Editing Casino game (section)

==House advantage==
{{anchor|House advantage}}
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Casino games typically provide a predictable long-term advantage to the casino, or "house", while offering the players the possibility of a short-term gain that in some cases can be large. Some casino games have a skill element, where the players' decisions have an impact on the results. Players possessing sufficient skills to eliminate the inherent long-term disadvantage (the ''house edge'' or [[vigorish]]) in a casino game are referred to as [[advantage play]]ers.

The players' disadvantage is a result of the casino not paying winning wagers according to the game's "true odds", which are the payouts that would be expected considering the odds of a wager either winning or losing. For example, if a game is played by wagering on the number that would result from the roll of one die, the true odds would be 6 times the amount wagered since there is a 1 in 6 chance of any single number appearing, assuming that the player gets the original amount wagered back. However, the casino may only pay 4 times the amount wagered for a winning wager.

The house edge, or vigorish, is defined as the casino profit expressed as a percentage of the player's original bet. (In games such as [[blackjack]] or [[Spanish 21]], the final bet may be several times the original bet, if the player doubles and splits.)

In American [[roulette]], there are two "zeroes" (0, 00) and 36 non-zero numbers (18 red and 18 black). This leads to a higher house edge compared to European roulette. The chances of a player, who bets 1 unit on red, winning are 18/38 and his chances of losing 1 unit are 20/38. The player's [[expected value]] is EV = (18/38 × 1) + (20/38 × (−1)) = 18/38 − 20/38 = −2/38 = −5.26%. Therefore, the house edge is 5.26%. After 10 spins, betting 1 unit per spin, the average house profit will be 10 × 1 × 5.26% = 0.53 units. European roulette wheels have only one "zero" and therefore the house advantage (ignoring the [[en prison]] rule) is equal to 1/37 = 2.7%.

The house edge of casino games varies greatly with the game, with some games having an edge as low as 0.3%. [[Keno]] can have house edges of up to 25%, slot machines having up to&nbsp;15%.

The calculation of the roulette house edge is a trivial exercise; for other games, this is not usually the case. Combinatorial analysis and/or computer simulation is necessary to complete the task.

In games that have a skill element, such as [[blackjack]] or [[Spanish 21]], the house edge is defined as the house advantage from optimal play (without the use of advanced techniques such as [[card counting]]), on the first hand of the shoe (the container that holds the cards). The set of optimal plays for all possible hands is known as "[[Blackjack#Basic strategy|basic strategy]]" and is highly dependent on the specific rules and even the number of decks used.

Traditionally, the majority of casinos have refused to reveal the house edge information for their slots games, and due to the unknown number of symbols and weightings of the reels, in most cases, it is much more difficult to calculate the house edge than in other casino games. However, due to some online properties revealing this information and some independent research conducted by [[Michael Shackleford]] in the offline sector, this pattern is slowly changing.<ref>{{cite news|title=Michael Shackleford is the wizard of odds|url=http://observer.com/2015/03/michael-shackleford-is-the-wizard-of-odds/|access-date=13 October 2015|work=[[observer.com|Observer]]|archive-date=18 October 2015|archive-url=https://web.archive.org/web/20151018084543/http://observer.com/2015/03/michael-shackleford-is-the-wizard-of-odds/|url-status=live}}</ref>

In games where players are not competing against the house, such as [[poker]], the casino usually earns money via a commission, known as a "[[Rake (poker)|rake]]".

===Standard deviation===
{{more citations needed|date=October 2015}}
The luck factor in a casino game is quantified using [[standard deviation]]s (SD).<ref>{{cite book|last1=Hagan|first1=general editor, Julian Harris, Harris|title=Gaming law : jurisdictional comparisons|date=2012|publisher=European Lawyer Reference Series/Thomson Reuters|location=London|isbn=978-0414024861|edition=1st}}</ref> The standard deviation of a simple game like roulette can be calculated using the [[binomial distribution]]. In the binomial distribution, SD = <math>\sqrt{npq}</math>, where ''n'' = number of rounds played, ''p'' = probability of winning, and ''q'' = probability of losing. The binomial distribution assumes a result of 1 unit for a win, and 0 units for a loss, rather than −1 units for a loss, which doubles the range of possible outcomes. Furthermore, if we flat bet at 10 units per round instead of 1 unit, the range of possible outcomes increases 10 fold.<ref>{{cite journal|last1=Gao|first1=J.Z.|last2=Fong|first2=D.|last3=Liu|first3=X.|title=Mathematical analyses of casino rebate systems for VIP gambling|journal=International Gambling Studies|date=April 2011|volume=11|issue=1|pages=93–106|doi=10.1080/14459795.2011.552575|s2cid=144540412}}<!--|access-date=13 October 2015--></ref>

: SD (roulette, even-money bet) = 2''b'' <math>\sqrt{npq}</math>, where ''b'' = flat bet per round, ''n'' = number of rounds, ''p'' = 18/38, and ''q'' = 20/38.

For example, after 10 rounds at 1 unit per round, the standard deviation will be 2 × 1 × <math>\sqrt{10 * 18/38 * 20/38}</math> = 3.16 units. After 10 rounds, the expected loss will be 10 × 1 × 5.26% = 0.53. As you can see, standard deviation is many times the magnitude of the expected loss.<ref>{{cite book|last1=Andrew|first1=Siegel|title=Practical Business Statistics|date=2011|publisher=[[Academic Press]]|isbn=978-0123877178|url=https://books.google.com/books?id=cyc_TWqZDO4C|access-date=13 October 2015}}</ref>

The standard deviation for [[pai gow poker]] is the lowest out of all common casino games. Many casino games, particularly slot machines, have extremely high standard deviations. The bigger size of the potential payouts, the more the standard deviation may increase.

As the number of rounds increases, eventually, the expected loss will exceed the standard deviation, many times over. From the formula, we can see that the standard deviation is proportional to the square root of the number of rounds played, while the expected loss is proportional to the number of rounds played. As the number of rounds increases, the expected loss increases at a much faster rate. This is why it is impossible for a gambler to win in the long term. It is the high ratio of short-term standard deviation to expected loss that fools gamblers into thinking that they can win.

It is important for a casino to know both the house edge and variance for all of their games. The house edge tells them what kind of profit they will make as a percentage of turnover, and the variance tells them how much they need in the way of cash reserves. The mathematicians and computer programmers that do this kind of work are called [[gaming mathematics|gaming mathematicians]] and gaming analysts. Casinos do not have in-house expertise in this field, so they outsource their requirements to experts in the gaming analysis field.