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.

Secret Formula Of Black Jack Winners

When someone plays a game, they play to win. One of the many simple games at the casino is Blackjack. For those who don’t know how to play the game, it’s always best to start by learning the basics.

When playing Blackjack the most important feature to remember is that there are four times as many cards that have a value of 10 (10, J,Q,K) as any other value. With this information, one must always assume that the next card to be drawn from the deck has a value of 10.

By keeping this in mind, one may realize that if one has any hand that has a value over 12 one will likely lose or in blackjack terms bust (go over 21). Likewise if the dealer’s card is a 2-6 it is likely to draw a 10 card which means he will have to draw again, likely causing him to bust.

Another way to make the odds more favorable to the player than the dealer is by counting cards. This of course requires many pages and charts that must be memorized. Having a professionally written guide is very helpful and there are many different authors that have different styles and different counting systems. It’s always best to get more than one opinion to find out which style works best.

Practice makes perfect. By investing time and money with online training software learning the game, it will eventually pay off at the casino.

Upon entering the casino, one must choose the right table to play at. When one observes that the players at a table seem to be winning a lot, it is a good time to join in the action. Likewise, if you notice the dealer has a lot of chips and the players seem to be doing more losing than winning, then look for a better table.

Reward the dealer by tipping when one is winning. If the dealer knows he will be tipped when the players wins, he may begin rooting and actually do subtle things to help such as performing less thorough shuffles and penetrating deeper into the shoe before he re-shuffles thus allowing the player to get a more accurate count of the cards.

Though if the dealer starts losing too much money for the casino the pit boss will likely change dealers.

Although it may be impossible to find the perfect game, one should be familiar with how certain house rules affect the player.

Keep in mind that as a rule of thumb, the fewer decks the casino uses the more advantageous it is for the player.

Information About Free Casino Slot

Most people believe that the slots are the worst bet there is in a casino. This is because earnings are not as high compared to other casino games. Some people say that this is the very reason why the slots are the favorite game of new casino players and women.nnBut this assumption is not true ? slots are not the worst bet, even when we are talking about free casino slot which can be played online. There are actually a lot of casino games which have higher house edge and lower payoffs.nnMost free casino slot machines have an average house advantage of 2% up to 15%. Looking at the reports coming from the Nevada Gaming Commission and online slots, the average is at around 8%. Now, here\’s a list of casino games and the corresponding average house advantage;nnBaccarat (tie bet): 14% Casino War (tie bet): 18% Craps (any):11% Sic Bo (depends on the bet): up to 33% Big Six: 11-24% Keno: 25-30%nnLooking from a purely mathematical point of view and ignoring player errors, we see that casinos have a higher advantage by offering the games above. Now, let\’s consider the way the games are being played and if a player is playing properly. Many games offer a low house advantage only if the game is being played properly. So if you love poker, you ought to know what kind of poker your favorite casino is offering. The better you are at a game, the higher your chances are of winning.nnThis is not true when it comes to free casino slots. Slots are so simple that the chance of making a mistake is so low. According to most \”Slots experts\” playing the maximum bet is a great idea since you will surely break even when you win – this is because no other game in a casino has a higher bet to win ratio than that of slots. Sure, you might not win every time you play free casino slot but you will always have a high chance of earning good money for a relatively low bet.nnNow let\’s talk about entertainment value – the very reason why gambling was invented in the first place. The penny slots you find at pubs and bars now have an online counterpart, and it also offers hours of entertainment – yes, this is true even if you have the smallest bankroll known to man. There are some free casino slot which offer bonuses and video slots have even more features that will keep you glued to the edge of your seats.nnSo, back to the question: are slots the worst bet in a casino? Well, if you want to get rich, then slots is certainly not the game for you. But if you are looking for entertainment and a way to earn a small amount of money, then slots are your perfect choice. Remember, however, that casino games are all games of chances. Don\’t go playing a free casino slot expecting to get a thousand dollars. Just try to relax and enjoy, who knows, you might just get a bonus.

Choosing The Right Casino In Albuquerque For Gambling

Today, there are a number of casinos that are in or near Albuquerque that gamblers can play at. With so many local choices, there is little need for gamblers to go to Las Vegas in order to get exciting gambling action. Because there are so many choices in the state of New Mexico for casino gambling, it is helpful for players to figure out which casino to patronize. Choosing the right casino can make playing a lot more fun. And it can make gambling more rewarding as well. That is why it is important to choose well when looking for the right casino in Albuquerque to play at.

First of all, the casino in Albuquerque should have the games that the player wants to play. For example, a video poker player will want to make sure that the casino has a good selection of video poker machines. Those who want to play with slot machines should choose a casino with many different choices. Table players will want to make sure that their favorite games are offered. Most casinos will offer blackjack, roulette, and poker at a minimum for table players. Many also offer craps for those who like that game.

Secondly, the casino in Albuquerque should have a decent player’s reward program. A good rewards program will make it more fun to gamble at a particular casino. Although a rewards program will help enhance a gambler’s return on playing, players should also concentrate on the fun aspects of the rewards program as well. Many programs offer free show tickets, free meals, and free tournaments in order to entice players to return. High rollers often great special treatment with most casino rewards programs. Being treated like a VIP can be a great way to enhance the experience of gambling at a casino. Of course, it is also important to evaluate the cash bonuses that are offered by the program as well. Some casinos will offer more generous cash bonuses than others.

Another important criteria when choosing a casino in Albuquerque is look at the returns of the games being played. Most gamblers know that the house has an advantage at most or all of the games. However, a gambler will have more enjoyment when the house edge is lower. On a slot machine, it is usually impossible to figure out what the return is. However, it is possible to calculate the house edge on games such as craps, blackjack, poker, roulette, and video poker by looking at the payouts. Going to a casino with better payouts means the gambler’s money will last longer. That means more enjoyment. And that also means a higher chance of hitting the royal flush jackpot for video poker players.

When choosing a casino, many people prefer to stay at the casino’s hotel. That makes it easy to access the gaming over multiple days. Many gamblers prefer to stay at hotels that have higher quality rooms such as suites. In many cases, the casino’s reward program may offer free rooms to frequent gamblers and high rollers. It would be better to choose a casino with nicer hotel rooms if the casino is giving away free rooms. Other hotel amenities such as a spa and restaurant options may be important to some gamblers as well. Many gamblers find that they need a break from the gaming from time to time. Choosing a hotel casino with extra amenities can be well worth it.

There is a lot to think about when gamblers are choosing a casino to play at. If gamblers concentrate their play at a single casino, they can often get better rewards from the casino’s frequent gambler program. That is why it is beneficial to choose a single best casino at the beginning. And the best part is that there are a lot of great casino choices that are in or near the Albuquerque area.

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).