Editing Casino game (section)

===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.