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This guide, written by casino math professor Robert Hannumcontains 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 slots на доллары в excel issues.

Statistical advantages associated with the major games are also provided. Selected Bibliography About the Author. At its core the business of casino gaming is pretty simple. Casinos make money on their games because of the mathematics behind the games. It is all mathematics. With a few notable exceptions, the house always wins - in the long run - because of the mathematical advantage the casino enjoys over the player. That is what Mario Puzo was referring to in his famous novel Fools Die when his fictional casino boss character, Gronevelt, commented: 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. Puzo is, of course, right on the money about casino gaming. 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 The answer, typically, was because the casino maintained "a house advantage.

Given that products offered by casinos are games, slots на доллары в excel 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 бонус на депозит ставки на спорт qiwi 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 slots на доллары в excel 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 this web page 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 learn more here too strong and the person loses his money too quickly, slots на доллары в excel 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 slots на доллары в excel. In recent years, casinos have debuted a variety of new games that attempt to garner деньги казино реальные которые выплачивают interest and keep their attention.

Regardless of whether a game is fun or interesting to play, most often slots на доллары в excel player will not slots на доллары в excel to play games where his money is lost slots на доллары в excel quickly or where he has a exceptionally remote chance of returning home with winnings.

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. He wrote "There is not, however, a more certain proposition in mathematics than that the more tickets [in a lottery] you advertise upon, the more likely you are a loser. Adventure upon all slots на доллары в excel tickets in the lottery, and you lose for certain; and the greater the number of your tickets, the nearer you approach to this certainty.

Even where the house has slots на доллары в excel advantage, however, a gambler may be justified if the amount lost means little to him, but the potential gain would elevate him to a higher standing of living.

He could slots на доллары в excel 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 slots на доллары в excel they meet regulatory standards. Equally important, casino executives should understand how government mandated rules would impact their gaming revenues. For the roulette example, the house advantage is 5. The formal calculation is as follows: When this EV calculation is performed for a 1-unit amount, the negative of the resulting value is the house edge.

Here are the calculations for bets on a single-number in double-zero and single-zero roulette. Double-zero roulette single number bet: Single-zero roulette single number bet: The house advantage represents the long run percentage of the wagered money that will be retained by the casino.

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, slots на доллары в excel the long run, losing battle. There are some exceptions.

Occasionally the casino will even offer a promotion that gives the astute player a positive expectation. But by and large the player will lose money in the long run, and the house edge is slots на доллары в excel measure of how fast the money will be lost.

The trick to intelligent casino gambling - at least slots на доллары в excel 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 unt в каком казино реально вывести деньги за Blut 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. 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. 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 slots на доллары в excel 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 represents the long run ratio of of times an outcome occurs to of times android казино на реальные деньги андроид 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 slots на доллары в excel 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 check this out 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.

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