Casino Games and the Central Limit Theorem
2013; Volume: 17; Issue: 2 Linguagem: Inglês
ISSN
1531-0930
AutoresAshok K. Singh, Anthony F. Lucas, Rohan J. Dalpatadu, Dennis J. Murphy,
Tópico(s)Sports Analytics and Performance
ResumoAbstractThe central limit theorem, in simple terms, states that the probability distribution of the mean of a random sample, for most probability distributions, can be approximated by a normal distribution when the number of observations in the sample is 'sufficiently' large. Most applied statistics books recommend using the normal approximation for the probability distribution of the sample mean when the number of observations exceeds 30. It is commonly known in the discipline of statistics that larger samples will be needed when the underlying probability distribution is heavily skewed. However, the minimum number of samples needed for the CLT to yield a reasonable approximation, when the distribution being sampled is heavily skewed, is not known. The Berry-Esseen theorem does provide an upper bound on the error in approximating the probability distribution of the sample mean by the normal distribution, but this upper bound turns out to be of no value when applied to slot games. The pay-out probability distributions of many casino games such as slots are heavily skewed, yet the CLT is used for calculating 'confidence limits' for total casino win, or rebates on losses, for these games. We will use Monte Carlo experiments to simulate the play of a few slot games and the table game of baccarat to estimate the probability distribution of the mean payout for sample sizes as large as 4,000, and compare it to the normal distribution.(ProQuest: ... denotes formulae omitted.)IntroductionThe law of large numbers (LLN) and the central limit theorem (CLT) are the cornerstones of inferential statistics. The LLN ensures that as the number of random samples collected from a probability distribution is increased, the sample mean converges to the true population mean, and the CLT guarantees that the sampling distribution of the mean will be Gaussian, provided there are a sufficient number of independent observations. The law oflarge numbers was utilizedby the Chevalier de Mere (1607 - 1684), an astute gambler (see Maxwell, 1999), and first proven by Jakob Bernoulli in 1713 (Bauer, 1996). In 1733, Abraham de Moivre introduced the concept of the Gaussian distribution as an approximation to the binomial distribution. This result, now called the theorem of de Moivre - Laplace (Feller, 1968), is a special case of the CLT. The CLT is invoked in deriving formulas for confidence intervals in estimation problems and critical regions in hypothesis testing problems.There are many practical applications of the central limit theorem. The CLT provides justification for many procedures in statistical process control and statistical quality control. The concept of Six Sigma derives from the central limit theorem. The best- known practical application of the LLN and the CLT is the casino gaming industry. The profitability of all of the casino games is guaranteed by the LLN and the CLT. In other words, these two theorems ensure that, in the long run, any casino game will result in a positive casino win, provided the game carries a positive expected value. For example, some video poker games have offered a positive expected value for the player, if played perfectly.Applications of the central limit theorem exist in the gaming literature as well. Johnson (2006) used the central limit theorem to calculate optimal keno strategies. Ethier and Levin (2005) have derived a variation of the central limit theorem and provided a simplified proof of Thorp and Walden's theorem of card counting.The approximate normality of the sample mean for large samples is routinely used for calculating confidence limits on actual casino win from a table game, or actual casino win from a given player (Hannum and Cabot,2005). The PAR (probability accounting report) sheet of every slot game on a casino floor includes, among other things, confidence limits of payback percentage for given numbers of games played (Hannum and Cabot, 2005; Harrigan and Dixon, 2009). …
Referência(s)