Casino Games And Mathematics. Part 2.

Thorp managed to find out that owners of gambling houses gave their officials rather strict directions with regard to the strategies which they should stick to in the game with visitors. Control over fulfillment of these directions had its initial aim to prevent from a frame-up of a croupier with the rest of the gamblers, a chance of which could not be excluded. Assigned for a croupier strict rules determining his game strategy really substantially reduced a probability of such a frame-up, but on the other hand, allowed an advanced” gambler to rather adequately reveal the essence of this strategy and effectively oppose it. For unlike a croupier a gambler needn’t show the first of the received cards, as well as isn’t enchained by any strict rules as regards his strategy, that is why flexibly changing his behavior he can confuse a croupier. For example, Thorp found out that practically in all gambling houses of Nevada State croupiers were strictly ordered to keep away from a widow in case the amount of points in his cards exceeded or was equal to 17, and a player, from our mathematician’s point of view did not have to miss an opportunity to make use of the knowledge of even some aspects of a croupier’s strategy for achievement of his aims. Thus, those advantages which had an official of a gambling house from the start (as we already know, he is not obliged to open his cards at the end of the game), can be compensated to a certain degree for the knowledge of a player about the strategic “tunnel vision” of a croupier.

Besides, as has been mentioned, Thorp, while building his strategy presumed that cards were not often shuffled, in particular, if after finishing of a regular game there were still cards left in a pack, a croupier did not collect the thrown-away by the gamblers cards but dealt them anew (and the next game was played), and only after complete exhaustion of a pack, an official of a gambling house collected all the cards, thoroughly shuffled them and a new cycle” began. Naturally, if a gambler had a good memory he could change his strategy depending on the knowledge of the cards which had gone out of the game, and what cards could still be counted upon. It is important to remember that a croupier himself who was to strictly follow the directions of the casino’s owners practically without changing his strategy!

Thorp set himself a task to formulate the rules which would allow him to calculate probabilities of taking out one or another card out of an incomplete pack. Knowing these probabilities a gambler could already with reasonable assurance draw cards from the widow without being too much afraid of a pip out”, and besides, on the basis of the knowledge of some aspects of a croupier’s strategy to make suppositions about those cards which he had, and other gamblers as well. Naturally, as a gambler was to make a decision with regard to a widow very quickly, the sought rules for calculation of probabilities were to be rather simple for a gambler to be able to use them in mind” with the help of neither a calculator, nor a pen and paper (even if we suppose that a gambler will be given a chance to do calculation on paper, it will certainly arise suspicion). Edward Thorp managed to solve this mathematical problem having created rather simple algorithms for calculation of probabilities of taking out of one or another card from a pack, and using them to build a strategy of the game of twenty-one which would not be very complicated, allowing a gambler to considerably increase his chances of winning!

As the Hungarian mathematician A.Reni states after a few days of presenting his report on the obtained results at the meeting of the American Maths Society in 1960 in Washington Thorp received from a businessman a letter with a check for 1 thousand dollars intended for checking of a winning strategy in practice. Thorp accepted the check and having learnt the formulated by him rules left for Nevada to try his discovery. The trial went well: less than after two hours Thorp won 17 thousand dollars.

Needless to say, the owner of a gambling house didn’t share Thorp and his companion’s delight with regard to a successful comeout of the trial and the next day did his best to prevent Thorp from joining in the game. Later on Thorp tried to penetrate into other gambling houses, but the news of him had already spread far and wide, so that the doors of all the gambling houses appeared to be closed for him. Several times having adjusted a fake beard or having got a make up of a Chinese, Thorp managed to get to the gaming-table, but in any disguise his constant gain invariably gave him away. Thorp had to refuse from further checking of the strategy developed by him”. Though “additional checks” were necessary” only to enrich the pockets of the talented mathematician. One could hardly doubt that E.Thorp managed to create a real winning strategy!

However, since he could no longer benefit from his discovery himself, he decided to render welfare assistance” to his colleagues having published in 1961 a small article in an American academic journal (Thorp E.O. “A favourable strategy for twenty-one”, Proc.Nat.Acad.Sci., 47, 110-112, (1961)). And despite the small size of the article and, consequently, an extremely condensed form of persentment, made it comprehensible for rather a narrow group of professionals, one can be sure that a number of American scientists and their friends certainly “improved” their material situation (owners of gambling houses were unlikely to read scientific magazines at that time).

Canadian Casinos North Of The Border For Fun & Games

Gambling in Canada expanded in leaps and bounds during the 1990′s.

Canada now boasts 59 casinos and 70 racetrack/slots venues in 7 provinces and NWT.

Provincial governments oversee gaming in each of the participating provinces.

The legal gambling age is 19 years of age. I suggest you call ahead for casino hours and location.

All information is correct at time of writing, but may change.

Nova Scotia

Casino Nova Scotia operates two locations in Halifax, 1983 Upper Water St. on the waterfront-Purdy’s Wharf and Sydney.

Quebec

All casinos have a dress code (no jeans), and no gratuities are accepted.

Casino Montreal: 1, Avenue du Casino, Montreal in Parc Des Iles. Nuance – 5 star restaurant.

Casino de Hull: 1, Blvd. du Casino, Hull. Across the river from Ottawa, Canada’s capital city.

Ontario

Contact Ontario Casino Corp. 416-326-0076 for current updates.

Charity Casinos: Four locations. All use The Winners Circle players card.

Brantford: 40 Icomm Drive, Brantford. Located 55 miles southwest of Toronto in downtown Brantford, ON. This celestial themed casino has a gift shop, a liquor license, Meteor Bar lounge and dining at the Cosmic Diner cafeteria. Open 24 hours.

Thunder Bay: 50 Cumberland St. S., Thunder Bay. Location: On the north shore of Lake Superior in Thunder Bay, ON. The area is rich in the history of mining, marine and fur trading. A sea faring themed casino of more than 14,000 square feet.

Point Edward: 2000 Venetian Blvd., Point Edward. Location: Ten blocks from Sarnia, Ont. near the Bluewater Bridge on the St. Clair River. Indoor sights of a ship’s hull, saltwater aquarium and glass bridge add to this nautical themed casino.

Casino Sault Ste. Marie: 30 Bay St. W., Sault Ste. Marie. Location: Northern Ontario approx. 400 miles north of Toronto. Great hunting and fishing in this area.

Eighteen racetracks have placed more than 9000 slots as per the provincial plan in partnership with the Ontario Lottery & Gaming Commission.

Manitoba

VLT’s available in many bars throughout Manitoba. Two of the best are:

Club Regent: 1425 Regent W., Winnipeg.
McPhillips Stn.: 484 McPhillips St., Winnipeg.

Saskatchewan

Casino Regina: Union Station, 1880 Saskatchewan Dr. is this province’s premiere casino.

Alberta

Casinos offer regular slots rather than VLT’s.

Edmonton is Canada’s fastest growing gaming city, and Casino Yellowhead is its jewel.

Home to the world’s largest shopping/entertainment complex – West Edmonton Mall.

Casino Yellowhead: Yellow Head Trail at 153rd St. located in the West Edmonton Mall.

At 73,500 sq. ft, Yellowhead is the largest casino in western Canada.

British Columbia

Great Canadian Casinos operates several casinos in BC.

Most locations offer these table games: Blackjack, Roulette, Mini-Baccarat, Red Dog, Let It Ride, Caribbean Stud Poker, Sic Bo, Texas Hold ‘Em, Seven-Card Stud Poker and slots.

Each casino is centrally located in large municipalities and offers currency exchange,food and beverage service, wheel chair accessibility and nearby public transportation.

Western Canada Lottery Corporation works in conjunction with the provinces of Alberta, Saskatchewan, Manitoba and territory of Yukon and Sport North Lottery Authority.

There are no taxes paid on lottery or casino winnings in Canada.

Being Canadian and born in Toronto, the research for this information was a labour (Canadian spelling) of love.

Casino Gambling

Casino gambling can be fun and exciting. At the same time expensive. Dependent on how much you are willing to gamble. It is important to keep in mind your specified budget. As it is very easy to get carried away when you are in the casino, you have to take care that the next thing you know, you haven’t gambled everything away and now you are down to your last penny. Before you get to this scenario, stop, think and walk away or quit casino gambling.

The first rule in casino gambling is to know the game. Casino gambling can be difficult if you are not sure about the rules of a game. For instance, if you want to play baccarat but don’t know exactly what baccarat means it is better to stand back, watch and learn the game first before you try your luck. Another golden rule that you should observe in casino gambling is to not gamble more than you can afford. Always know how much you can spend before you step foot in a casino and don’t risk any more than that.

Do not indulge in casino gambling with the aim to get rich. If you do, you will only be disappointed. And you may lose more money than you care to. Casino gambling should be for fun and entertainment. In casino gambling you should not hand a dealer the loose change. Willingly you place it in the table. If you find that your dealer is rude, then leave the table and play elsewhere. You should not put up with bad service. You can avail yourself of free casino gambling lessons. You could try casino gambling in the morning when there are less people in the casino. You could try casino gambling in the morning when there are lesser people in the casino. This way the dealer could offer you assistance when you are new to the game. In casino gambling you should not up your stakes to cover your losses. You may lose big time if you do. You can up your stakes when you are winning so you use the casino’s money instead of your own.

There are strategies you can use that can reduce the house’s edge but you can never beat the house. Another to watch out in casino gambling is the game called keno. This may be a simple game but the odds are always in favor of the house. If you are new to casino gambling, you should look for a casino that lets you play with lower stakes. This way if you commit mistakes, your losses will be smaller. The posh hotel-casinos located on the Strip in Las Vegas generally have higher minimums compared to those in the downtown area.

Casino gambling is a good way to add some exitement and break the monotony of your life. Casino gambling gives you an opportunity to engage your senses including your mental abilities to focus on the game. Casino gambling can be a good source of entertainment. You should not view casino gambling as a get-rich-quick scheme. But just like any endeavor, it should be taken in moderation. Anything that is done in extremes could be detrimental to the purpose casino gambling is aimed for- which is to provide release of stress and pleasure. Gambling, done in the extreme can become an addiction, so play in moderation and have some fun.

Casino Is The Temple Of Games.

An individual plays only when he is human in the full sense of the word, and he can be wholly human only when he is playing”. Friedrich Schiller

The history tells us that in 49 B.C. the famous ancient Roman commander Julius Caesar, having won brilliant victories over barbarian galls, wanted to seize supreme power in the Eternal City. Then the frightened senators forbade him and his troops to return to Italy.

Without hesitations, the commander announced about his decision regarding the senate ban by saying “Alea jacta est” (“The die has been cast”) and crossed the border river Rubicon. This led to lingering civil wars in Rome, which became one of the most significant events in ancient history.

The words of the ferocious dictator became proverbial, however, today there is hardly a person who thinks about its real meaning. As it turned out, the Great Caesar actually cast the dice. So great was his passion to gambling, that he deeply believed in the magic ability of the dice to predict the future.

The public opinion hardly favors gamble and everything connected with it. Conversely, in those times the word “player” was nearly a swearword – so huge was the contempt of the majority of people to those who gave the game its due.

But at the same time ancient people understood perfectly well that the thirst for gambling could not be eradicated. The Greeks with their characteristic rich imagination invented a myth about the goddess of the fate Tyche (the Romans called her Fortune”), who gave birth to Zeus’ daughter, and this girl was endowed with the gift of inventing various dangerous amusements, which caused the people to lose a lot of money, cheat, scuffle and committed suicides.

Tyche loved her daughter and thus winked at her cruel pranks. She even presented her with a large beautiful house, to which her daughter allured the most credulous players to make them miserable.
More than two thousand years passed since those times, and today hardly anyone believes in fortune-telling by throwing lot and myths about the goddess Tyche, but there is one thing that has not changed. It is the human need for the game. This unquenchable desire stipulated for the fact that in the course of time special premises were built for gambling as if the ancient Greek legend came true.

In these establishments visitors played with each other in these and paid a part of their winnings to the owner – or played with owner and then, if they lost, they were to pay the whole amount of the bet to the casino owner. Approximately in the 16th century such establishments came to be called by the Italian word “casino”, which has not changed its meaning up to now.

Gambling houses irrepressibly attracted people with different characters, different talents and varying financial possibilities. The list of famous casino frequenters, compiled by the largest casinos in Europe, includes such celebrities as chancellor Bismarck, composers Berlioz and Brahms, the writer Dostoyevskiy, the poet Mayakovsky and the automobile king Citroen.

Reverberating fame, however, did not prevent these people from insidious tricks of Fortune. Admittedly, some celebrities were often lucky and they won a lot. For example, Citroen was such a lucky player. He loved to play for high stakes, in order to impress other rich men. Journalists never grew tired of writing that the automobile king is as lucky on the green cloth, as in business.

Others mostly lost. For instance, Mayakovskiy was such an unlucky fellow. He loved billiards, cards and particularly the roulette. During his trips abroad the poet run into unprecedented debts, because he was lucky only at billiard table, but by no means in the roulette.

Gambling houses are known not only by their frequenters, but also by various legends that surround these establishments. The most enduring is the story how a certain Frenchman monsieur Blanchard won twice in “Casino Monte-Carlo”. When he intended to enter this casino for the first time, his hat was spoilt by the dove. Blanchard interpreted this as a good sign and was right. The player managed to win several thousands. Then he intended to go to casino once again, but on condition that a bird would spoil his hat for one more time.

He had to wait for the new dove for several days, but his expectations paid for themselves. The Frenchman was lucky that time and he won even more than before. After this, the doves displayed no interest in Blanchard and he could not win. However, all inveterate players believe that if the bird marks you before the visit to the gambling house is a true sign of good luck.

Casino Games And Mathematics. Part 3.

After one more year Thorp published a book (I mentioned it at the beginning of the article) in which he rather in details, in the form comprehensible to any even a slightly literate and sensible person, set the rules of formation of a winning strategy. But the publication of the book did not only cause a quick growth of those willing to enrich themselves at the cost of gambling houses’ owners, as well as allowed the latter ones to understand the main reason of effectiveness of the developed by Thorp strategy.

First of all, casinos’ owners understood at last that it was necessary to introduce the following obligatory point into the rules of the game: cards are to be thoroughly shuffled after each game! If this rule is rigorously observed, then a winning strategy of Thorp cannot be applied, since the calculation of probabilities of extracting one or another card from a pack was based on the knowledge of the fact that some cards would already not appear in the game!

But what does it mean to have thoroughly shuffled” cards? Usually in gambling houses the process of thoroughly shuffling” presupposes the process when a croupier, one of the gamblers or, that is still oftener seen of late, a special automatic device makes a certain number of more or less monotonous movements with a pack (the number of which varies from 10 to 20-25, as a rule). Each of these movements changes the arrangement of cards in a pack. As mathematicians say, as a result of each movement with cards a kind of substitution” is made. But is it really so that as a result of such 10-25 movements a pack is thoroughly shuffled, and in particular, if there are 52 cards in a pack then a probability of the fact that, for instance, an upper card will appear to be a queen will be equal to 1/13? In other words, if we will, thus, for example, shuffle cards 130 times, then the quality of our shuffling will turn out to be more thorough” if the number of times of the queen’s appearance on top out of these 130 times will be closer to 10.

Strictly mathematically it is possible to prove that in case our movements appear to be exactly similar (monotonous) then such a method of shuffling cards is not satisfactory. At this it is still worse if the so called “order of substitution” is less, i.e. less is the number of these movements (substitutions) after which the cards are located in the same order they were from the start of a pack shuffling. In fact, if this number equals to t, then repeating exactly similar movements any number of times we, for all our wish, can not get more t different positioning of cards in a pack, or, using mathematical terms, not more t different combinations of cards.

Certainly, in reality, shuffling of cards does not come down to recurrence of the same movements. But even if we assume that a shuffling person (or an automatic device) makes casual movements at which there can appear with a certain probability all possible arrangements of cards in a pack at each single movement, the question of “quality” of such mixing turns out to be far from simple. This question is especially interesting from the practical point of view that the majority of notorious crooked gamblers achieve phenomenal success using the circumstance, that seemingly “careful shuffling” of cards actually is not such!

Mathematics helps to clear a situation with regard to this issue as well. In the work Gambling and Probability Theory” A.Reni presents mathematical calculations allowing him to draw the following practical conclusion: ” If all movements of a shuffling person are casual, so, basically, while shuffling a pack there can be any substitution of cards, and if the number of such movements is large enough, reasonably it is possible to consider a pack “carefully reshuffled”. Analyzing these words, it is possible to notice, that, firstly, the conclusion about “quality” of shuffling has an essentially likelihood character (“reasonably”), and, secondly, that the number of movements should be rather large (A.Reni prefers not to consider a question of what is understood as “rather a large number”). It is clear, however, that the necessary number at least a sequence higher than those 10-25 movements usually applied in a real game situation. Besides, it is not that simple “to test” movements of a shuffling person (let alone the automatic device) for “accidence”!

Summing it all up, let’s come back to a question which has been the headline of the article. Certainly, it would be reckless to think that knowledge of maths can help a gambler work out a winning strategy even in such an easy game like twenty-one. Thorp succeeded in doing it only by using imperfection (temporary!) of the then used rules. We can also point out that one shouldn’t expect that maths will be able to provide a gambler at least with a nonlosing strategy. But on the other hand, understanding of mathematical aspects connected with gambling games will undoubtedly help a gambler to avoid the most unprofitable situations, in particular, not to become a victim of fraud as it takes place with the problem of cards shuffling”, for example. Apart from that, an impossibility of creation of a winning strategy for all “cases” not in the least prevents a mathematically advanced” gambler to choose whenever possible the best” decision in each particular game situation and within the bounds allowed by “Dame Fortune” not only to enjoy the very process of the Game, as well as its result.