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Casino Mathematics This guide, written here casino math professorcontains a brief, 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 blackjack or craps house edge 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 would 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 gaming 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 the player or the dealer came closest to 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 click here ordinary life and from societal and personal pressures.
As an entertainment alternative, however, players may consider the value of blackjack cheating 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 may 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 kelly criterion blackjack 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 check this out 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.
If 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 blackjack review draftkings 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 using 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 to mistakes and deviations from basic strategy.
Complete basic strategy tables can be found in many books and many 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 cost 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, https://tayorindustry.com/blackjack/hit-someone-with-a-blackjack.html actual win percentage will differ from the theoretical win percentage the magnitude of this deviation can be predicted from blackjack or craps house edge 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 blackjack or craps house edge 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 atlantic city arena football />Those familiar with the game know that the player begins with three equal base bets, but may withdraw one or two of these initial units.
The final amount 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 blackjack guide to betting which representation is chosen, the same truth blackjack or craps house edge money emerges: expected win is the same.
Volatility and Risk Statistical theory can be used to predict the see more 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 much 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 blackjack or craps house edge win percentage should get 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 these programs will not be addressed in this article.
are onyx blackjack windows 10 think Pricing Mistakes In an effort to entice players and increase business, casinos occasionally offer novel wagers, side bets, increased payoffs, or rule variations.
These promotions have the effect of lowering the house advantage and the effective price 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% really. blackjack deck rules situation />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 push ups blackjack 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 be 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 Association 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.
Follow on Twitter and on Facebook © 2019 University of Nevada, Las Vegas.
Do not copy or reuse without permission.

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In gambling, there's one certainty—one thing not left to chance: The house always comes out the winner in the end.
A casino is a business, not a charitable organization blackjack or craps house edge free blackjack edge away.
Like any other business, it has a in place designed to ensure its profitability.
A casino has a number of built-in advantages that insure it, and not the players overall, will always come out a winner in the end.
The House Edge No matter what game you choose to play, the of the casino winning your money are greater than the odds of you winning the casino's money.
That's because all casino games are designed to provide the house with a built-in edge, diminishing the chances and the size of potential payouts.
For example, in roulette, the highest payout for a single number bet is 36 to 1.
However, roulette blackjack or craps house edge, besides having the numbers 1 to 36, also have a 0 and sometimes a 00 as well.
The true odds of winning are 37 to 1 or 38 to 1, not the 36 to 1 that is the most the player can get paid on a winning bet.
The house edge, the odds advantage in its favor, represents the average the casino can reliably expect to make from each game.
On the games with the lowest house edge, the smallest advantage, a casino might only be generating about a 1% to 2% profit.
On other games, it may make profits of up to 15 %to 25% or more.
The house edge on a 00 roulette wheel is 5.
The casino isn't aiming to bankrupt a player in one sitting — it just wants to make sure that in the long run, the players walk out with a little less money than they came in with, leaving money in the casino's pocket.
Many people who are aware of the house edge still don't really grasp its implications for their bankrolls.
They fail to understand that the house edge doesn't apply to their starting bankroll, but to the total amount they wager.
The longer you play, the greater the odds are that the result of your play will match up with the house edge.
In the short term, a player blackjack or craps house edge well be ahead; over the long haul, the house edge will eventually grind the player down into unprofitability.
Knowing this, casinos do all they can to keep you playing longer.
Casinos are famous for lacking clocks and windows.
They're designed that way to keep players unaware of the passage of time.
Many first-time players are pleasantly surprised at being offered free drinks by the management.
Those complimentary libations will cost you, though: being inebriated doesn't usually improve their judgment when it comes to betting.
Although all the laws of are in the casino's favor, the house edge varies significantly among the different casino games.
The game with the lowest advantage to the casino is blackjack; if a player follows blackjack or craps house edge perfect betting strategy, the house edge is only 0.
At some very liberal casinos, the house edge at blackjack may even be as low as 0.
Craps offers the next lowest edge, 0.
The smallest edge only applies if the player is which few people do.
The house edge increases as players wager less expertly.
Roulette remains one of the most popular casino games, but it carries a high 5.
The house edge on slot machines goes as high as 17%; for keno, it is a massive 25%.
The offers that appear in this table are from partnerships from which Investopedia receives compensation.
Gambling income refers to any money that is generated from games of chance or wagers on events with uncertain outcomes.
Bitcoin is a digital or virtual currency created in link that uses peer-to-peer technology to facilitate instant payments.
It follows the ideas set out in a whitepaper by the mysterious Satoshi Nakamoto, whose true identity has yet to be verified.
A guide to help you understand what blockchain is and how it can be used https://tayorindustry.com/blackjack/how-much-money-to-bring-to-blackjack.html industries.
Certificates of deposit CDs pay more interest than standard savings accounts.
Find the blackjack or craps house edge nationally available rates for each CD term here from federally insured banks and credit unions.
Value investors like Warren Buffett select undervalued stocks trading at less than their intrinsic book value that have long-term potential.
Brexit refers to Britain's leaving the European Union, which was slated to happen at the end of October, but has been delayed again.

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Instead, they hone the sharpest legal edge they can.
On Flamingo Road in Blackjack online Vegas, James Grosjean sat at a steel table outside a Starbucks.
In the near distance stood a sign for blackjack or craps house edge free blackjack demo casino, the Palms, where he has been shown the door more than once.
Being run out of casinos is an occupational hazard for Grosjean, a professional gambler who majored in applied here at Harvard and briefly considered careers on Wall Street and in academia.
He sipped from a venti-size container of coffee and typed rapidly on his laptop computer.
The opportunity was in Shawnee, Okla.
He uses his programming skills to divine the odds in various situations and then develops strategies for exploiting them.
Only two questions seemed to temper his confidence in taking on this particular game.
How long would they be allowed to play before being asked to leave?
How much money would they be able to win?
When Grosjean first reconnoitered the game, he saw that the 12 playing cards used to simulate a pair of craps dice were being shuffled by a machine designed to speed up play and randomize the order of the cards.
But Grosjean knew that shuffling machines are computer driven and therefore only as good as they are programmed and used: Sometimes, in fact, the devices are surprisingly predictable.
That was true in Shawnee.
After each round, the dealer there swept up the cards and put them in the shuffler without mixing them by hand.
Grosjean discovered that he could see the identity and order of at least three cards entering the machine, the bottom one held by the dealer and the two that were exposed during game play.
Because he has examined these shuffling machines and knows how they work, he could reliably judge the likelihood that certain cards would be excluded from play.
Armed with that knowledge, he spent several months simulating the game in software; com cardplayerpoker computer mimicked the shuffling algorithm and played the game millions of times.
His findings would give him a significant edge playing the card-based craps game in Shawnee.
It would be equivalent to gambling at standard craps with dice and knowing which three dice faces — out of 12 possible — would have a reduced probability of coming up on any roll.
Many casino executives despise gamblers like Grosjean.
They accuse him blackjack or craps house edge cheating.
Yet what he does is entirely legal.
Whiting acknowledges that they do not deserve to be arrested.
That is usually illegal, the sort of thing that can result in jail time.
I view it all as preventable loss.
Cumulatively, they rake in large profits from games that were designed to be unbeatable: While some bettors might get lucky and win in the short run, over time they are supposed to lose and the casinos are expected to win, statistically speaking.
In recent years, however, Whiting says the ranks of advantage players have swelled.
Several factors are responsible.
One is the ease with which gamblers can find each other online and share tactics.
Grosjean has a blog called Beyond Numbers, for example.
And because regulated casino gambling now takes place in at least 40 states, casinos compete for customers in part by introducing new games, some of which turn out to be vulnerable.
Munchkin, whose real first name is Richard, chose his professional surname because of the fact that he stands slightly taller than five feet.
I know people who clock roulette wheels and others who can control a single die at craps.
Elbows were thrown and there was a lot of jostling around the table.
An older civilian accidentally got in the middle of it.
His son thought I had hit him, and the son jumped on my back.
Following all the tumult, the table was being watched and somebody figured out what was going on.
As I walked beside him, I tried to appear casual, blackjack or craps house edge the tail of my untucked shirt covering the notepad in the back pocket of my slacks.
Grosjean passed an escalator and headed down a back staircase.
To experienced surveillance people, he is a known advantage player; at any time he could be spotted, matched to his picture in a database of such players and asked to leave a casino.
If that happens, the security guard could also read him the trespass act, meaning Grosjean would risk arrest if he tried to return.
Getting away, on the other hand, would give him an blackjack or craps house edge to come back on some future day and perhaps go unnoticed.
So if security was waiting for him at the bottom, Grosjean needed to be able blackjack or craps house edge run back up in the opposite direction with the hope of avoiding a confrontation.
Down below on the gaming floor, ringed by wall-mounted TV monitors silently showing a sporting event, slot machines chirped and crowded blackjack tables article source with action.
Grosjean had explained earlier the reason for this quirk: The Grand happens this web page be located in a jurisdiction where it is illegal for dice to determine financial outcomes in games of chance.
Two sets of six playing cards, numbered one through six, one set with red backs, the other with blue backs, serve as de facto dice.
A player rolls a giant numbered cube, apparently made from plastic foam.
The cube determines which cards are turned over.
It is a way to make the game feel like craps without dice directly producing a monetary outcome.
After that, standard rules apply.
A gambler might bet, for example, that the sum of the first two cards in play will total 7 or 11.
If the sum equals 2, 3 or 12, he loses.
If a total of 7 comes first, he loses.
Over the course of the game, players can wager on other combinations, like two 5s turned over which pays out 7 to 1.
Such proposition, or prop, bets favor the casino.
After every two-card set is turned over, the cards were machine-shuffled before the next roll.
Play had been temporarily halted to accommodate another recent arrival at the felt-covered table — Richard Munchkin.
Two regular gamblers, whom Grosjean had noticed on numerous occasions in Shawnee, watched in slack-jawed surprise: Patrons rarely played for such large sums there.
Dealers on the table clearly knew him — he had been establishing his presence here for the past week, getting used to the game and figuring out its subtleties — and they good-naturedly commiserated with him over his propensity for losing.
The motions were signals to Munchkin: With a split-second glance, he gained the statistically significant advantage of knowing numbers likely to be excluded.
Munchkin was less than friendly, tipped modestly, demanded a lot of service.
Call me Little Joe.
His signaling had not been perfected yet — some gestures were being missed.
That, combined with a bit of bad luck, had put Munchkin into a difficult situation.
Their scheme, their play against the casino, was perhaps too complicated to pull off, Grosjean would later concede.
Still, he kept feeding Munchkin information about the cards.
Slowly, as the night grinded on, some of the high-returning prop bets started to pay off.
When they came up, Munchkin shot a fist in the air and shouted.
The thrill was feigned.
An observant dealer, a man who appeared to be in his early 30s, discerned something fishy.
Maybe he had noticed the discreet glances at Grosjean.
After all, Little Joe was bleeding money.
He learned as much when he spent the next two days gambling on his own.
Her name was Cheung Yin Sun.
Elite Western gamblers and rattled casino bosses know her as Kelly, a fast-talking, sunglasses-wearing advantage player in her 40s.
Her https://tayorindustry.com/blackjack/blackjack-strain-nirvana.html, now deceased, was a wealthy factory owner based in Hong Kong.
She claims to be unbothered by having learn more here a fortune in gambling dens around the world.
In fact, she all but brags about her losses.
We were sitting in a back booth in https://tayorindustry.com/blackjack/bet365-live-dealer-blackjack.html coffee shop of SLS Las Vegas, a stylish casino on the north end of the Strip.
Sun wore a neon green approximation of a tennis dress.
I decided that one day I would get back the money by playing goes pokerstars client software 32 bit opinion MGM properties.
They look identical to those used at the gaming tables but have holes punched through their centers to prevent cheaters from slipping a souvenir ace of spades, say, into a poker game.
Sun had no such intention.
She scrutinized the backs of the cards.
Some had crisscrossing patterns that went right to all four edges.
She spent around a thousand hours, over four years, training herself to recognize the minute variations on particular cards.
Sun figured out how she could leverage these differences that were almost imperceptible and acceptable by industry standards.
Mini-baccarat is played with eight decks of cards.
To begin, four cards are dealt face down onto the table.
Patrons never touch the cards.
Before the cards are dealt, gamblers bet on banker or player or a tie.
Whichever side gets closer to 9 is the winner.
Tens and picture-cards count as zero; aces count as 1.
If the sum exceeds 9, then only the second digit is recognized for example, a 9 and 6 add up to 15 but count as 5.
From her years as a losing high-stakes gambler, she knew that casino executives will accommodate even outlandish requests from this web page who wager huge sums of money.
She also believed that Asian gamblers were viewed as superstitious.
In October 2011, having trained herself to edge-sort, Sun decided to exploit both the servility and the stereotype.
At a reserved table, they played mini-baccarat straightforwardly, as typical high-stakes gamblers might.
But they had a request: to be allowed to make their bets after the four cards used in each mini-baccarat hand were dealt.
Casino representatives approved the requests.
On that second day, Sun pulled a rolling Louis Vuitton suitcase behind her.
Once play began, they instructed the dealer to turn certain cards half a rotation.
Luck, of course, had nothing to do with their request.
Edgy casino personnel crowded the table.
Sun recalls six men in suits watching her intensely.
When a partner failed to bet quickly enough, she grabbed his chips and made the wager herself.
Sun wanted to get through the eight-deck stack of 416 cards before Aria personnel could recognize what was going on.
To ease her nerves, she says she rubbed a finger along the outline of a freshly inked jaguar tattoo on her right thigh.
Few gamblers take such substantial sums in cash, usually preferring a check instead.
Sun blackjack or craps house edge her team disappeared into the Las Vegas dusk with their winnings in the suitcase.
According to a surveillance officer who witnessed the play, Aria employees spent two days piecing together what had happened.
They made a trip to Foxwoods Resort Casino in Mashantucket, Conn.
Eventually Sun recruited the celebrity poker pro Phil Ivey, who is also known as a high-stakes gambler at craps and baccarat.
During the next year, he wired seven-figure sums to various casinos and did the betting.
Sun did the edge-sorting of the cards and tipped off Ivey whether to wager on banker or player.
Their combined winnings in Atlantic City, London and other places were in the eight figures.
A couple weeks ago, I spoke to Sun on the phone.
She told me she was in Macau, presumably beating baccarat games there.
But her success has exacted a price.
She is currently linked to three lawsuits with casinos that insist she used deceptive practices.
Ivey is appealing a case in London against a casino that withheld their winnings.
A similar dispute not involving Ivey with Foxwoods is also under appeal.
Ted Whiting, who works for MGM, one of the owners of Aria, will not comment on her.
When I last saw Grosjean in action, he had returned to Shawnee and was passing himself off as A.
The cover story explained why a newcomer would suddenly show up and spend multiple nights in an out-of-the-way casino, gambling more money blackjack or craps house edge anyone else in the place.
Sanchez bet aggressively, tipped generously and never missed a signal.
Time to move East.
Meanwhile, the card-craps table in Shawnee continues to simmer.
The casino has taken several procedural measures, like extra shuffling by hand, that makes its craps iteration more difficult to beat.
If you have a well producing a billion barrels and another one producing only 50 million, you still pump the smaller well because eventually the big one will run out of oil.

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