Casino Games Expected Value Probability Calculator
This guide, written by casino math professor Robert Hannumcontains a brief, nontechnical 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. 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 Casino Games Expected Value Probability Calculator "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 Reel Power Casino Igri Free 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, managers must understand why the games provide the expected revenues.
Researchers the Honda Casino Probability Value Expected Calculator Games accept
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.
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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 Rave Casino X Slots Plus Bonus 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 of new games that attempt to garner player interest and keep Casino Games Expected Value Probability Calculator 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.
If gambling involves rational decisionmaking, 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 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.
Even where the house has an 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.
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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 Casino Games Expected Value Probability Calculator. 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. Jackpot Casino Aberdeen Sd Social Security 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 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. The formal calculation is as follows: When this EV calculation is performed for a 1unit amount, the negative of the resulting value is the house edge. Here are the calculations for bets on a singlenumber in doublezero and singlezero roulette.
Doublezero roulette single number bet: Singlezero 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, in 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. 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.
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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 bigsix 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 pokerbased table games: 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 singledeck game and only a 0.
Complete basic strategy tables can be found in many books and many casinohotel gift shops sell colorcoded credit card size versions.
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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 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.
Expected profit from lottery ticket
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.
Value Probability Calculator Casino Games Expected
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It can become harder in games where you don't have all the information you need to make the calculations; in games like that (such as poker), expected value often requires a little estimation along with the hard math. In order to make an expected value calculation, you'll need to know four things: the probability of winning.:Now turn to the casino. In the same way as before we can calculate the expected value of games of chance such as roulette. In the U.S. a roulette wheel has 38 numbered slots from 1 to 36, 0 and Half of the are red, half are black. Both 0 and 00 are green. A ball randomly lands in one of the slots. The purpose of this page is to explain the relationship between expected value (EV) and casino deposit bonuses. Expected value is a concept in probability that describes the average outcome of a random event. To calculate the expected value of any gambling game, you need just two key pieces of information. The Expected Value is a powerful tool to determine house advantage (a player's disadvantage) of any wager in any casino game, but it is also essential for mathematically perfect play of Poker. The risk, in contrast to uncertainty, can be measured by probability, e.g. we know what the probability to lose a stake is.

Tool to compute an expected value for a game. In mathematics, the probability of winning that indicates the chances of winning a given game, while expected value helps to know how much a player can earn (on average).:How to calculate expected value. nguyensan.me?! Oh Chillax. Obviously I'm going to explain. Expected Value is actually a pretty simple concept to grasp. All it does is use probability theory to break down your 'average' win/lose to give you a mathematical determination of exactly how much you're making or losing on that bet. The player's disadvantage is a result of the casino not paying winning wagers according to the game's "true odds," which are the payouts expected Calculate the house edge for American Roulette, which contains two zeros and 36 nonzero numbers (18 red and 18 black). Expected Value = (1)(18/38) + (−1)(20/38). I'm writing a post with 14 gambling probability examples because I think that examples are one of the easiest ways to teach something. .. If the game were mechanical, you'd be able to calculate the odds just by knowing the number of symbols on each reel, because each symbol would have an equal.
The basic Casino Games Expected Value Probability Calculator are lots
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, 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 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 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. If gambling involves rational decisionmaking, 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 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. Even where the house has an 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 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 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 EV , or 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. The formal calculation is as follows: When this EV calculation is performed for a 1unit amount, the negative of the resulting value is the house edge.
Here are the calculations for bets on a singlenumber in doublezero and singlezero roulette. Doublezero roulette single number bet: Singlezero 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 living , gamblers are faced with an uphill and, in 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. 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.
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 bigsix 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 pokerbased table games: 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 singledeck game and only a 0. Complete basic strategy tables can be found in many books and many casinohotel gift shops sell colorcoded 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 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 allimportant percentage that explains how casinos make money  is also called the house edge, the theoretical win percentage, and expected win percentage. In doublezero 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. The drop and hold percentage are affected by many factors; we won't delve into these nor the associated management issues. 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.
Furthermore, the house advantage is itself subject to varying interpretations. In Let It Ride, for example, the casino advantage is either 3. 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 In the long run, the casino will win 3. The expected value is a mean value, not necessarily the most probable outcome! The general formula of the expected value , EV , is the following:. The expected value can be both positive and negative.
A rationally acting person makes such decisions where the expected value is positive and refuses those decisions that bring negative expected value. However decisions with negative expected value can be admitted in case there is nothing better and we have to choose the lesser evil — we go for a decision with the least expected loss.
This strategy applies to Poker and is the key to longterm success. In terms of casino games, whereas the result depends solely on chance, the expected value is almost always in player's disfavor — it secures casino's longterm profit. The house edge comes from a difference between real and fair payout.
The real payout declared and paid out by a casino is lower than the fair payout. The fair payout is such a one when the expected value of a wager is 0, in other words when casino's longterm profit is 0. In the following exhibits we presume to bet one dollar.
There are typically only two possible outcomes of a wager — either a win or a loss. The win is then one dollar times the payout and the loss is always the one dollar.
We are able to determine the probabilities of winning and losing , therefore we can mark the following:. There are 18 red numbers, 18 black numbers and a zero in French Roulette , that is 37 numbers in total. There is an extra number in American Roulette — so called double zero — which makes the total of 38 numbers.
First let us have a look and the variables and the calculation for the French Roulette:. A player's disadvantage i. In case of this single number bet, a casino has an edge over the player 2. Let us see the expected value in the American Roulette with a double zero, i. Now 37 out of 38 numbers lose one dollar and only 1 number of of 38 numbers wins 35 dollars, so the expected value is then:.
The house advantage of the single number bet in the American Roulette is almost twice higher than in case of the French Roulette! If you feel like playing Roulette, which one of them will you choose? Evenmoney bets are characterized by the payout 1: The following calculation applies to the French Roulette. If you bet e.
If you're trying to make money, is it in your interest to play the game? To answer a question like this we need the concept of expected value. The expected value can really be thought of as the mean of a random variable.
This means that if you ran a probability experiment over and over, keeping track of the results, the expected value is the average of all the values obtained. The expected value is what you should anticipate happening in the long run of many trials of a game of chance. The carnival game mentioned above is an example of a discrete random variable. The variable is not continuous and each outcome comes to us in a number that can be separated out from the others.
To find the expected value of a game that has outcomes x 1 , x 2 ,. Why 8 and not 10? Now plug these values and probabilities into the expected value formula and end up with: This means that over the long run, you should expect to lose on average about 33 cents each time you play this game.
Yes, you will win sometimes. But you will lose more often. Now suppose that the carnival game has been modified slightly. In the long run you won't lose any money, but you won't win any. Don't expect to see a game with these numbers at your local carnival. If in the long run you won't lose any money, then the carnival won't make any. Now turn to the casino.
Here are the calculations for bets on a singlenumber in doublezero and singlezero roulette. Doublezero roulette single number bet: Singlezero 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 living , gamblers are faced with an uphill and, in 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. 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. 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 bigsix 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 pokerbased table games: 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 singledeck game and only a 0. Complete basic strategy tables can be found in many books and many casinohotel gift shops sell colorcoded 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 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 allimportant percentage that explains how casinos make money  is also called the house edge, the theoretical win percentage, and expected win percentage.
In doublezero 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. The drop and hold percentage are affected by many factors; we won't delve into these nor the associated management issues.
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.
Furthermore, the house advantage is itself subject to varying interpretations. In Let It Ride, for example, the casino advantage is either 3. 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 In the long run, the casino will win 3.
So what's the house edge for Let It Ride? Some prefer to say 3. Either way, the bottom line is the same either way: 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: 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 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 numbers , the standard deviation SD can be used to determine confidence limits with the following volatility guidelines: Obviously a key to using these guidelines is the value of the SD. Computing the SD value is beyond the scope of this article, but to get an idea behind confidence limits, consider a series of 1, 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. Note that if the volatility analysis is done in terms of the percentage win rather than the number of units or amount won , the 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 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. This player earning potential also called player value, player worth, or theoretical win can be calculated by the formula: 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. 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. This is easy to see using the wellknown probabilities of winning and losing the banker bet: A casino in Biloxi, Mississippi gave players a Again, this is an easy calculation.
Usual 60 to 1 payoff: Promotional 80 to 1 payoff: 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. A small Las Vegas casino offered a blackjack rule variation called the "Free Ride" in which players were given a free righttosurrender 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. Gaming Regulation and Mathematics. Casino gaming is one of the most regulated industries in the world. Most gaming regulatory systems share common objectives: 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? 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.
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