Introduction to Casino Mathematics

Casino Mathematics 1

introduction to game expectancy

RAMACHANDAR SIVA. ALL RIGHTS RESERVED.2006.

Casino Mathematics 1Introduction to Game Expectancy

Presented by : Ramachandar Siva

Duration : 3 hours


COURSE OBJECTIVES

To provide basic understanding of the functions of Probabilities & Statistics

To provide a mathematical explanation of the casino games

To understand the application of mathematical tools in day to day operations of a casino.



AT THE END OF LESSON STUDENTS WILL

Understand basics of probabilities and expectations

Learn about basic statistics that would be useful in casino table games operations.

Learn about Theoretical Advantage of Games



AT THE END OF LESSON STUDENTS WILL

Understand the application of House Edge, Theoretical Win and Hold in managing casino table games.

Understand how Theoretical Win works as the management tool for casino managers



Basics of Probability & Statistics

1. PROBABILITY “The probability of an event is defined as

the ratio of the number of favourable cases to the total number of possible cases”

INTERNATIONAL CLUB GAMES TRAINING CENTRE PTE. LTD,ALL RIGHTS RESERVED.2006.



The chance of ‘23’ being drawn is 1 out of 37.

1/37 = 0.02703 = 2.703 %

You have a 2.703 % chance that you will win

the bet on number 23.

The chances that you will loose the bet :

0.02703 – 1 = 0.9729 = 97.29%

You have a 97.29% chance of losing

that bet.



Basics of Probability & Statistics

2. STATISTICS “The mathematics of the collection,

organization, and interpretation of numerical data, especially the analysis of population characteristics by inference from sampling.”



2. STATISTICS

MEASURE OF CENTRAL TENDENCY

POPULATION

A population, or universe, is defined as an entire group of persons, things, or events having at least one trait in common.



2. STATISTICS


POPULATION SAMPLE AVERAGE OR MEAN MEDIAN MODE



2. STATISTICS


POPULATION

A population, or universe, is defined as an entire group of persons, things, or events having at least one trait in common.

Example 1: All the students of Batch 1 Example 2: All employees of Casino De Genting

who hold the position of Croupier



2. STATISTICS


AVERAGE OR MEAN The average is the total of all elements of the population or sample divided by

the number of elements of the population or sample. The number of elements in a population is denoted by N The number of elements in a sample is denoted by n The Greek letter µ ( mu ) denotes the population mean. The symbol X ( “X bar” ) denotes sample mean. µ = average of all elements in the population µ = Σ X¡ N X = average of sample elements selected form the population. X = Σ X¡ n



2. STATISTICS


MEDIAN 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.



2. STATISTICS


MODE Is the most frequent element in the sample. In

the sample of example 4, the mode is 167. If two numbers occur more frequently than all

other numbers in the group, the distribution is described as bimodal.



2. STATISTICS

MEASURES OF DISPERSION Measurement of dispersion allows greater

understanding of dispersion of values from which the average came.

Range, variance and standard deviation are measurements that will give the casino executive a better idea of the differences in the data.



Fundamental Principles of a Theory of Gambling

The essence of the phenomenon of gambling is decision making.

The act of making decision consists of selecting one course of action, or strategy, from among the set of admissible strategies.

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.



Decision making can be categories:

1. Decision making under certainty. Each specific strategy leads to a specific outcome.

2. Decision making under risk. Each specific strategy leads to one of a set of possible

outcomes with known probability distributions.

3. Decision making under uncertainty. Each specific strategy has as its consequence a set of

possible specific outcomes whose priori probability distribution is unknown or is meaningless.




Fundamental Principles of a Theory of Gambling

There are no conventional games involving conditions of uncertainty without risk.

Gambling theory, then is primarily concerned with decision making under conditions of risk.



HOW CASINOS MAKE MONEY

HOUSE EDGEThe mathematical advantage

that the casino has over player in casino games is

known as the House edge or Theoretical House Advantage.



THEORETICAL WIN & HOLD PERCENTAGE

Theoretical WinSince 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’.



THEORETICAL WIN & HOLD

PERCENTAGE formula 1

Theoretical Win (TW) = Total Wager x House Edge %




PERCENTAGE

HANDLEIs 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.




PERCENTAGE formula 2

Handle (Player) = Number of hands played x Average bet




DROPActual amount of cash sales of chips at a

gaming table (or the whole casino). Includes warrant, marker and program chip sales.




PERCENTAGE HOLD PERCENTAGEThe net amount of money a particular

game takes in (wins) as a percentage of the table drop over a certain period of

time.




formula 4

Hold Percentage = Actual win X 100%

Actual Drop



Theoretical win & hold percentage as management tools

Casino Profit ↑ = Win ↑= Handle↑ x House Edge %




formula 5

Win ↑= (Hands/hour ↑x Duration of play ↑x Average Bet↑) x House Edge↑ %



Theoretical win & hold percentage as management toolsEFFECTIVE MANAGEMENT OF WIN & ITS

FUNCTIONS

•Hands per hour ( or Games/hour )•Duration of play (Length of play•Average Bet•House Edge




a. HANDS/HOUR ( or GAMES/HOUR )Factors that contribute towards greater number of hands/hour:

• Highly skilled dealers.

• Experienced dealers, higher game pace and lower errors.

• Standard operating Game Procedures set standards. • Employee motivation, enforcement and supervision.

• Gaming equipment, innovative methodology and quality standards




b. DURATION OF PLAY (LENGTH OF PLAY)Factors that contribute towards greater duration of play:

• Facility comfort level, Temperature, Noise and Lights.

• Service in the casino and at the table.

• Repeat Customers – loyalty programmes to increase player visits.




c. AVERAGE BETFactors that contribute to greater average bets:

• Table betting limits management.

• Player Development, having in-house loyalty programmes.

• Targeting marketing for higher income groups.

• Premium player market development.




d. HOUSE EDGEFactors that effect House Edge:

• Theft and pilferation render the House Edge factor.

• Gaming scams have damaging effect on short term casino hold and win.

• Optimising the offer of games of higher House Edge.


Introduction to Casino Mathematics

Education

Transcript of Introduction to Casino Mathematics