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# Introduction to Casino Mathematics

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This topic provides a basic introduction to casino mathematics and discusses key formulas that define all casino games. Factors that control or provide avenues for casino revenue management as covered in basic form. This is suitable for new all casino workers who need a quick introduction to the topic.

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• Explain : In this lesson, you will cover the basic job functions of a Dealer, What skills/knowledge and ability a dealer needs to have to perform the job function, and define key performance areas which are used to evaluate the performance of a Dealer. Inform class that a copy of generic Job Description is provided for later reading.
• Tell students: “ This is the generic job description for a Croupier in a casino. Specific tasks are basically conducting casino games where a croupier is assigned to. Apart from dealing the games, Croupiers are required to ensure security of the game and service by following the work procedures. A main aspect of the job is to always ensure customer satisfaction. You may ask the class : “ How can you ensure satisfaction if the player has lost his money ?” Seek some responses from students. To conclude, tell students that the casino’s customer service is not about guaranteed winning, but to make sure the player was provided friendly services even when his losing, ESPECIALLY when losing.
• Tell students: “ This is the generic job description for a Croupier in a casino. Specific tasks are basically conducting casino games where a croupier is assigned to. Apart from dealing the games, Croupiers are required to ensure security of the game and service by following the work procedures. A main aspect of the job is to always ensure customer satisfaction. You may ask the class : “ How can you ensure satisfaction if the player has lost his money ?” Seek some responses from students. To conclude, tell students that the casino’s customer service is not about guaranteed winning, but to make sure the player was provided friendly services even when his losing, ESPECIALLY when losing.
• Tell students: “ This is the generic job description for a Croupier in a casino. Specific tasks are basically conducting casino games where a croupier is assigned to. Apart from dealing the games, Croupiers are required to ensure security of the game and service by following the work procedures. A main aspect of the job is to always ensure customer satisfaction. You may ask the class : “ How can you ensure satisfaction if the player has lost his money ?” Seek some responses from students. To conclude, tell students that the casino’s customer service is not about guaranteed winning, but to make sure the player was provided friendly services even when his losing, ESPECIALLY when losing.
• ### Introduction to Casino Mathematics

2. 2. Casino Mathematics 1 Introduction to Game Expectancy <ul><li>Presented by : Ramachandar Siva </li></ul><ul><li>Duration : 3 hours </li></ul>RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
3. 3. <ul><li>COURSE OBJECTIVES </li></ul><ul><li>To provide basic understanding of the functions of Probabilities & Statistics </li></ul><ul><li>To provide a mathematical explanation of the casino games </li></ul><ul><li>To understand the application of mathematical tools in day to day operations of a casino. </li></ul>Casino Mathematics 1 RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
4. 4. <ul><li>AT THE END OF LESSON STUDENTS WILL </li></ul><ul><li>Understand basics of probabilities and expectations </li></ul><ul><li>Learn about basic statistics that would be useful in casino table games operations. </li></ul><ul><li>Learn about Theoretical Advantage of Games </li></ul>Casino Mathematics 1 RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
5. 5. <ul><li>AT THE END OF LESSON STUDENTS WILL </li></ul><ul><li>Understand the application of House Edge, Theoretical Win and Hold in managing casino table games. </li></ul><ul><li>Understand how Theoretical Win works as the management tool for casino managers </li></ul>Casino Mathematics 1 RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
6. 6. <ul><li>Basics of Probability & Statistics </li></ul><ul><li>1. PROBABILITY </li></ul><ul><li>“ The probability of an event is defined as the ratio of the number of favourable cases to the total number of possible cases” </li></ul>INTERNATIONAL CLUB GAMES TRAINING CENTRE PTE. LTD,ALL RIGHTS RESERVED.2006. Casino Mathematics 1 RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
7. 7. <ul><li>The chance of ‘23’ being drawn is 1 out of 37. </li></ul><ul><li>1/37 = 0.02703 = 2.703 % </li></ul><ul><li>You have a 2.703 % chance that you will win </li></ul><ul><li>the bet on number 23. </li></ul><ul><li>The chances that you will loose the bet : </li></ul><ul><li>0.02703 – 1 = 0.9729 = 97.29% </li></ul><ul><li>You have a 97.29% chance of losing </li></ul><ul><li>that bet. </li></ul>Casino Mathematics 1 RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
8. 8. <ul><li>Basics of Probability & Statistics </li></ul><ul><li>2. STATISTICS </li></ul><ul><li>“ The mathematics of the collection, organization, and interpretation of numerical data, especially the analysis of population characteristics by inference from sampling.” </li></ul>Casino Mathematics 1 RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
9. 9. <ul><li>2. STATISTICS </li></ul><ul><li>MEASURE OF CENTRAL TENDENCY </li></ul><ul><li>POPULATION </li></ul><ul><li>A population, or universe, is defined as an entire group of persons, things, or events having at least one trait in common. </li></ul>Casino Mathematics 1 RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
10. 10. <ul><li>2. STATISTICS </li></ul><ul><li>MEASURE OF CENTRAL TENDENCY </li></ul><ul><li>POPULATION </li></ul><ul><li>SAMPLE </li></ul><ul><li>AVERAGE OR MEAN </li></ul><ul><li>MEDIAN </li></ul><ul><li>MODE </li></ul>Casino Mathematics 1 RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
11. 11. <ul><li>2. STATISTICS </li></ul><ul><li>MEASURE OF CENTRAL TENDENCY </li></ul><ul><li>POPULATION </li></ul><ul><li>A population, or universe, is defined as an entire group of persons, things, or events having at least one trait in common. </li></ul><ul><li>Example 1: All the students of Batch 1 </li></ul><ul><li>Example 2: All employees of Casino De Genting who hold the position of Croupier </li></ul>Casino Mathematics 1 RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
12. 12. <ul><li>2. STATISTICS </li></ul><ul><li>MEASURE OF CENTRAL TENDENCY </li></ul><ul><li>AVERAGE OR MEAN </li></ul><ul><li>The average is the total of all elements of the population or sample divided by the number of elements of the population or sample. </li></ul><ul><li>The number of elements in a population is denoted by N </li></ul><ul><li>The number of elements in a sample is denoted by n </li></ul><ul><li>The Greek letter µ ( mu ) denotes the population mean. </li></ul><ul><li>The symbol X ( “X bar” ) denotes sample mean. </li></ul><ul><li>µ = average of all elements in the population </li></ul><ul><li>µ = Σ X¡ </li></ul><ul><li>N </li></ul><ul><li>X = average of sample elements selected form the population. </li></ul><ul><li>X = Σ X¡ </li></ul><ul><li>n </li></ul>Casino Mathematics 1 RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
13. 13. <ul><li>2. STATISTICS </li></ul><ul><li>MEASURE OF CENTRAL TENDENCY </li></ul><ul><li>MEDIAN </li></ul><ul><li>The median is the middle value in a population or sample. If the sample contained an odd number of elements, the median would be the middle value. If the sample contained an even number of elements, as in the example 4, the median is the average of the two middle elements. </li></ul>Casino Mathematics 1 RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
14. 14. <ul><li>2. STATISTICS </li></ul><ul><li>MEASURE OF CENTRAL TENDENCY </li></ul><ul><li>MODE </li></ul><ul><li>Is the most frequent element in the sample. In the sample of example 4, the mode is 167. </li></ul><ul><li>If two numbers occur more frequently than all other numbers in the group, the distribution is described as bimodal. </li></ul>Casino Mathematics 1 RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
15. 15. <ul><li>2. STATISTICS </li></ul><ul><li>MEASURES OF DISPERSION </li></ul><ul><li>Measurement of dispersion allows greater understanding of dispersion of values from which the average came. </li></ul><ul><li>Range, variance and standard deviation are measurements that will give the casino executive a better idea of the differences in the data. </li></ul>Casino Mathematics 1 RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
16. 16. <ul><li>Fundamental Principles of a Theory of Gambling </li></ul><ul><li>The essence of the phenomenon of gambling is decision making. </li></ul><ul><li>The act of making decision consists of selecting one course of action, or strategy, from among the set of admissible strategies. </li></ul><ul><li>Associated with the decision making process are the questions of preference, utility, and evaluation criteria, inter alia. Together, these concepts constitute the end result for a sound gambling-theory superstructure. </li></ul>Casino Mathematics 1 RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
17. 17. <ul><li>Decision making can be categories: </li></ul><ul><li>1. Decision making under certainty. </li></ul><ul><ul><li>Each specific strategy leads to a specific outcome. </li></ul></ul><ul><li>2. Decision making under risk. </li></ul><ul><ul><li>Each specific strategy leads to one of a set of possible outcomes with known probability distributions. </li></ul></ul><ul><li>3. Decision making under uncertainty. </li></ul><ul><ul><li>Each specific strategy has as its consequence a set of possible specific outcomes whose priori probability distribution is unknown or is meaningless. </li></ul></ul>Casino Mathematics 1 RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
18. 18. <ul><li>Fundamental Principles of a Theory of Gambling </li></ul><ul><li>There are no conventional games involving conditions of uncertainty without risk. Gambling theory, then is primarily concerned with decision making under conditions of risk. </li></ul>Casino Mathematics 1 RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
19. 19. Casino Mathematics 1 HOW CASINOS MAKE MONEY HOUSE EDGE The mathematical advantage that the casino has over player in casino games is known as the House edge or Theoretical House Advantage. RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
20. 20. Casino Mathematics 1 THEORETICAL WIN & HOLD PERCENTAGE Theoretical Win Since the potential win is calculated using a mathematical house edge which is derived from theory of probabilities, it remains theoretical win and not ‘actual win’. RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
21. 21. Casino Mathematics 1 THEORETICAL WIN & HOLD PERCENTAGE formula 1 Theoretical Win (TW) = Total Wager x House Edge % RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
22. 22. Casino Mathematics 1 THEORETICAL WIN & HOLD PERCENTAGE HANDLE Is defined the total amount of wager placed by player or players at a game or all gaming tables of the casino. It’s known as DROP when casinos assume all chip purchase are wagered. Casinos don’t keep track of total wagers placed for each game by every player. RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
23. 23. Casino Mathematics 1 THEORETICAL WIN & HOLD PERCENTAGE formula 2 Handle (Player) = Number of hands played x Average bet RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
24. 24. Casino Mathematics 1 THEORETICAL WIN & HOLD PERCENTAGE DROP Actual amount of cash sales of chips at a gaming table (or the whole casino). Includes warrant, marker and program chip sales. RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
25. 25. Casino Mathematics 1 THEORETICAL WIN & HOLD PERCENTAGE HOLD PERCENTAGE The net amount of money a particular game takes in (wins) as a percentage of the table drop over a certain period of time. RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
26. 26. Casino Mathematics 1 THEORETICAL WIN & HOLD PERCENTAGE formula 4 Hold Percentage = Actual win X 100% Actual Drop RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
27. 27. Casino Mathematics 1 Theoretical win & hold percentage as management tools Casino Profit ↑ = Win ↑= Handle↑ x House Edge % RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
28. 28. Casino Mathematics 1 Theoretical win & hold percentage as management tools formula 5 Win ↑ = (Hands/hour ↑ x Duration of play ↑ x Average Bet ↑ ) x House Edge ↑ % RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
29. 29. Casino Mathematics 1 Theoretical win & hold percentage as management tools <ul><li>EFFECTIVE MANAGEMENT OF WIN & ITS FUNCTIONS </li></ul><ul><li>Hands per hour ( or Games/hour ) </li></ul><ul><li>Duration of play (Length of play </li></ul><ul><li>Average Bet </li></ul><ul><li>House Edge </li></ul>RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
30. 30. Casino Mathematics 1 Theoretical win & hold percentage as management tools <ul><li>a. HANDS/HOUR ( or GAMES/HOUR ) </li></ul><ul><li>Factors that contribute towards greater number of hands/hour: </li></ul><ul><li>Highly skilled dealers. </li></ul><ul><li>Experienced dealers, higher game pace and lower errors. </li></ul><ul><li>Standard operating Game Procedures set standards. </li></ul><ul><li>Employee motivation, enforcement and supervision. </li></ul><ul><li>Gaming equipment, innovative methodology and quality standards </li></ul>RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
31. 31. Casino Mathematics 1 Theoretical win & hold percentage as management tools <ul><li>b. DURATION OF PLAY (LENGTH OF PLAY) </li></ul><ul><li>Factors that contribute towards greater duration of play: </li></ul><ul><li>Facility comfort level, Temperature, Noise and Lights. </li></ul><ul><li>Service in the casino and at the table. </li></ul><ul><li>Repeat Customers – loyalty programmes to increase player visits. </li></ul>RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
32. 32. Casino Mathematics 1 Theoretical win & hold percentage as management tools <ul><li>c. AVERAGE BET </li></ul><ul><li>Factors that contribute to greater average bets: </li></ul><ul><li>Table betting limits management. </li></ul><ul><li>Player Development, having in-house loyalty programmes. </li></ul><ul><li>Targeting marketing for higher income groups. </li></ul><ul><li>Premium player market development . </li></ul>RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.
33. 33. Casino Mathematics 1 Theoretical win & hold percentage as management tools <ul><li>d. HOUSE EDGE </li></ul><ul><li>Factors that effect House Edge: </li></ul><ul><li>Theft and pilferation render the House Edge factor. </li></ul><ul><li>Gaming scams have damaging effect on short term casino hold and win. </li></ul><ul><li>Optimising the offer of games of higher House Edge. </li></ul>RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.