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).

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.

A Closer Look At Positive Expectation

What do I mean when I refer to an over 100% machine?” An over 100% machine simply means a machine that pays back more than you put in. Fantastic, except for one thing: this is factored over a long period of time. It does not mean over 100% in every session.” In fact, an over 100% machine isn’t guaranteed to earn you a profit even over what most would consider a long period of time. And it’s certainly not graphed by a steady upward curve. The only sure thing I’ve ever found in casino gambling is streakiness, and video poker results can be as streaky as it gets.
Also, and this is a very important concept to grasp: whether or not you achieve that over 100% return depends on the number of royals you hit. A royal contributes approximately 1.7%-2% (depending on the game) to the total payback percentage, and you only hit a royal about once every 35,000- 45,000 handssome 80 hours of play. So during those other 79 hours, even if you’re playing a deuces wild machine with a 100.76% payback percentage and accruing .5% slot club cash back for your play, that 1.8% you lose for not hitting the royal pretty much wipes out your advantage during the periods in between the big jackpot. During these periods, you will most likely lose money (positive expectation is sweet, but it usually doesn’t come easily). That’s why it’s necessary to analyze situations in terms of what mathematicians call expected value” or expected return,” which assumes an infinite period of play.
Okay, with that groundwork laid, here are a couple of examples of positive expectation readily available in any number of casinos in Las Vegas at any given time.
Playing full-pay” deuces wild with computerperfect (optimal”) strategy, the return percentage is 100.76%. Another way to say this is: for every $100
you put into a full-pay deuces wild video poker machine, you’ll get back $100.76. It may look skinny,” but that’s a pretty good edge. Unfortunately, only a computer can play computer-perfect strategy. Humans make mistakes. They get tired of sitting and staring at a screen. They get a little fuzzy from the free drinks. The guys get distracted by short skirts on cocktail waitresses. I figure skilled human strategy on a full-pay deuces wild machine is more like 100.5%. In these examples, my numbers are based on that one-half-percentage-point advantage.
Playing 25 full-pay deuces, you’re making $1.25 bets (five quarters). If you play slowly (say 320 hands per hour), you’ll put about $400 through the deuces wild machine in 60 minutes. At our skilled human capability of 100.5%, over that hour your expected return is $402 (1.005 X $400 = $402). Thus, your expected return (or win) is $2 per hour. Don’t quit your day job!
Example 2. This time you’re playing $1 deuces wild with skilled human strategy, but you’re playing fast (600 hands per hour). Betting $5 per hand, you’re putting $3,000 through the machine per hour. Now your .5% positive expected return produces $3,015; over the long term you’ll take in $15 per hour. That’s like a $30,000-a-year paycheck.

A Closer Look at Negative Expectation

The flip side of positive expectation is, logically enough, negative expectation. An under 100% machine” is a machine that pays back less than you put in. Let’s say you find a 25 deuces wild machine that has only one change in the pay table from the full- pay deuces schedule. A key changeit pays only four coins for four-of-a-kind instead of five. That’s just one tiny alteration, but it gives the house a whopping 5.7% edge. This means that you’re playing a game that returns only 94.3%. Now if you play a slow 320 hands per hour, you’re losing $22.80 per hour (.057 X $400)and that’s on a 25 machine.
Does the $2-per-hour profit from full-pay deuces look better to you now? And does the $22.80-per- hour loss convince you to read pay tables carefully? The main reason for all the new video poker variations is that casino know there are a lot of players with a little bit of video poker knowledge. Many people now know that full-pay jacks or better and deuces wild are good games, but they’re not real careful about reading the pay tables to make sure that full-pay is what they’re playing. Point them toward any jacks or better or deuces machine and they assume it pays liberally. The casinos, of course, make sure that they aren’t: Play this game. It’s almost like the other one that pays a lot more.”
This occasionally backfires on a casino when it monkeys with a pay schedule of a well-known machine and the changes result in a larger payback percentage. If you become a true student of the game, you’ll be able to spot these situations, know what changes to make in the strategy, and take advantage to make a lot of money on these machines, until the casino wises up and fixes the schedule or removes the machines entirely.

Jacks or Better

Jacks or better is the basic video poker game. One of the first to come out, it’s considered something of a video poker standard. The full-pay version is known as 9/6 jacks or better, meaning the payback (with one coin in) for a full house is 9 coins and for a flush is 6 coins. Sad to say, there are not a lot of them outside of Las Vegas. Full-pay jacks or better has a return percentage of 99.5%. It’s not a positive game; as we’ve seen, the casino has a .5% advantage. But because it’s readily available in Vegas and a lot of other video poker variations are derived from it, I also consider it the standard. Often you get enough cash back from the slot club or a promotion to make jacks or better an over 100% game.The first chart is what the pay table for a 9/6 jacks or better video poker machine looks like. The second is also a jacks or better pay table, but with a crib- cal difference: it’s an 8/5 machine. Note the difference in the payouts for full house and flush; this lowers the return to 97.3%.
As I mentioned earlier, I’m not going to provide you with the strategies, which are readily available in any number of books and computer software programs. See the Appendix for recommendations.

Slot Machines For Casino Gambling People

There’s a lot of available games to play inside a casino whenever somebody wants to go Casino Gambling. A gambler could pick to play card games, roulettes, and other games that have evolved over time. But among these, slot machines remain to be one of the favorites of people who love Casino Gambling.

Slot machines have been around almost instantaneously as casinos. The very first machine was created in 1887. It featured the commonly known configuration of the machine. This one-armed bandit” was also able to feature an automated system that enabled it to simplify the payout mechanism of the machine. This became popular and soon found its way into every local bar or saloon at that time. Thus, possibly initiating the beginning of Casino Gambling.

Today, variations in the popular machine have become an act of creativity for its developers. Different machines come up with very different themes. The variations could go as creative as using pirate themes, geometric shapes, just to name a few. This is probably the reason why in Britain, slot machines are more commonly called as the fruit machines”. But whatever theme developers might come up with, the basic principle behind it still remains. The characters and the number of lines might change, but nevertheless, they still follow the original game structuring of its origin. It adds to the flavor of course, that Casino Gambling people are able to enjoy the machines they are betting on, given that they have evolved to become more fun.

The development of the slot machine did not come without myths surrounding it. Some Casino Gambling people believe that certain machines become either hot” or cold”, depending on the number hits or wins a particular machine has given. Although myths such as this have crept its way throughout different places, the odds of winning on a standard slot machine is determined by an RNG or Random Number Generator. This means that on standardized machines, the odds of winning is always the same.

There are however, some machines that are specifically programmed to increase the probability of winning. Depending on the state laws that govern a particular casino, Casino Gambling on slot machines are required to give out what is called a payout percentage. Usually, the payout percentage is somewhere between 82 to 98 percent. But in the case of Nevada where state laws are not the same as others, the baseline payout percentage of machines is set to 75. This means that supposing one spin costs $1, after 1,000,000 spins have been played, the machine should have at least returned an average of $750,000 to the players. And so goes for other machines that have a different payout percentage set.

Other machines that are commonly seen when going Casino Gambling are those that give out huge jackpot prizes. The machines that could give out as much as $7,000,000, are linked in a network of different machines. The network could also extend to other casinos, enabling a link between one casino’s slot machines and the other’s. However, because of the large jackpot prize being allotted by these systems, the usual case is that the payout percentage is smaller than the one from a single machine.

Casino Gambling is still predominantly based on probability. Although this is sometimes preset by programs that are usually required by state laws, the odds of winning on a slot machine and any other Casino Gambling endeavor still requires a certain amount of luck to pull off.