of events, but will be certain to make a profit on each event no matter what the outcome. This process can also be investigated via a spreadsheet, and some useful insights learnt. To a spreadsheet storing the outcomes and payouts, add a column that takes the amount bet on each outcome, and calculates the bookmakers return if that outcome occurs. Much like a simple profit and loss spreadsheet, this can be compared with the total amount bet to find the profit on each outcome. The MINIMUM function can be used to create a cell that calculates the bookmaker s worst result. A bookmaker tries to steer money away from his worst result (or any result on which he makes a loss) by adjusting the odds and making that bet less attractive. Suppose a bookmaker holds $100 on a race. What is the optimal amount for bookmaker to hold on each outcome? Students might try to adjust the amounts bet to ensure the bookmaker s worst result is as good as possible. How does this amount compare to the expected profit? In fact solver is the perfect tool to apply here. Maximise the worst result, by altering the amounts bet, subject to the constraint that the total bet is $100. In this case students find the optimum bets for each outcome are 100/price, the return for each outcome is exactly the same, and the worst result (in fact common
9 result for all outcomes) for the bookmaker is a profit equal to his house percentage. The book is balanced. A Totalisator produces a balanced book automatically, by a similar process, by determining the payouts after all the bets have been placed. Conclusion The use of gambling applications can enhance the interest and usefulness of probability and statistics courses. Haigh (1999), Orkin (2000) and Croucher (2001) are examples of texts that could be used with this approach. In addition, the pervasiness of gambling means there is no shortage of real life examples that can be analysed. Real and web casinos, and the betting pages in newspapers would be the first port of call. There are many other interesting classical problems in statistics that can be investigated via gambling applications - betting systems, runs, gambler s ruin problem, optimal bet size. Croucher (2000) and Clarke & Bedford (2002) are examples of interesting applications that arise in the press from time to time. The capabilities of Excel make it a useful tool to use in this context, and allows real problems to be analyzed. With thought in setting up the spreadsheet, new events can be analysed with a minimum of structural alteration and data entry. These spreadsheets can be used to give students an appreciation of the principles, and hopefully the pitfalls, of gambling. References Clarke, S. R., & Bedford, A. (2002). Statistics going to the dogs. Teaching Statistics, 24(1), 6-9. Croucher, J. S. (2000). Using probability intervals to evaluate long-term gambling success. Teaching Statistics, 22(2), Croucher, J. S. (2001). The Statistics of Gambling : Understanding Tactics and Risk. North Ryde: Macquarie University Lighthouse Press. Haigh, J. (1999). Taking chances: Winning with Probability. New York: Oxford University Press. Orkin, M. (2000). What are the Odds? Chance in Every day Life. New York: W.H. Freeman and Company.
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
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
(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 state-of-the-art and controlled
REWARD System For Even Money Bet in Roulette By Izak Matatya By even money betting we mean betting on Red or Black, High or Low, Even or Odd, because they pay 1 to 1. With the exception of the green zeros,
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
Lotto Master Formula (v.) The Formula Used By Lottery Winners I. Introduction This book is designed to provide you with all of the knowledge that you will need to be a consistent winner in your local lottery
Expected Value 24 February 2014 Expected Value 24 February 2014 1/19 This week we discuss the notion of expected value and how it applies to probability situations, including the various New Mexico Lottery
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
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
IMPORTANT: This document contains 100% FREE gambling systems designed specifically for ROULETTE, and any casino game that involves even money bets such as BLACKJACK, CRAPS & POKER. Please note although
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
Week 5: Expected value and Betting systems Random variable A random variable represents a measurement in a random experiment. We usually denote random variable with capital letter X, Y,. If S is the sample
STRIKE FORCE ROULETTE Cycles, cycles, cycles... You cannot get away from them in the game of Roulette. Red, black, red, black... Red, red, red, red. Black, black, black... Red, red, black, black... 1st
Full name: Class teacher: Due date: Orange High School Year 7, 05 Mathematics Assignment Instructions All work must be your own. You are encouraged to use the internet but you need to rewrite your findings
Evaluating Trading Systems By John Ehlers and Ric Way INTRODUCTION What is the best way to evaluate the performance of a trading system? Conventional wisdom holds that the best way is to examine the system
STUDENT MODULE 12.1 GAMBLING PAGE 1 Standard 12: The student will explain and evaluate the financial impact and consequences of gambling. Risky Business Simone, Paula, and Randy meet in the library every
An Unconvered Roulette Secret designed to expose any Casino and make YOU MONEY spin after spin! Even in the LONG Run... The UNFAIR Casino advantage put BACK into the HANDS of every Roulette player! INTRODUCTION...2
Law, Probability and Risk (2010) 9, 139 147 Advance Access publication on April 27, 2010 doi:10.1093/lpr/mgq002 Applying the Kelly criterion to lawsuits TRISTAN BARNETT Faculty of Business and Law, Victoria
Learn How to Use The Roulette Layout To Calculate Winning Payoffs For All Straight-up Winning Bets Understand that every square on every street on every roulette layout has a value depending on the bet
IMA Journal of Management Mathematics (2005) 16, 113 120 doi:10.1093/imaman/dpi001 Combining player statistics to predict outcomes of tennis matches TRISTAN BARNETT AND STEPHEN R. CLARKE School of Mathematical
Online Gambling The main forms of online gambling are online wagering and online gaming. Online wagering is comprised of betting on racing (thoroughbred, harness and dog), sports betting (such as the outcome
Martin J. Silverthorne How to Beat Online Roulette! Silverthorne Publications, Inc. How to Beat Online Roulette! COPYRIGHT 2015 Silverthorne Publications Inc. All rights reserved. Except for brief passages
You are about to learn the very best method there is to beat an even-money bet ever devised. This works on almost any game that pays you an equal amount of your wager every time you win. Casino games are
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
Russell Hunter Street Smart Roulette Automatic Bet Tracker! Russell Hunter Publishing, Inc. Street Smart Roulette Automatic Bet Tracker 2015 Russell Hunter and Russell Hunter Publishing, Inc. All Rights
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
MATH 110 Test Three Outline of Test Material EXPECTED VALUE (8.5) Super easy ones (when the PDF is already given to you as a table and all you need to do is multiply down the columns and add across) Example:
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
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
Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a real-valued function defined on the sample space of some experiment. For instance,
THE CHAOS THEORY ROULETTE SYSTEM Please note that all information is provided as is and no guarantees are given whatsoever as to the amount of profit you will make if you use this system. Neither the seller
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.
What constitutes bingo? Advice note, January 2014 1 Summary 1.1 Bingo is a traditional form of gambling that has seen considerable innovation in recent years. It is also the only form of gambling recognised
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
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
Gambling participation: activities and mode of access January 2014 1 Key findings 1.1 The following findings are based on a set of questions commissioned by the Gambling Commission in omnibus surveys conducted
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
The Mathematician s Wastebasket Volume 1, Issue 4 Stephen Devereaux April 28, 2013 Beating Roulette? An analysis with probability and statistics. Every time I watch the film 21, I feel like I ve made the
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
Inside the pokies - player guide 3nd Edition - May 2009 References 1, 2, 3 Productivity Commission 1999, Australia s Gambling Industries, Report No. 10, AusInfo, Canberra. 4 Victorian Department of Justice,
Rules of core casino games in Great Britain June 2011 Contents 1 Introduction 3 2 American Roulette 4 3 Blackjack 5 4 Punto Banco 7 5 Three Card Poker 9 6 Dice/Craps 11 2 1 Introduction 1.1 This document
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,
Introduction to the Rebate on Loss Analyzer Contact: JimKilby@usa.net 702-436-7954 One of the hottest marketing tools used to attract the premium table game customer is the "Rebate on Loss." The rebate
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
GAMING MANAGEMENT SEMINAR SERIES Prices of Casino Games Jason Zhicheng Gao 高 志 成 Casinos-Games-Players Casinos make money by offering games to the players. Games are casinos major products, and players
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
VISUAL GUIDE to RX Scripting for Roulette Xtreme - System Designer 2.0 UX Software - 2009 TABLE OF CONTENTS INTRODUCTION... ii What is this book about?... iii How to use this book... iii Time to start...
If you like a flutter on the horses or any other sport then I would strongly recommend Betfair to place those bets. I find it amazing the number of people still using high street bookmakers which offer
HONORS STATISTICS Mrs. Garrett Block 2 & 3 Tuesday December 4, 2012 1 Daily Agenda 1. Welcome to class 2. Please find folder and take your seat. 3. Review OTL C7#1 4. Notes and practice 7.2 day 1 5. Folders
Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables Discrete vs. continuous random variables Examples of continuous distributions o Uniform o Exponential o Normal Recall: A random
TEACHER GUIDE 12.1 GAMBLING PAGE 1 Standard 12: The student will explain and evaluate the financial impact and consequences of gambling. Risky Business Priority Academic Student Skills Personal Financial
Gambling participation: activities and mode of access July 2013 1 Key findings 1.1 The following findings are based on a set of questions commissioned by the Gambling Commission in omnibus surveys conducted
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
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:
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:
How to Win the Stock Market Game 1 Developing Short-Term Stock Trading Strategies by Vladimir Daragan PART 1 Table of Contents 1. Introduction 2. Comparison of trading strategies 3. Return per trade 4.
Table of Contents Casino Help... 1 Introduction... 1 Introduction... 1 Using This Help Guide... 1 Getting Started... 2 Getting Started... 2 Web Casino... 3 Game Client... 4 To Play the Games... 4 Common
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
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
Strike Rate!! Introduction Firstly, congratulations of becoming an owner of this fantastic selection system. You ll find it difficult to find a system that can produce so many winners at such low liability.
Disclaimer Please make sure you read this disclaimer all the way through and contact us if there is anything you don't understand before you use the contents of this guide. We always recommend that you
CHAPTER 5 Probability Distributions CHAPTER OUTLINE 5.1 Probability Distribution of a Discrete Random Variable 5.2 Mean and Standard Deviation of a Probability Distribution 5.3 The Binomial Distribution
Martin J. Silverthorne Triple Win Roulette A Powerful New Way to Win $4,000 a Day Playing Roulette! Silverthorne Publications, Inc. By Martin J. Silverthorne COPYRIGHT 2011 Silverthorne Publications, Inc.
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,
The Queensland Responsible Gambling Strategy Responsible Gambling Education Unit: Mathematics A & B Outline of the Unit This document is a guide for teachers to the Responsible Gambling Education Unit:
Roulette: On an American roulette wheel here are 38 compartments where the ball can land. They are numbered 1-36, and there are two compartments labeled 0 and 00. Half of the compartments numbered 1-36
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
Executive summary This report presents results from the British Gambling Prevalence Survey (BGPS) 2010. This is the third nationally representative survey of its kind; previous studies were conducted in
Man Vs Bookie The 3 ways to make profit betting on Football Sports Betting can be one of the most exciting and rewarding forms of entertainment and is enjoyed by millions of people around the world. In
EASY MONEY? Compiled by Campbell M Gold (2008) CMG Archives http://campbellmgold.com Important Note The Author in no way recommends, underwrites, or endorses the "money making system" described herein.
Random Fibonacci-type Sequences in Online Gambling Adam Biello, CJ Cacciatore, Logan Thomas Department of Mathematics CSUMS Advisor: Alfa Heryudono Department of Mathematics University of Massachusetts
When Intuition Chapter 18 Differs from Relative Frequency Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc. Thought Question 1: Do you think it likely that anyone will ever win a state lottery
Advice on non-commercial and private gaming and betting November 2012 Contents 1 Introduction 3 2 Defining non-commercial and private gaming and betting 3 3 Non-commercial prize gaming 4 4 Non-commercial
Gambling participation: activities and mode of access January 2015 1 Key findings 1.1 The following findings are based on a set of questions commissioned by the Gambling Commission in omnibus surveys conducted
GLRE-2012-1655-ver9-Auer_1P.3d 04/09/12 2:46pm Page 1 GAMING LAW REVIEW AND ECONOMICS Volume 16, Number 5, 2012 Ó Mary Ann Liebert, Inc. DOI: 10.1089/glre.2012.1655 GLRE-2012-1655-ver9-Auer_1P Type: research-article
William Chon 22600008 Stat 157 Project, Fall 2014 Near Misses in Bingo I. Introduction A near miss is a special kind of failure to reach a goal, in which one is close to being successful. Near misses can
Fixed Odds Betting Terminals and the Code of Practice A report for the Association of British Bookmakers Limited SUMMARY ONLY Europe Economics Chancery House 53-64 Chancery Lane London WC2A 1QU Tel: (+44)
A THEORETICAL ANALYSIS OF THE MECHANISMS OF COMPETITION IN THE GAMBLING MARKET RORY MCSTAY Senior Freshman In this essay, Rory McStay describes the the effects of information asymmetries in the gambling