Optimal play mathematical studies of games and gambling

From an elementary probability standpoint, every bet except one in American roulette has a house advantage of 5. Not that the basic house advantage changes, of course — the game is still unbeatable in the long run. Mark Bollman mbollman albion. His mathematical interests include number theory, probability, and geometry. His claim to be the only Project NExT fellow Forest dot, who has taught both English composition and organic chemistry to college students has not, to his knowledge, been successfully contradicted.

If it ever is, he is sure that his experience teaching introductory geology will break the deadlock. See the table of contents in pdf format. Skip to main content. Search form Search. Login Join Give Shops. Halmos - Lester R. Ford Awards Merten M. Stewart N. Ethier and William R. 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 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 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 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.

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 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 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, 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. 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.

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. No matter. 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 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.