Probability Commons^™

A Modified Maximum Likelihood Estimation Algorithm For Modeling Threshold Exceedances With The Generalized Pareto Distribution, Jeffrey Harkness

UNLV Theses, Dissertations, Professional Papers, and Capstones

A modified maximum likelihood estimation (MLE) algorithm is proposed for modeling threshold exceedances with the generalized Pareto distribution (GPD). The algorithm addresses multiple issues with an approach originally published in the Journal Computational Statistics and Data Analysis (Castillo and Serra, 2015). The modified algorithm is intended to be comparatively simple to understand and implement, accurate in the handling of boundary conditions, relatively fast and reliable for most data sets, and relatively easy to transfer between computer languages by leveraging existing optimization routines.

A reproducibility study of work in recent literature published in the journal Extremes (Belzile, et al., 2023) is …

A Mechanistic Model Of The Wash Shuffle And Monte Carlo Exploration Of Its Impact On Card Shuffling In Texas Hold’Em, Michael A. Alexeev, Peter B. Chi

UNLV Gaming Research & Review Journal

In casino games using a standard deck of cards, a wash shuffle is sometimes performed prior to the rest of the card shuffling procedure. Unlike other methods of shuffling, the wash shuffle has not yet been well studied. To this end, we first develop a mechanistic model of the wash shuffle based on our observation of how cards tend to move when a wash shuffle is being performed. Then, we use this model to simulate the card shuffling procedure used in casino poker rooms, and explore the resulting impact on where the cards are dealt in the context of Texas …

Cash Or Crash: Return To Player Percentages And Expected Value Of Crash Games, Robert H. Scott Iii, Mikhail M. Sher, Jonathan Daigle

UNLV Gaming Research & Review Journal

Crash games are a new type of casino game offered on some online gambling sites. The first crash game was created by the online cryptocurrency casino Bustabit. Other cryptocurrency casinos started offering their own versions of crash games. Now mainstream online casinos have developed their own crash games—notably Rocket by DraftKings. Crash games are easy to learn and play. They offer the opportunity to increase your bet by many hundreds of multiples—though, as we show when deriving expected values, these outcomes are rare. In this paper, we study the history of crash games and calculate theoretical values of crash game …

Payments Data In Gambling Research, Kasra Ghaharian, Mana Azizsoltani

International Conference on Gambling & Risk Taking

A considerable body of gambling-related research has leveraged gamblers' behavioral tracking data to address a broad set of research questions. These data have typically comprised of gamblers' betting-related behaviors including, for example, the frequency and volume of betting. The analysis of gamblers' payment-related behavioral data is far less common, but provides a fruitful avenue gambling-related research.

In this presentation we discuss a selection of potential research opportunities that payments transaction data presents. We supplement this discussion with specific analyses that have been performed by our research group. We also discuss knowledge gaps and areas for future research.

A Game-Theoretic Analysis Of Baccara Chemin De Fer, Ii, Stewart N. Ethier, Jiyeon Lee

International Conference on Gambling & Risk Taking

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Snackjack: A Toy Model Of Blackjack, Stewart N. Ethier, Jiyeon Lee

UNLV Gaming Research & Review Journal

Snackjack is a highly simplified version of blackjack that was proposed by Ethier (2010) and given its name by Epstein (2013). The eight-card deck comprises two aces, two deuces, and four treys, with aces having value either 1 or 4, and deuces and treys having values 2 and 3, respectively. The target total is 7 (vs. 21 in blackjack), and ace-trey is a natural. The dealer stands on 6 and 7, including soft totals, and otherwise hits. The player can stand, hit, double, or split, but split pairs receive only one card per paircard (like split aces in blackjack), and …

Teaching A University Course On The Mathematics Of Gambling, Stewart N. Ethier, Fred M. Hoppe

UNLV Gaming Research & Review Journal

Courses on the mathematics of gambling have been offered by a number of colleges and universities, and for a number of reasons. In the past 15 years, at least seven potential textbooks for such a course have been published. In this article we objectively compare these books for their probability content, their gambling content, and their mathematical level, to see which ones might be most suitable, depending on student interests and abilities. This is not a book review (e.g., none of the books is recommended over others) but rather an essay offering advice about which topics to include in a …

A Statistical Analysis Of The Roulette Martingale System: Examples, Formulas And Simulations With R, Peter Pflaumer

International Conference on Gambling & Risk Taking

Some gamblers use a martingale or doubling strategy as a way of improving their chances of winning. This paper derives important formulas for the martingale strategy, such as the distribution, the expected value, the standard deviation of the profit, the risk of a loss or the expected bet of one or multiple martingale rounds. A computer simulation study with R of the doubling strategy is presented. The results of doubling to gambling with a constant sized bet on simple chances (red or black numbers, even or odd numbers, and low (1 – 18) or high (19 – 36) numbers) and …

Optimal Conditional Expectation At The Video Poker Game Jacks Or Better, Stewart N. Ethier, John J. Kim, Jiyeon Lee

UNLV Gaming Research & Review Journal

There are 134,459 distinct initial hands at the video poker game Jacks or Better, taking suit exchangeability into account. A computer program can determine the optimal strategy (i.e., which cards to hold) for each such hand, but a complete list of these strategies would require a book-length manuscript. Instead, a hand-rank table, which fits on a single page and reproduces the optimal strategy perfectly, was found for Jacks or Better as early as the mid 1990s. Is there a systematic way to derive such a hand-rank table? We show that there is indeed, and it involves finding the exact optimal …

Golden Arm: A Probabilistic Study Of Dice Control In Craps, Donald R. Smith, Robert Scott Iii

UNLV Gaming Research & Review Journal

This paper calculates how much control a craps shooter must possess on dice outcomes to eliminate the house advantage. A golden arm is someone who has dice control (or a rhythm roller or dice influencer). There are various strategies for dice control in craps. We discuss several possibilities of dice control that would result in several different mathematical models of control. We do not assert whether dice control is possible or not (there is a lack of published evidence). However, after studying casino-legal methods described by dice-control advocates, we can see only one realistic mathematical model that describes the resulting …

Mathematical Models Of Games Of Chance: Epistemological Taxonomy And Potential In Problem-Gambling Research, Catalin Barboianu

UNLV Gaming Research & Review Journal

Games of chance are developed in their physical consumer-ready form on the basis of mathematical models, which stand as the premises of their existence and represent their physical processes. There is a prevalence of statistical and probabilistic models in the interest of all parties involved in the study of gambling – researchers, game producers and operators, and players – while functional models are of interest more to math-inclined players than problem-gambling researchers. In this paper I present a structural analysis of the knowledge attached to mathematical models of games of chance and the act of mathematical modeling, arguing that such …

Exact Statistical Inferences For Functions Of Parameters Of The Log-Gamma Distribution, Joseph F. Mcdonald

UNLV Theses, Dissertations, Professional Papers, and Capstones

The log-gamma model has been used extensively for flood frequency analysis and is an important distribution in reliability, medical and other areas of lifetime testing. Conventional methods fails to provide exact solutions for the log-gamma model while asymptotic methods provide approximate solutions that often have poor performance for typical sample sizes. The two parameter log-gamma distribution is examined using the generalized p-value approach. The methods are exact in the sense that the tests and the confidence intervals are based on exact probability statements rather than on asymptotic approximations. Exact tests and exact confidence intervals for the parameter of interest based …

A Gaming Application Of The Negative Hypergeometric Distribution, Steven Norman Jones

UNLV Theses, Dissertations, Professional Papers, and Capstones

The Negative Hypergeometric distribution represents waiting times when drawing from a finite sample without replacement. It is analogous to the negative binomial, which models the distribution of waiting times when drawing with replacement. Even though the Negative Hypergeometric has applications it is typically omitted from textbooks on probability and statistics and is not generally known. The main purpose of this thesis is to derive expressions for the mean and variance of a new application of the Negative Hypergeometric to gaming and gambling. Other applications are described as well.

General Coupon Collecting Models And Multinomial Games, James Y. Lee

UNLV Theses, Dissertations, Professional Papers, and Capstones

The coupon collection problem is one of the most studied problems in statistics. It is the problem of collecting r (r<∞) distinct coupons one by one from k different kinds (k<∞) of coupons. We note that this is equivalent to the classical occupancy problem which involves the random allocation of r distinct balls into k distinct cells. Although the problem was first introduced centuries ago, it is still actively investigated today. Perhaps its greatest feature is its versatility, numerous approaches, and countless variations. For this reason, we are particularly interested in creating a classification system for the many generalizations of the coupon collection problem. In this thesis, we will introduce models that will be able to categorize these generalizations. In addition, we calculate the waiting time for the models under consideration. Our approach is to use the Dirichlet Type II integral. We compare our calculations to the ones obtained through Monte Carlo simulation. Our results will show that our models and the method used to find the waiting times are ideal for solving problems of this type.

A Monte Carlo Analysis Of Hedonic Models Using Traditional And Spatial Approaches, Helen R. Neill, David M. Hassenzahl, Djeto D. Assane

Public Policy and Leadership Faculty Research

Hedonic regression analysis of single family homes typically includes structural variables, locational variables and neighborhood quality characteristics. When nearby properties are related, Dubin (1988) reports that error terms are spatially autocorrelated. Estimation methods for these spatially autocorrelated error terms or hereafter, spatial approaches, include maximum likelihood estimation (MLE) and kriging techniques such as kriged maximum likelihood estimation (KMLE). Unfortunately these spatial methods require massive computer resources and are limited to significantly fewer observations than traditional ordinary least squares (OLS). This paper investigates the combination of spatial approaches and Monte Carlo analysis, a method that approximates large data sets. A question …