Casino Games and the Central Limit Theorem

Abstract

The 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.