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• Casino Math Formulas
• The Hangover Math Casino Scene
• Casino Dealer Math Practice Test
• Math Casino Games
• Math Igler's Casino

The LIFE Science Library about several broad topics, published in the 1960s, included the tome Mathematics. The section on probability and combinatorics featured games with playing cards. The study of games of chance have a long and venerable history as a gateway to statistical sciences.

Slot Machine Math All gaming machines are designed to pay the player back a percentage of what is played. The amounts vary from machine to machine and from casino to casino. All machines have one thing in common: The longer the machine is played, the closer the actual payouts will be to. For slot free play, you can look at the casino house edge on the denomination of slot you’re going to play (which for some states is listed online, and averages/reporting available in some places) and do some quick math. But if you presume a 85-95% average payback based. Mathematics in casino gaming management. The reader will gain an understanding of how and why casino games produce the expected revenues. Topics discussed in this text range from the basic principles of probability, odds, expectation, house advantage, and the law of averages, to price setting using game odds, gaming.

In addition to games of pure chance, we have games of skill, often with an element of luck vs. misfortune. The nature of casino gambling is not fixed for all time and in this curriculum segment our games may well serve a philanthropic purpose, in that winnings are committed to some worthy cause or charity.

Casino Math, by virtue of including the topic of permutations, may be extended to include simple group theory concepts.

Note to teachers: one option is to introduce the J language (a computer language) and its native primitive for outputting in cyclic notation. Introducing more than one computer language is in keeping with the overall agenda for these four interconnected curriculum segments: Casino Math, Supermarket Math, Neolithic Math and Martian Math.

Note to teachers: playing games of chance (and some skill) need not imply one is gambling for money (consider solitaire). If your community actively discourages gambling and expects you to share this attitude, then you should feel free to defend your ethical position without neglecting your responsibility to model random or chance events, accidents, preferably in the context of simulations and/or games, to reduce the chance of injury.

Historical note: a casino need not be for gambling, may be a gathering place or public hall. For example, the casino on Catalina Island in the town of Avalon, is a movie palace at the time of this writing. Math games played in a casino setting might have more to do with monitoring work flow in some settings (like mission control?).

Depending on the needs of the surrounding curriculum, you may want to build this as a straight statistics course.

On the other hand, you may live in a subculture that actually staffs real casinos, or trains some of its people to program a next generation of game.

Either way, Casino Math may become your laboratory for discovering object oriented programming and its potential to satisfactorily define simplified (better understood) simulations of real, naturally occurring systems.

Chaotic Sequences

An original purpose of the RAND Corporation was to publish sets of random numbers. However, any sequence generated by a computer program that might be duplicated is deterministic and in that sense pseudo-random. Chaotic sequences have statistical properties without being precisely predictable over the long run by means of algorithmic short cuts.

Because these limits to precision may have major long term consequences (the butterfly effect), the idea of a strange attractor (e.g. the Lorenz Attractor) became important. Chaotic sequences are not without order, even where the algorithms, often dependent on previous results, provide no short cut to the future.

Studies of chaotic sequences and their properties may involve forays into cellular automata, fractal geometry. A library of machine executable pseudo-random number generators should be harnessed and pressed into service, if the requisite hardware and electricity is available. Python's random module might be a place to start.

Card Playing Objects

Games of cards provide a rich coding environment for object oriented languages in that one expects Card, Deck and Player objects at least.

• A deck comes with methods for shuffling and dealing, keeping track of how many cards have been dealt. A discard pile may also be modeled as a deck.

• A card object has attributes of suit and rank, and methods for comparing its value with other cards.

• Player objects would typically consist of hands, rules for their evaluation.

A player might also have methods, based on probablistic analysis, for guessing the contents of other player hands (or hidden cards in the player's own hand).

The object oriented apparatus of polymorphism, inheritance, getters and setters, private and public variables, class blueprints versus objects (instances), has the potential to come through clearly provided the student has some previous grounding in card games.

The rules of chance are best explored in non-punitive environments, which is why the practice of gambling for money may be excluded by one's subculture as too likely to cause harm (too risky in a foolhardy sense). Please note that dissecting (analyzing) games of chance, coding them in software, does not constitute an endorsement for gambling or investing in the stock market.

Unlocking Codes

Information Theory suggests that random noise and high bandwidth communication share the same lack of redundancy or predictability.

In other words, an information stream such as the digits of pi, with completely predictable content, might be considered low in information content in the sense of highly compressible. The same information could be packaged far more efficiently.

On the other hand, information with no discernable rules for lossless compression, have the appearance of being maximally dense, which may mean they're maximally informative communications, or they may be random noise.

Casino Math includes the idea of making some information secret, perhaps through cryptographic techniques, with the flip side being the game of deciphering secrets, recovering lost and/or hidden messages. These topics are sometimes packaged under Spy Math by some authors.

Note to teachers: the topic of public or symmetric key cryptography, RSA for example, might be undertaken in connection with deciphering, the related statistical studies around word and/or vowel sound frequencies (cite Who is Fourier?). On the other hand, this topic might be relegated to Supermarket Math, in connection with eCommerce.

Investing in Possible Futures

Casino Math provides students with the means to appreciate the scientific method. Thanks to educated guesses or hypotheses, either purely speculative or reality-checked by experiment, one potentially reduces one's reliance on pure luck or miracles and maximizes one's chances of realizing one's plans.

Work consistent with nature's principles (e.g. in cahoots with the weather) is more likely to bear fruit than work undertaken blindly and/or with only minimal appreciation for the underlying patterns and systems.

Thanks to scientific study, many natural systems have yielded enough secrets to have become both predictable and sometimes even manageable to some degree. This frees us to devote more attention to as yet poorly understood yet often critical systems.

Engineering is about leaving no more to chance than is necessary, is about optimizing the trade-offs, maybe getting a better outcome in exchange for concerted study. As our knowledge and level of mastery increases, we're able to do more with less, creating more life support (wealth) for more people, for more days into the future.

We look at Neolithic Math (one of three related modules) as marking the advent of a long epic saga about learning to work more sustainably and effectively within our shared natural environment, Spaceship Earth and surroundings.

Albert Einstein supposedly once said: “No one can win at roulette unless he steals money from the table while the croupier isn’t looking.”

Although I wouldn’t normally question Einstein, this statement isn’t true. In fact, you can use Einstein’s specialist subject, physics, to help you win. Or you can find a biased wheel that makes some numbers more likely to come up.

What Einstein actually meant was that there is no mathematical trick that can help you win at roulette. Each spin is an independent trial and, in the long run, the casino will win. This is different to a game such as Blackjack where the probabilities change as cards are dealt.

But some believe that it is possible to exploit the way the roulette wheel, and the betting cloth, is laid out to give themselves an advantage. The idea is that you can make bets on the layout in a way that you are guaranteed to win. But is this really possible?

Roulette wheel layout

Like a dartboard, the layout of a roulette wheel did not come about by accident. It was carefully planned and exhibits certain properties. In fact, there are two different layouts. An American wheel and a European wheel. The two layouts are shown below.

Notice that the American wheel has two zeroes. This is important as it doubles the advantage for the casino. On a European wheel you would expect to lose, in the long run, 2.7% of any money you bet with. On an American wheel you can expect to lose 5.26% (if you are interested in the mathematics of roulette, the video at the end will show you how these odds are calculated).

Casino Math Formulas

The numbers are arranged in a different order on each wheel but there are some similarities in the patterns. On both wheels, the red and black numbers alternate around the wheel, although if you removed the zeroes, the American wheel would have consecutive reds and blacks. The wheels are also structured so that the low numbers (1-18) and the high numbers (19-36) should alternate as much as possible.

On a European wheel, this is only violated where the 5 sits next to the 10 (both low numbers). On the American wheel, there are many examples where this rule is violated. It is for this reason that the American wheel is considered not as balanced as the European wheel. Both wheels also try to distribute odd and even numbers as evenly as possible. But again there are a number of violations of this rule on both wheels.

On the European wheel there are two other interesting symmetries. First, all the low red numbers and black high numbers are on one side of the zero, and the high red numbers and low black numbers are on the other side. Second, the sequence 29-7-28-12-35-3-26-0-32 contains no numbers between 13 and 24 (the second dozen). You can place a bet on the whole of the second dozen, with odds of 2-1.

The Hangover Math Casino Scene

So, can we beat the maths?

A simple search on Google will return many (possibly millions) of systems for playing (and supposedly winning) roulette. Some easy, some complicated, some well described, some not so.

Casino Dealer Math Practice Test

A system should really be a combination of a playing strategy and a money management strategy. Perhaps the best known money management strategy is the Martingale system. This system is guaranteed to win money as long as you have enough of a bankroll to double your bet after every loss and you do not hit the table limit, which you will quickly do so. The Martingale system is probably the quickest way to bankruptcy known to man.

Math Casino Games

Whatever betting strategy, and money management strategy, you choose, they all suffer from the same fate. Assuming that each number on the wheel has the same probability of being selected – meaning the wheel is not biased – the maths means the casino will always win. The system may look good, and may work in the short term, but when one of the numbers comes up that you have not bet on you will lose and the casino will move towards its win expectation (2.7% or 5.26%).

Some systems involve betting on many numbers, perhaps 20. In this case, you will win quite often as you are covering more than half of the numbers. But when one of the numbers does not turn up (and it will almost half the time) you lose all of the 20 bets you have made. This will often wipe out any wins to date.

Any system, so far devised, can be analysed to show that there is a win expectation for the casino. The following video shows the maths.

You might as well place a single chip on the same number every time and hope that it appears more than it should during the short time that you are playing.

Math Igler's Casino

We can dress up the layout of the wheel, the layout of the betting cloth, our number selection and our money management system however we like, but the maths is always there, quietly working against us. You might as well just have fun, pick random numbers and trust to Lady Luck. Either that, or do as Einstein suggested and steal chips (not that we’d recommend it).