Casino Games And Mathematics. Part 1.

Can the knowledge of mathematics help a gambler to win?

One can often hear that the best piece of advice given by a mathematician to a lover of gambling games is an assertion which lies in the fact that the best strategy in gambling games is complete abstention from participation in them. A lot of mathematicians consider that the most which the theory of probability and the theory of games can give a gambler are the strategies following which he won’t lose too much.

It is difficult to predict whether the American mathematician Edward Thorp shared this view, when once spending winter holidays in Las-Vegas, he, having entered a casino, decided to try his luck in the game of twenty-one. As it turned out, Dame Fortune” was extremely unkind to him. We do not know for sure what amount of money this teacher of mathematics of one of American universities lost that winter night at the end of the 50-s the beginning of the 60-s of the last century, however, judging by the following events the amount was not small. Otherwise, how can we account for the fact that development of an optimal strategy of this game became for a number of years an idte fixe” of our hero. Besides, the matter was not only in the quantity of money lost by the mathematician. Perhaps, Thorp was simply an extremely venturesome person, and his pride both of a gambler and an expert-mathematician was hurt. Besides, he could suspect a croupier of dishonesty, since, as he had noticed, cards were not shuffled after each game. Though, during the game itself it did not make him very uneasy. However, afterwards, having visited casinos a number of times, he noticed that as the rules did not presuppose obligatory shuffling of cards after each game, so it was difficult to accuse a croupier of anything. Anyway, he managed to develop a winning strategy in the game of twenty-one.

This strategy among other things was based on the same very aspect which had put a defeated mathematician on his guard cards were not shuffled too often. At that, this, apparently, as a rule, was done not because of some evil design, but in order to avoid, so to say, unnecessary slowdowns in the game. The results of his studies Edward Thorp put forth in a book published in 1962 (Thorp E.O Beat the dealer. A winning strategy for the game of twenty one. – New York: Blaisdell,1962.) which made owners of gambling houses in the state of Nevada essentially change the rules of the game of twenty-one. But let’s not ride before the hounds.

In accordance with the game rules of twenty-one of that time one croupier dealt gamblers two cards each out of a thoroughly shuffled pack consisting of 52 cards. Gamblers themselves did not show their cards to a dealing croupier. At the same time out of two cards taken for himself an official of a casino showed one of them (usually the first one) to gamblers. Gamblers evaluate their cards according to the following scale. Jacks, queens and kings have a value equal to 10 points, an ace could be assigned either 1 point or 11 points, the value of the rest of the cards coincided with their numerical value (eights had 8 points, nines took 9, and etc). That gambler was considered a winner who had cards on hand with the sum of points closest to 21 from the bottom. At that, having assessed the received cards every gambler (including a croupier) had a right to take from a pack or putting it simpler, take a widow”, any amount of cards. However, if, as a result, the total number of points after a widow, will exceed 21 points then a gambler must drop out of a game having shown his cards.

Special rules were established with regard to stakes. Initially, upper and lower bounds were set, and every gambler had a right of choice of a specific stake (within these bounds) depending on the evaluation of his position. If, as a result, it turned out that in accordance with the game rules a casino’s visitor had a “better” number of points on hand than a croupier had, he received a gain in the amount of the stake that he had made, otherwise, this gambler lost his stake. In case of an equal number of points of a gambler and a croupier, the game ended in peace, that is the result of the game is considered harmless” both for a gambler and a casino.

Let’s point out that unlike ordinary gamblers a croupier is not obliged to open his cards in that case if the number of points in these cards exceeds 21. Moreover, after all the gamblers have opened their cards, and therefore, all the stakes go to a casino gamblers cannot practically find out what was the number of points of a croupier, in order to build their game strategy for the next game (whether to risk or not to stand pat, and etc). It goes without saying, it gives a croupier considerable advantages. Besides, all the gamblers are surely aware of this, and, … continue to play. Nothing can be done about it, who does not take risks, as is known, does not win.

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.

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).

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.

Best Casino Gambling Pay-outs

Nobody comes to a casino simply because they want to have fun. Underneath this vague reason is the desire to bring home cash, and lots of it. Indeed, casino gambling is a way to get your hands on lots of cash in a short time. However, it can also be the way to lose cash in an even shorter time. The thing with casino gambling is that the games have different pay-out rates. For example, spending half an hour in the slots will not help you reach the hundred dollar mark unless you win the jackpot. But if you go and spend time at the roulette table, you can reach the 200-dollar mark without waiting for 30 minutes. Likewise, you could lose all your casino gambling money at the roulette table in a shorter time than it will take you to lose $10 at the slots.

The trick to maximize casino gambling profits is to choose the right games to play. High pay-out games usually give you more money for limited funds. However, skill is highly needed in playing these games. Being a novice and attempting this game with the more experienced players only make you lose your funds. The rules may vary in different casinos, but still the most favored high-paying games are blackjack and video poker.

Depending on your casino’s blackjack rules, blackjack is a game wherein a player can expect high returns providing he plays wisely. There are many variations in blackjack rules like the number of card decks, splitting and doubling bets, and other rule variations. Years back, most casino gambling websites offer great odds. If played strategically, the player even gets valuable advantage. However, rules change and now, the odds are in the casino’s side regardless of the player’s strategy. So to ensure good pay-outs in playing blackjacks, it is to your favor to seriously consider being an advance blackjack player.

Of course, not everyone is born with jacks and aces in their hands. But learning good blackjack strategy is easy. There are many free and downloadable blackjack software tools that you can download and practice on. Also, blackjack books are available at most bookstores. These books are usually authored by top blackjack players who have gained fame and fortune in playing the game. Don’t frown because nothing is easy these days. Being a winner requires effort, so that at least you can enjoy yourself doing it. With regular blackjack practice, you could soon be the next casino gambling blackjack king.

Video poker is another casino gambling game that has a reputation for high pay-outs. However, like blackjacks, video poker rules and pay-out schemes vary among casinos. Examining the casino’s pay tables is important before starting video poker, or any game for that matter. Usually, there are different video poker machines at a given casino. Be sure to choose the one with best pay tables, otherwise, you’re just giving the casino your money without a fight. Good pay tables are assurance that the game you’re playing actually favors you.

Good video poker machines usually allow over a hundred percent return the longer you play the game. Also, despite being similar to slot machines, video poker is not a game of chance. This means that you have an actual advantage depending on your skills. Consequently, you cannot play the game and expect a high pay-out without understanding the basic rules and arming yourself with useful techniques. Casino gambling is fun, but it does entail some work.

Most casino gambling websites and traditional casinos offer both games. Betting experience and game skills improve your chances of winning the games and getting high pay-outs. Though playing requires effort in learning the games and mastering the skills, it does have its rewards. Besides, casino gambling is more fun if you go home laden with cash than big debts.