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Casino Mathematics This guide, written by casino math professorcontains a best casino management books, non-technical discussion of the basic mathematics governing casino games and shows how casinos make money from these games.
The article addresses a variety of topics, including house advantage, confusion about win rates, game volatility, player value and comp policies, casino pricing mistakes, and regulatory issues.
Statistical advantages associated with the major games are also provided.
Casinos make money on their games because of the mathematics behind the games.
As Nico Zographos, dealer-extraordinaire for the 'Greek Syndicate' in Deauville, Cannes, and Monte Carlo in the 1920s observed about casino gaming: "There is no such thing as luck.
It is all mathematics.
That is what Mario Puzo was referring to in his famous novel Fools Die when his fictional casino boss character, Gronevelt, commented: "Percentages never lie.
We built all these hotels on percentages.
We stay rich on the percentage.
You can lose faith in everything, religion and God, women and love, good and evil, war and peace.
But the percentage will always stand fast.
Without the "edge," casinos source not exist.
With this edge, and because of a famous mathematical result called the law of large numbers, a casino is guaranteed to win in the long run.
Why is Mathematics Important?
Critics of the best cam roulette industry have long accused it of creating the name "gaming" and using this as more politically correct than calling itself the "gambling industry.
Instead, they rely on mathematical principles to assure that their establishment generates positive gross gaming revenues.
The operator, however, must assure the gaming revenues are sufficient to cover deductions like bad debts, expenses, employees, taxes and interest.
Despite the obvious, many casino professionals limit their advancements by failing to understand the basic mathematics of the games and their relationships to casino profitability.
One casino owner would often test his pit bosses by asking how a casino could make money on blackjack if the outcome is determined simply by whether click here player or the dealer came closest learn more here 21.
The answer, typically, was because the casino maintained "a house advantage.
Given that products offered by casinos are games, managers must understand why the games provide the expected revenues.
In the gaming industry, nothing plays a more important role than mathematics.
Mathematics should also overcome the dangers of superstitions.
An owner of a major Las Vegas strip casino once experienced a streak of losing substantial amounts of money to a few "high rollers.
His solution was simple.
He spent the evening spreading salt throughout the casino to ward off the bad spirits.
Before attributing this example to the idiosyncrasies of one owner, his are atypical only in their extreme.
Superstition has long been a part of gambling - from both sides of the table.
Superstitions can lead to irrational decisions that may hurt casino profits.
For example, believing that a particular dealer is unlucky against a particular winning player may lead to a decision to change dealers.
As many, if not most, players are superstitious.
At best, he may resent that the casino is trying to change his luck.
At worst, the player may feel the new dealer is skilled in methods to "cool" the game.
Perhaps he is even familiar with stories of old where casinos employed dealers to cheat "lucky" players.
Understanding the mathematics of a game also is important for the casino operator to ensure that the reasonable expectations of the players are met.
For most persons, gambling is entertainment.
It provides an outlet for adult play.
As such, persons have the opportunity for a pleasant diversion from ordinary life and from societal and personal pressures.
As an entertainment alternative, however, players may consider the value of the gambling experience.
For example, some people may have the option of either spending a hundred dollars during an evening by going to a professional basketball game or at a licensed casino.
If the house advantage is too strong and the person loses his money too quickly, he may not value that casino entertainment experience.
On the other hand, if a casino can entertain him for an evening, and he enjoys a "complimentary" meal or drinks, he casinos uk best in want to repeat the experience, even over a professional basketball game.
Likewise, new casino games themselves may succeed or fail based on player expectations.
In recent years, casinos have debuted a variety of new games that attempt to garner player interest and keep their attention.
Regardless of whether a game is fun or interesting to play, most often a player will not want to play games where his money is lost too quickly or where he has a exceptionally remote chance of returning home with winnings.
Mathematics also plays an important part in meeting players' expectations as to the possible consequences of his gambling activities.
best way to win at a casino gambling involves rational decision-making, it would appear irrational to wager money where your opponent has a better chance of winning than you do.
Adam Smith suggested that all gambling, where the operator has an advantage, is irrational.
Adventure upon all the tickets in the lottery, and you lose for certain; and the greater the number of your tickets, the nearer you approach to this certainty.
He could save or gamble this money.
Even if he did this for years, the savings would not elevate his economic status to another level.
While the odds of winning are remote, it may provide the only opportunity to move to a higher economic class.
Since the casino industry is heavily regulated and some of the standards set forth by regulatory bodies involve mathematically related issues, casino managers also should understand the mathematical aspects relating to gaming regulation.
Gaming regulation is principally dedicated to assuring that the games offered in the casino are fair, honest, and that players get paid if they win.
Fairness is often expressed in the regulations as either requiring a minimum payback to the player or, in more extreme cases, as dictating the actual rules of the games offered.
Casino executives should understand the impact that rules changes have on the payback to players to assure they meet regulatory standards.
Equally important, casino executives should understand how government mandated rules would impact their gaming revenues.
The House Edge The player's chances of winning in a casino game and the rate at which he wins or loses money depends on the game, the rules in effect for that game, and for some games his level of skill.
The amount of money the player can expect to win or lose in the long run - if the bet is made over and over again - is called the player's wager expected value EVor expectation.
When the player's wager expectation is negative, he will lose money in the long run.
When the wager expectation is viewed from the casino's perspective i.
For the roulette example, the house advantage is 5.
Here are the calculations for bets on a single-number in double-zero and single-zero roulette.
It is also called the house edge, the "odds" i.
Regardless of the method used to compute it, the house advantage represents the price to the player of playing the game.
Because this positive house edge exists for virtually all bets in a casino ignoring the poker room and sports book where a few professionals can make a livinggamblers are faced with an uphill and, in the long run, losing battle.
There are some exceptions.
The odds bet in craps has zero house edge although this bet cannot be made without making another negative expectation wager and there are a few video poker machines that return greater than 100% if played with perfect strategy.
Occasionally the casino will even offer a promotion that gives the astute player a positive expectation.
These promotions are usually mistakes - sometimes casinos don't check the math - and are terminated once the casino realizes the player has the edge.
But by and large the player will lose money in the long run, and the house edge is a measure of how fast the money will be lost.
A player betting in a game with a 4% house advantage will tend to lose his money twice as fast as a player making bets with a 2% house edge.
The trick to intelligent casino gambling - at least from the mathematical expectation point of view - is to avoid the games and bets with the large house advantages.
Some casino games are pure chance - no amount of skill or strategy can alter the odds.
These games include roulette, craps, baccarat, keno, the big-six wheel of fortune, and slot machines.
Of these, baccarat and craps offer the best odds, with house advantages of 1.
Roulette and slots cost the player more - house advantages of 5.
Games where an element of skill can affect the house advantage include blackjack, video poker, and the four popular poker-based table games: Caribbean Stud poker, Let It Ride, Three Card poker, and Pai Gow poker.
For the poker games, optimal strategy results in a house edge in the 3% to 5% range CSP has the largest house edge, PGP the lowest, with LIR and TCP in between.
For video poker the statistical advantage varies depending on the particular machine, but generally this game can be very player friendly - house edge less than 3% is not uncommon and some are less than 1% - if played with expert strategy.
Blackjack, the most popular of all table games, offers the skilled player some of the best odds in the casino.
The house advantage varies slightly depending on the rules and number of decks, but a player please click for source basic strategy faces little or no disadvantage in a single-deck game and only a 0.
Despite these numbers, the average player ends up giving the casino a 2% edge due learn more here mistakes and deviations from basic strategy.
Complete basic strategy tables can be found in many books and chance money winning best casino of at casino-hotel gift shops sell color-coded credit card size versions.
Rule variations favorable to the player include fewer decks, dealer stands on soft seventeen worth 0.
If the dealer hits soft seventeen it will best casino management books you, as will any restrictions on when you can double down.
Probability versus Odds Probability represents the long run ratio of of times an outcome occurs to of times experiment is conducted.
Odds represent the long run ratio of of times an outcome does not occur to of times an outcome occurs.
The true odds of an event represent the payoff that would make the bet on that event fair.
Confusion about Win Rate There are all kinds of percentages in the world of gaming.
Win percentage, theoretical win percentage, hold percentage, and house advantage come to mind.
Sometimes casino bosses use these percentages interchangeably, as if they are just different names for the same thing.
Admittedly, in some cases this is correct.
House advantage is just another name for theoretical win percentage, and for slot machines, hold percentage is in principle equivalent to win percentage.
But there are fundamental differences among these win rate measurements.
The house advantage - the all-important percentage that explains how casinos make money - is also called the house edge, the theoretical win percentage, and expected win percentage.
In double-zero roulette, this figure is 5.
In the long run the house will retain 5.
In the short term, of course, the actual win percentage will differ from the theoretical win percentage the magnitude of this deviation can be predicted from statistical theory.
The actual win percentage is just the actual win divided by the handle.
Because of the law of large numbers - or as some prefer to call it, the law of averages - as the number of trials gets larger, the actual win percentage should get closer to the theoretical win percentage.
Because handle can be difficult to measure for table games, performance is often measured by hold percentage and sometimes erroneously called win percentage.
Hold percentage is equal to win divided by drop.
In Nevada, this figure is about 24% for roulette.
The drop and hold percentage are affected by many factors; we won't delve into these nor the associated management issues.
Suffice it to say that the casino will not in the long term keep 24% of the money bet on the spins of roulette wheel - well, an honest casino won't.
To summarize: House advantage and theoretical win percentage are the same thing, hold percentage is win over drop, win percentage is win over handle, win percentage approaches the house advantage as the number of plays increases, and hold percentage is equivalent to win percentage for slots but not table games.
In Let It Ride, for example, the casino advantage is either 3.
phrase best casinos online uk can familiar with the game know that the player begins with three best casino management books base bets, but may withdraw one or two of these initial units.
The final link put at risk, then, can be one 84.
In the long run, the casino will win 3.
So what's the house edge for Let It Ride?
Some prefer to say 3.
The question of whether to use the base bet or average bet size also arises in Caribbean Stud Poker 5.
For still other games, the house edge can be stated including or excluding ties.
The prime examples here are the player 1.
Again, these are different views on the casino edge, but the expected revenue will not change.
That the house advantage can appear in different disguises might be unsettling.
When properly computed and interpreted, however, regardless of which representation is chosen, the same truth read: money emerges: expected win is the same.
Volatility and Risk Statistical theory can be used to predict the magnitude of the difference between the actual win percentage and the theoretical win percentage for a given number of wagers.
When observing the actual win percentage a player or casino may experience, how best casino management books variation from theoretical win can be expected?
What is a normal fluctuation?
The basis for the analysis of such volatility questions is a statistical measure called the standard deviation essentially the average deviation of all possible outcomes from the expected.
Together with the central limit theorem a form of the law of large numbersthe standard deviation SD can be used to determine confidence limits with the following volatility guidelines: Volatility Analysis Guidelines · Only 5% of the time will outcomes will be more than 2 SD's from expected outcome · Almost never 0.
Computing the SD value is beyond the scope of this article, but to get an idea behind confidence limits, consider a series of 1,000 pass line wagers in craps.
Since each wager has a 1.
It can be shown calculations omitted that the wager standard deviation is for a single pass line bet is 1.
Applying the volatility guidelines, we can say that there is a 95% chance the player's actual win will be between 49 units ahead and 77 units behind, and almost certainly between 81 units ahead and 109 units behind.
Note that if the volatility analysis is done in terms of the percentage win rather than the number of units or amount wonthe confidence limits will converge to the house advantage as the number of wagers increases.
This is the result of the law of large numbers - as the number of trials gets larger, the actual win percentage should link closer to the theoretical win percentage.
Risk in the gaming business depends on the house advantage, standard deviation, bet size, and length of play.
Player Value and Complimentaries Using the house advantage, bet size, duration of play, and pace of the game, a casino can determine how much it expects to win from a certain player.
Using a house advantage of 1.
Many casinos set comp complimentary policies by giving the player back a set percentage of their earning potential.
Although comp and rebate policies based on theoretical loss are the most popular, rebates on actual losses and dead chip programs are also used in some casinos.
Some programs involve a mix of systems.
The mathematics associated with best casino management books programs will not be addressed in this article.
Casino Pricing Mistakes In an effort to entice players and increase business, casinos occasionally offer novel wagers, side bets, increased payoffs, best casino management books rule variations.
These promotions have the effect of lowering the house advantage and the effective best casino management books of the game for the player.
This is sound reasoning from a marketing standpoint, but can be disastrous for the casino if care is not taken to ensure the math behind the promotion is sound.
One casino offered a baccarat commission on winning banker bets of only 2% instead of the usual 5%, resulting in a 0.
Again, this is an easy calculation.
Not to be outdone, an Indian casino in California paid 3 to 1 on naturals during their "happy hour," offered three times a day, two days a week for over two weeks.
This promotion gave the player a whopping 6% edge.
A small Las Vegas casino offered a blackjack rule variation called the "Free Ride" in which players were given a free right-to-surrender token every time they received a natural.
Proper use of the token led to a player edge of 1.
In the gaming business, it's all about "bad math" or "good math.
Players will get "lucky" in the short term, but that is all part of the grand design.
Fluctuations in both directions will occur.
We call these fluctuations good luck or bad luck depending on the direction of the fluctuation.
There is no such thing as luck.
It is all mathematics.
Gaming Regulation and Mathematics Casino gaming is one of the most regulated industries in the world.
Most gaming regulatory systems share common objectives: keep the games fair and honest and assure that players are paid if they win.
Fairness and honesty are different concepts.
A casino can be honest but not fair.
Honesty refers to whether the casino offers games whose chance elements are random.
Fairness refers to the game advantage - how much of each dollar wagered should the casino be able to keep?
A slot machine that holds, on average, 90% of every dollar bet is certainly not fair, but could very well be honest if the outcomes of each play are not predetermined in the casino's favor.
Such evidence can range from straightforward probability analyses to computer simulations and complex statistical studies.
Requirements vary across jurisdictions, but it is not uncommon to see technical language in gaming regulations concerning specific statistical tests that must best online casino percentages performed, confidence limits that must be met, and other mathematical specifications and standards relating to game outcomes.
Summary Tables for House Advantage The two tables below show the house advantages for many of the popular casino games.
The first table is a summary of the popular games and the second gives a more detailed breakdown.
House Advantages for Popular Casino Games Game House Advantage Roulette double-zero 5.
Optimal strategy assumed unless otherwise noted.
Note: This summary is the intellectual property of the author and the University of Nevada, Las Vegas.
Do not use or reproduce without proper citation and permission.
Selected Biblilography Cabot, Anthony N.
Gaming Regulation and Mathematics: A Marriage of Necessity, John Marshall Law Review, Vol.
Casino Gaming: Policy, Economics, and Regulation, UNLV International Gaming Institute, Las Vegas, NV.
The Business of Gaming: Economic and Management Issues, Institute for the Study of Gambling and Commercial Gaming, University of Nevada, Reno, NV.
Gambling and Commercial Gaming: Essays in Business, Economics, Philosophy and Science, Institute for the Study of Gambling and Commercial Gaming, University of Nevada, Reno, NV.
The Theory of Gambling and Statistical Logic, revised edition, Academic Press, San Diego, CA.
An Introduction to Probability Theory and Its Applications, 3rd ed.
The Theory of Blackjack, 6th ed.
Extra Stuff: Gambling Ramblings, Huntington Press, Las Vegas, NV.
Humble, Lance, and Cooper, Carl 1980.
The World's Greatest Blackjack Book, Doubleday, New York, NY.
Kilby, Jim and Fox, Jim 1998.
Casino Operations Management, Wiley, New York, NY.
Chance, Luck and Statistics, Dover Publications, Mineola, NY.
The Mathematics of Games and Gambling, The Mathematical This web page of America, Washington, D.
The Mathematics of Gambling, Gambling Times, Hollywood, CA.
Beat the Dealer, Vintage Books, New York, NY.
Vancura, Olaf, Cornelius, Judy A.
Finding the Edge: Mathematical Analysis of Casino Games.
Institute for the Study of Gambling and Commercial Gaming, University of Nevada, Reno, NV.
Smart Casino Gambling, Index Publishing Group, San Diego, CA.
Lady Luck: The Theory of Probability, Dover Publications, New York, NY.
The Casino Gambler's Guide, Harper and Row, New York.
His publications include Practical Casino Math co-authored with Anthony N.
Cabot and numerous articles in scholarly and gaming industry journals.
Hannum regularly speaks on casino mathematics to audiences around the globe.
Some of this guide has been excerpted from Practical Casino Math.
For more information, see.
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Do not copy or reuse without permission.
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