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Most people/players, IMO--have no idea what casino mathematics really are.  They also take published statistics and believe those apply to whatever you sit down to play.  False and wrong. 

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Casino Mathematics

This guide, written by casino math professor Robert Hannum, contains a brief, non-technical discussion of the basic mathematics governing casino games and shows how casinos make money from these games. The article addresses a variety of topics, including house advantage, confusion about win rates, game volatility, player value and comp policies, casino pricing mistakes, and regulatory issues. Statistical advantages associated with the major games are also provided.

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Selected Bibliography | About the Author
Understanding Casino Math

    Introduction
    Why is Mathematics Important?
    The House Edge
    Probability versus Odds
    Confusion about Win Rate
    Volatility and Risk
    Player Value and Complimentaries
    Gaming Regulation and Mathematics
    Summary Tables for House Advantage

Introduction

At its core the business of casino gaming is pretty simple. Casinos make money on their games because of the mathematics behind the games. As Nico Zographos, dealer-extraordinaire for the 'Greek Syndicate' in Deauville, Cannes, and Monte Carlo in the 1920s observed about casino gaming: "There is no such thing as luck. It is all mathematics."

With a few notable exceptions, the house always wins - in the long run - because of the mathematical advantage the casino enjoys over the player. That is what Mario Puzo was referring to in his famous novel Fools Die when his fictional casino boss character, Gronevelt, commented: "Percentages never lie. We built all these hotels on percentages. We stay rich on the percentage. You can lose faith in everything, religion and God, women and love, good and evil, war and peace. You name it. But the percentage will always stand fast."

Puzo is, of course, right on the money about casino gaming. Without the "edge," casinos would not exist. With this edge, and because of a famous mathematical result called the law of large numbers, a casino is guaranteed to win in the long run.


Why is Mathematics Important?

Critics of the gaming industry have long accused it of creating the name "gaming" and using this as more politically correct than calling itself the "gambling industry." The term "gaming," however, has been around for centuries and more accurately describes the operators' view of the industry because most often casino operators are not gambling. Instead, they rely on mathematical principles to assure that their establishment generates positive gross gaming revenues. The operator, however, must assure the gaming revenues are sufficient to cover deductions like bad debts, expenses, employees, taxes and interest.

Despite the obvious, many casino professionals limit their advancements by failing to understand the basic mathematics of the games and their relationships to casino profitability. One casino owner would often test his pit bosses by asking how a casino could make money on blackjack if the outcome is determined simply by whether the player or the dealer came closest to 21. The answer, typically, was because the casino maintained "a house advantage." This was fair enough, but many could not identify the amount of that advantage or what aspect of the game created the advantage. Given that products offered by casinos are games, managers must understand why the games provide the expected revenues. In the gaming industry, nothing plays a more important role than mathematics.

Mathematics should also overcome the dangers of superstitions. An owner of a major Las Vegas strip casino once experienced a streak of losing substantial amounts of money to a few "high rollers." He did not attribute this losing streak to normal volatility in the games, but to bad luck. His solution was simple. He spent the evening spreading salt throughout the casino to ward off the bad spirits. Before attributing this example to the idiosyncrasies of one owner, his are atypical only in their extreme. Superstition has long been a part of gambling - from both sides of the table. Superstitions can lead to irrational decisions that may hurt casino profits. For example, believing that a particular dealer is unlucky against a particular (winning) player may lead to a decision to change dealers. As many, if not most, players are superstitious. At best, he may resent that the casino is trying to change his luck. At worst, the player may feel the new dealer is skilled in methods to "cool" the game. Perhaps he is even familiar with stories of old where casinos employed dealers to cheat "lucky" players.

Understanding the mathematics of a game also is important for the casino operator to ensure that the reasonable expectations of the players are met. For most persons, gambling is entertainment. It provides an outlet for adult play. As such, persons have the opportunity for a pleasant diversion from ordinary life and from societal and personal pressures. As an entertainment alternative, however, players may consider the value of the gambling experience. For example, some people may have the option of either spending a hundred dollars during an evening by going to a professional basketball game or at a licensed casino. If the house advantage is too strong and the person loses his money too quickly, he may not value that casino entertainment experience. On the other hand, if a casino can entertain him for an evening, and he enjoys a "complimentary" meal or drinks, he may want to repeat the experience, even over a professional basketball game. Likewise, new casino games themselves may succeed or fail based on player expectations. In recent years, casinos have debuted a variety of new games that attempt to garner player interest and keep their attention. Regardless of whether a game is fun or interesting to play, most often a player will not want to play games where his money is lost too quickly or where he has a exceptionally remote chance of returning home with winnings.

Mathematics also plays an important part in meeting players' expectations as to the possible consequences of his gambling activities. If gambling involves rational decision-making, it would appear irrational to wager money where your opponent has a better chance of winning than you do. Adam Smith suggested that all gambling, where the operator has an advantage, is irrational. He wrote "There is not, however, a more certain proposition in mathematics than that the more tickets [in a lottery] you advertise upon, the more likely you are a loser. Adventure upon all the tickets in the lottery, and you lose for certain; and the greater the number of your tickets, the nearer you approach to this certainty."

Even where the house has an advantage, however, a gambler may be justified if the amount lost means little to him, but the potential gain would elevate him to a higher standing of living. For example, a person with an annual income of $30,000 may have $5 in disposable weekly income. He could save or gamble this money. By saving it, at the end of a year, he would have $260. Even if he did this for years, the savings would not elevate his economic status to another level. As an alternative, he could use the $5 to gamble for the chance to win $1 million. While the odds of winning are remote, it may provide the only opportunity to move to a higher economic class.

Since the casino industry is heavily regulated and some of the standards set forth by regulatory bodies involve mathematically related issues, casino managers also should understand the mathematical aspects relating to gaming regulation. Gaming regulation is principally dedicated to assuring that the games offered in the casino are fair, honest, and that players get paid if they win. Fairness is often expressed in the regulations as either requiring a minimum payback to the player or, in more extreme cases, as dictating the actual rules of the games offered. Casino executives should understand the impact that rules changes have on the payback to players to assure they meet regulatory standards. Equally important, casino executives should understand how government mandated rules would impact their gaming revenues.


The House Edge

The player's chances of winning in a casino game and the rate at which he wins or loses money depends on the game, the rules in effect for that game, and for some games his level of skill. The amount of money the player can expect to win or lose in the long run - if the bet is made over and over again - is called the player's wager expected value (EV), or expectation. When the player's wager expectation is negative, he will lose money in the long run. For a $5 bet on the color red in roulette, for example, the expectation is -$0.263. On the average the player will lose just over a quarter for each $5 bet on red.

When the wager expectation is viewed from the casino's perspective (i.e., the negative of the player's expectation) and expressed as a percentage, you have the house advantage. For the roulette example, the house advantage is 5.26% ($0.263 divided by $5). The formal calculation is as follows:

EV = (+5)(18/38) + (-5)(20/38) = -0.263
(House Advantage = 0.263/5 = 5.26%)

When this EV calculation is performed for a 1-unit amount, the negative of the resulting value is the house edge. Here are the calculations for bets on a single-number in double-zero and single-zero roulette.

Double-zero roulette (single number bet):
EV = (+35)(1/38) + (-1)(37/38) = -0.053
(House Advantage = 5.3%)

Single-zero roulette (single number bet):
EV = (+35)(1/37) + (-1)(36/37) = -0.027
(House Advantage = 2.7%)

The house advantage represents the long run percentage of the wagered money that will be retained by the casino. It is also called the house edge, the "odds" (i.e., avoid games with bad odds), or just the "percentage" (as in Mario Puzo's Fools Die). Although the house edge can be computed easily for some games - for example, roulette and craps - for others it requires more sophisticated mathematical analysis and/or computer simulations. Regardless of the method used to compute it, the house advantage represents the price to the player of playing the game.

Because this positive house edge exists for virtually all bets in a casino (ignoring the poker room and sports book where a few professionals can make a living), gamblers are faced with an uphill and, in the long run, losing battle. There are some exceptions. The odds bet in craps has zero house edge (although this bet cannot be made without making another negative expectation wager) and there are a few video poker machines that return greater than 100% if played with perfect strategy. Occasionally the casino will even offer a promotion that gives the astute player a positive expectation. These promotions are usually mistakes - sometimes casinos don't check the math - and are terminated once the casino realizes the player has the edge. But by and large the player will lose money in the long run, and the house edge is a measure of how fast the money will be lost. A player betting in a game with a 4% house advantage will tend to lose his money twice as fast as a player making bets with a 2% house edge. The trick to intelligent casino gambling - at least from the mathematical expectation point of view - is to avoid the games and bets with the large house advantages.

Some casino games are pure chance - no amount of skill or strategy can alter the odds. These games include roulette, craps, baccarat, keno, the big-six wheel of fortune, and slot machines. Of these, baccarat and craps offer the best odds, with house advantages of 1.2% and less than 1% (assuming only pass/come with full odds), respectively. Roulette and slots cost the player more - house advantages of 5.3% for double-zero roulette and 5% to 10% for slots - while the wheel of fortune feeds the casino near 20% of the wagers, and keno is a veritable casino cash cow with average house advantage close to 30%.

Games where an element of skill can affect the house advantage include blackjack, video poker, and the four popular poker-based table games: Caribbean Stud poker, Let It Ride, Three Card poker, and Pai Gow poker. For the poker games, optimal strategy results in a house edge in the 3% to 5% range (CSP has the largest house edge, PGP the lowest, with LIR and TCP in between). For video poker the statistical advantage varies depending on the particular machine, but generally this game can be very player friendly - house edge less than 3% is not uncommon and some are less than 1% - if played with expert strategy.

Blackjack, the most popular of all table games, offers the skilled player some of the best odds in the casino. The house advantage varies slightly depending on the rules and number of decks, but a player using basic strategy faces little or no disadvantage in a single-deck game and only a 0.5% house edge in the common six-deck game. Despite these numbers, the average player ends up giving the casino a 2% edge due to mistakes and deviations from basic strategy. Complete basic strategy tables can be found in many books and many casino-hotel gift shops sell color-coded credit card size versions. Rule variations favorable to the player include fewer decks, dealer stands on soft seventeen (worth 0.2%), doubling after splitting (0.14%), late surrender (worth 0.06%), and early surrender (uncommon, but worth 0.24%). If the dealer hits soft seventeen it will cost you, as will any restrictions on when you can double down.


Probability versus Odds

Probability represents the long run ratio of (# of times an outcome occurs) to (# of times experiment is conducted). Odds represent the long run ratio of (# of times an outcome does not occur) to (# of times an outcome occurs). If a card is randomly selected from a standard deck of 52 playing cards, the probability it is a spade is 1/4; the odds (against spade) are 3 to 1. The true odds of an event represent the payoff that would make the bet on that event fair. For example, a bet on a single number in double-zero roulette has probability of 1/38, so to break even in the long run a player would have to be paid 37 to 1 (the actual payoff is 35 to 1).

Confusion about Win Rate

There are all kinds of percentages in the world of gaming. Win percentage, theoretical win percentage, hold percentage, and house advantage come to mind. Sometimes casino bosses use these percentages interchangeably, as if they are just different names for the same thing. Admittedly, in some cases this is correct. House advantage is just another name for theoretical win percentage, and for slot machines, hold percentage is (in principle) equivalent to win percentage. But there are fundamental differences among these win rate measurements.

The house advantage - the all-important percentage that explains how casinos make money - is also called the house edge, the theoretical win percentage, and expected win percentage. In double-zero roulette, this figure is 5.3%. In the long run the house will retain 5.3% of the money wagered. In the short term, of course, the actual win percentage will differ from the theoretical win percentage (the magnitude of this deviation can be predicted from statistical theory). The actual win percentage is just the (actual) win divided by the handle. Because of the law of large numbers - or as some prefer to call it, the law of averages - as the number of trials gets larger, the actual win percentage should get closer to the theoretical win percentage.

Because handle can be difficult to measure for table games, performance is often measured by hold percentage (and sometimes erroneously called win percentage). Hold percentage is equal to win divided by drop. In Nevada, this figure is about 24% for roulette. The drop and hold percentage are affected by many factors; we won't delve into these nor the associated management issues. Suffice it to say that the casino will not in the long term keep 24% of the money bet on the spins of roulette wheel - well, an honest casino won't.

To summarize: House advantage and theoretical win percentage are the same thing, hold percentage is win over drop, win percentage is win over handle, win percentage approaches the house advantage as the number of plays increases, and hold percentage is equivalent to win percentage for slots but not table games.

· Hold % = Win/Drop
· Win % (actual) = Win/Handle
· H.A. = Theoretical Win % = Limit(Actual Win %) = Limit(Win/Handle)
· Hold Percentage ¹ House Edge

Furthermore, the house advantage is itself subject to varying interpretations. In Let It Ride, for example, the casino advantage is either 3.51% or 2.86% depending on whether you express the advantage with respect to the base bet or the average bet. Those familiar with the game know that the player begins with three equal base bets, but may withdraw one or two of these initial units. The final amount put at risk, then, can be one (84.6% of the time assuming proper strategy), two (8.5%), or three units (6.9%), making the average bet size 1.224 units. In the long run, the casino will win 3.51% of the hands, which equates to 2.86% of the money wagered. So what's the house edge for Let It Ride? Some prefer to say 3.51% per hand, others 2.86% per unit wagered. No matter. Either way, the bottom line is the same either way: assuming three $1 base bets, the casino can expect to earn 3.5¢ per hand (note that 1.224 x 0.0286 = 0.035).

The question of whether to use the base bet or average bet size also arises in Caribbean Stud Poker (5.22% vs. 2.56%), Three Card Poker (3.37% vs. 2.01%), Casino War (2.88% vs. 2.68%), and Red Dog (2.80% vs. 2.37%).

For still other games, the house edge can be stated including or excluding ties. The prime examples here are the player (1.24% vs. 1.37%) and banker (1.06% vs. 1.17%) bets in baccarat, and the don't pass bet (1.36% vs. 1.40%) in craps. Again, these are different views on the casino edge, but the expected revenue will not change.

That the house advantage can appear in different disguises might be unsettling. When properly computed and interpreted, however, regardless of which representation is chosen, the same truth (read: money) emerges: expected win is the same.

Volatility and Risk

Statistical theory can be used to predict the magnitude of the difference between the actual win percentage and the theoretical win percentage for a given number of wagers. When observing the actual win percentage a player (or casino) may experience, how much variation from theoretical win can be expected? What is a normal fluctuation? The basis for the analysis of such volatility questions is a statistical measure called the standard deviation (essentially the average deviation of all possible outcomes from the expected). Together with the central limit theorem (a form of the law of large numbers), the standard deviation (SD) can be used to determine confidence limits with the following volatility guidelines:

Volatility Analysis Guidelines
· Only 5% of the time will outcomes will be more than 2 SD's from expected outcome
· Almost never (0.3%) will outcomes be more than 3 SD's from expected outcome

Obviously a key to using these guidelines is the value of the SD. Computing the SD value is beyond the scope of this article, but to get an idea behind confidence limits, consider a series of 1,000 pass line wagers in craps. Since each wager has a 1.4% house advantage, on average the player will be behind by 14 units. It can be shown (calculations omitted) that the wager standard deviation is for a single pass line bet is 1.0, and for 1,000 wagers the SD is 31.6. Applying the volatility guidelines, we can say that there is a 95% chance the player's actual win will be between 49 units ahead and 77 units behind, and almost certainly between 81 units ahead and 109 units behind.

A similar analysis for 1,000 single-number wagers on double-zero roulette (on average the player will be behind 53 units, wager SD = 5.8, 1,000 wager SD = 182.2) will yield 95% confidence limits on the player win of 311 units ahead and 417 units behind, with win almost certainly between 494 units ahead and 600 units behind.

Note that if the volatility analysis is done in terms of the percentage win (rather than the number of units or amount won), the confidence limits will converge to the house advantage as the number of wagers increases. This is the result of the law of large numbers - as the number of trials gets larger, the actual win percentage should get closer to the theoretical win percentage. Risk in the gaming business depends on the house advantage, standard deviation, bet size, and length of play.

Player Value and Complimentaries

Using the house advantage, bet size, duration of play, and pace of the game, a casino can determine how much it expects to win from a certain player. This player earning potential (also called player value, player worth, or theoretical win) can be calculated by the formula:

Earning Potential = Average Bet ´ Hours Played ´ Decisions per Hour ´ House Advantage

For example, suppose a baccarat player bets $500 per hand for 12 hours at 60 hands per hour. Using a house advantage of 1.2%, this player's worth to the casino is $4,320 (500 ´ 12 ´ 60 ´ .012). A player who bets $500 per spin for 12 hours in double-zero roulette at 60 spins per hour would be worth about $19,000 (500 ´ 12 ´ 60 ´ .053).

Many casinos set comp (complimentary) policies by giving the player back a set percentage of their earning potential. Although comp and rebate policies based on theoretical loss are the most popular, rebates on actual losses and dead chip programs are also used in some casinos. Some programs involve a mix of systems. The mathematics associated with these programs will not be addressed in this article.


Casino Pricing Mistakes

In an effort to entice players and increase business, casinos occasionally offer novel wagers, side bets, increased payoffs, or rule variations. These promotions have the effect of lowering the house advantage and the effective price of the game for the player. This is sound reasoning from a marketing standpoint, but can be disastrous for the casino if care is not taken to ensure the math behind the promotion is sound. One casino offered a baccarat commission on winning banker bets of only 2% instead of the usual 5%, resulting in a 0.32% player advantage. This is easy to see (using the well-known probabilities of winning and losing the banker bet):

EV = (+0.98)(.4462) + (-1)(.4586) = 0.0032
(House Advantage = -0.32%)

A casino in Biloxi, Mississippi gave players a 12.5% edge on Sic Bo bets of 4 and 17 when they offered 80 to 1 payoffs instead of the usual 60 to 1. Again, this is an easy calculation. Using the fact that the probability of rolling a total of 4 (same calculation applies for a total of 17) with three dice is 1/72 (1/6 x 1/6 x 1/6 x 3), here are the expected values for both the usual and the promotional payoffs:

Usual 60 to 1 payoff: EV = (+60)(1/72) + (-1)(71/72) = -0.153
(House Advantage = 15.3%)

Promotional 80 to 1 payoff: EV = (+80)(1/72) + (-1)(71/72) = +0.125
(House Advantage = -12.5%)

In other promotional gaffes, an Illinois riverboat casino lost a reported $200,000 in one day with their "2 to 1 Tuesdays" that paid players 2 to 1 (the usual payoff is 3 to 2) on blackjack naturals, a scheme that gave players a 2% advantage. Not to be outdone, an Indian casino in California paid 3 to 1 on naturals during their "happy hour," offered three times a day, two days a week for over two weeks. This promotion gave the player a whopping 6% edge. A small Las Vegas casino offered a blackjack rule variation called the "Free Ride" in which players were given a free right-to-surrender token every time they received a natural. Proper use of the token led to a player edge of 1.3%, and the casino lost an estimated $17,000 in eight hours. Another major Las Vegas casino offered a "50/50 Split" blackjack side bet that allowed the player to stand on an initial holding of 12-16, and begin a new hand for equal stakes against the same dealer up card. Although the game marketers claimed the variation was to the advantage of the casino, it turned out that players who exercised the 50/50 Split only against dealer 2-6 had a 2% advantage. According to one pit boss, the casino suffered a $230,000 loss in three and a half days.

In the gaming business, it's all about "bad math" or "good math." Honest games based on good math with positive house advantage minimize the short-term risk and ensure the casino will make money in the long run. Players will get "lucky" in the short term, but that is all part of the grand design. Fluctuations in both directions will occur. We call these fluctuations good luck or bad luck depending on the direction of the fluctuation. There is no such thing as luck. It is all mathematics.

Gaming Regulation and Mathematics

Casino gaming is one of the most regulated industries in the world. Most gaming regulatory systems share common objectives: keep the games fair and honest and assure that players are paid if they win. Fairness and honesty are different concepts. A casino can be honest but not fair. Honesty refers to whether the casino offers games whose chance elements are random. Fairness refers to the game advantage - how much of each dollar wagered should the casino be able to keep? A slot machine that holds, on average, 90% of every dollar bet is certainly not fair, but could very well be honest (if the outcomes of each play are not predetermined in the casino's favor). Two major regulatory issues relating to fairness and honesty - ensuring random outcomes and controlling the house advantage - are inextricably tied to mathematics and most regulatory bodies require some type of mathematical analysis to demonstrate game advantage and/or confirm that games outcomes are random. Such evidence can range from straightforward probability analyses to computer simulations and complex statistical studies. Requirements vary across jurisdictions, but it is not uncommon to see technical language in gaming regulations concerning specific statistical tests that must be performed, confidence limits that must be met, and other mathematical specifications and standards relating to game outcomes.

Summary Tables for House Advantage

The two tables below show the house advantages for many of the popular casino games. The first table is a summary of the popular games and the second gives a more detailed breakdown.
House Advantages for Popular Casino Games
Game
   House Advantage
Roulette (double-zero)    5.3%
Craps (pass/come)    1.4%
Craps (pass/come with double odds)    0.6%
Blackjack - average player    2.0%
Blackjack - 6 decks, basic strategy*    0.5%
Blackjack - single deck, basic strategy*    0.0%
Baccarat (no tie bets)    1.2%
Caribbean Stud*    5.2%
Let It Ride*    3.5%
Three Card Poker*    3.4%
Pai Gow Poker (ante/play)*    2.5%
Slots    5% - 10%
Video Poker*    0.5% - 3%
Keno (average)    27.0%
*optimal strategy


House Advantages for Major Casino Wagers
Game    Bet    HA*
Baccarat    Banker (5% commission)    1.06%
Baccarat    Player    1.24%
Big Six Wheel    Average    19.84%
Blackjack    Card-Counting    -1.00%
Blackjack    Basic Strategy    0.50%
Blackjack    Average player    2.00%
Blackjack    Poor Player    4.00%
Caribbean Stud    Ante    5.22%
Casino War    Basic Bet    2.88%
Craps    Any Craps    11.11%
Craps    Any Seven    16.67%
Craps    Big 6, Big 8    9.09%
Craps    Buy (any)    4.76%
Craps    C&E    11.11%
Craps    don't pass/Don't Come    1.36%
Craps    don't pass/Don't Come w/1X Odds    0.68%
Craps    don't pass/Don't Come w/2X Odds    0.45%
Craps    don't pass/Don't Come w/3X Odds    0.34%
Craps    don't pass/Don't Come w/5X Odds    0.23%
Craps    don't pass/Don't Come w/10X Odds    0.12%
Craps    Don't Place 4 or 10    3.03%
Craps    Don't Place 5 or 9    2.50%
Craps    Don't Place 6 or 8    1.82%
Craps    Field (2 and 12 pay double)    5.56%
Craps    Field (2 or 12 pays triple)    2.78%
Craps    Hard 4, Hard 10    11.11%
Craps    Hard 6, Hard 8    9.09%
Craps    Hop Bet - easy (14-1)    16.67%
Craps    Hop Bet - easy (15-1)    11.11%
Craps    Hop Bet - hard (29-1)    16.67%
Craps    Hop Bet - hard (30-1)    13.89%
Craps    Horn Bet (30-1 & 15-1)    12.50%
Craps    Horn High - any (29-1 & 14-1)    16.67%
Craps    Horn High 2, Horn High 12 (30-1 & 15-1)    12.78%
Craps    Horn High 3, Horn High 11 (30-1 & 15-1)    12.22%
Craps    Lay 4 or 10    2.44%
Craps    Lay 5 or 9    3.23%
Craps    Lay 6 or 8    4.00%
Craps    Pass/Come    1.41%
Craps    Pass/Come w/1X Odds    0.85%
Craps    Pass/Come w/2X Odds    0.61%
Craps    Pass/Come w/3X Odds    0.47%
Craps    Pass/Come w/5X Odds    0.33%
Craps    Pass/Come w/10X Odds    0.18%
Craps    Place 4 or 10    6.67%
Craps    Place 5 or 9    4.00%
Craps    Place 6 or 8    1.52%
Craps    Three, Eleven (14-1)    16.67%
Craps    Three, Eleven (15-1)    11.11%
Craps    Two, Twelve (29-1)    16.67%
Craps    Two, Twelve (30-1)    13.89%
Keno    Typical    27.00%
Let It Ride    Base bet    3.51%
Pai Gow    Poker Skilled player (non-banker)    2.54%
Pai Gow Poker    Average player (non-banker)    2.84%
Red Dog    Basic bet (six decks)    2.80%
Roulette    Single-zero    2.70%
Roulette    Double-zero (except five-number)    5.26%
Roulette    Double-zero, five-number bet    7.89%
Sic Bo    Big/Small    2.78%
Sic Bo    One of a Kind    7.87%
Sic Bo    7, 14    9.72%
Sic Bo    8, 13    12.50%
Sic Bo    10, 11    12.50%
Sic Bo    Any three of a kind    13.89%
Sic Bo    5, 16    13.89%
Sic Bo    4, 17    15.28%
Sic Bo    Three of a kind    16.20%
Sic Bo    Two-dice combination    16.67%
Sic Bo    6, 15    16.67%
Sic Bo    Two of a kind    18.52%
Sic Bo    9, 12    18.98%
Slots    Dollar Slots (good)    4.00%
Slots    Quarter Slots (good)    5.00%
Slots    Dollar Slots (average)    6.00%
Slots    Quarter Slots (average)    8.00%
Sports Betting    Bet $11 to Win $10    4.55%
Three Card Poker    Pair Plus    2.32%
Three Card Poker    Ante    3.37%
Video Poker    Selected Machines    -0.50%
*House Advantages under typical conditions, expressed "per hand" and including ties, where appropriate. Optimal strategy assumed unless otherwise noted.

Selected Biblilography

Cabot, Anthony N., and Hannum, Robert C. (2002). Gaming Regulation and Mathematics: A Marriage of Necessity, John Marshall Law Review, Vol. 35, No. 3, pp. 333-358.

Cabot, Anthony N. (1996). Casino Gaming: Policy, Economics, and Regulation, UNLV International Gaming Institute, Las Vegas, NV.

Eadington, William R., and Cornelius, Judy (eds.) (1999). The Business of Gaming: Economic and Management Issues, Institute for the Study of Gambling and Commercial Gaming, University of Nevada, Reno, NV.

Eadington, William R., and Cornelius, Judy (eds.) (1992). Gambling and Commercial Gaming: Essays in Business, Economics, Philosophy and Science, Institute for the Study of Gambling and Commercial Gaming, University of Nevada, Reno, NV.

Epstein, Richard A. (1995). The Theory of Gambling and Statistical Logic, revised edition, Academic Press, San Diego, CA.

Feller, William (1968). An Introduction to Probability Theory and Its Applications, 3rd ed., Wiley, New York, NY.

Griffin, Peter A. (1999). The Theory of Blackjack, 6th ed., Huntington Press, Las Vegas, NV.

Griffin, Peter (1991). Extra Stuff: Gambling Ramblings, Huntington Press, Las Vegas, NV.

Hannum, Robert C. and Cabot, Anthony N. (2001). Practical Casino Math, Institute for the Study of Gambling & Commercial Gaming, University of Nevada, Reno.

Humble, Lance, and Cooper, Carl (1980). The World's Greatest Blackjack Book, Doubleday, New York, NY.

Kilby, Jim and Fox, Jim (1998). Casino Operations Management, Wiley, New York, NY.

Levinson, Horace C. (1963). Chance, Luck and Statistics, Dover Publications, Mineola, NY.

Millman, Martin H. (1983). "A Statistical Analysis of Casino Blackjack," American Mathematical Monthly, 90, pp. 431-436.

Packel, Edward (1981). The Mathematics of Games and Gambling, The Mathematical Association of America, Washington, D.C.

Thorp, Edward O. (1984). The Mathematics of Gambling, Gambling Times, Hollywood, CA.

Thorp, Edward O. (1966). Beat the Dealer, Vintage Books, New York, NY.

Vancura, Olaf, Cornelius, Judy A., and Eadington, William R. (eds.) (2000). Finding the Edge: Mathematical Analysis of Casino Games. Institute for the Study of Gambling and Commercial Gaming, University of Nevada, Reno, NV.

Vancura, Olaf (1996). Smart Casino Gambling, Index Publishing Group, San Diego, CA.

Weaver, Warren (1982). Lady Luck: The Theory of Probability, Dover Publications, New York, NY.

Wilson, Allan (1970). The Casino Gambler's Guide, Harper and Row, New York.

[Blue_Angel]

Nice copy paste!

[alrelax]

Nice copy paste!

Never claimed it was anything but.  Even left the credits and the UNLV reference.  What are you insinuating? 

[Blue_Angel]

What are you insinuating? 

I'm not insinuating but incinerating your plagiarism down to the ground! 

[alrelax]

Set the record straight.  You accuse me of passing off a written work by UNLV as my own and I never did or attempt to. 

To me, you are instigating problems here on this board.  I am as honest and straight up as they come.  You have a hidden agenda and it will come about and known. 

There is no reason for you to go onto my threads/posts here after this point.  Thank you.

[alrelax]

Due to Blue Angel, this post is closed.

[alrelax]

Such a shame, that if something doesn't fit somebody's role play garbage, it is attempted to have it discredited as plagiarism.  One of the most ludercious things I have been inflicted with in sometime. 

I remember this motorcycle the other week, squeezing between my vehicle and one running parrall with me on a tour lane divided highway.  The bike squeezes past on the yellow line, clearly making his own lane.  I thought he was waving as he slipped by but his hand gesture was about the same as a player wagering table max on baccarat and another wagering against him with clear intention. As the $10.00 wager wins, he gets the finger as the $2,000.00
Wagered player walks away.  My five year old in the car says, too bad, to the guy on the bike as he screams by.  A couple of miles up the road the bike is pulled over by a police officer.

My son and I are coming out off the shopping plaza we went to and this is about 45 mins later we pass the bike and 4 cops and a tow truck.  Now the guy is in handcuffs against one of the police cars and the wrecker is picking up his bike.  My little kid says, "why did he drive like that'?  I told him that the guy was probably in a hurry to get to a casino.

[alrelax]

To Answer 'Mike' I offer the Following:
_________________________________________________________________________________________________________
Quote from: alrelax on Today at 03:19:56 AM

    I estimate off the top of my head, without getting to nitty-gritty, I played at least, no less than 25,500 shoes of baccarat since I started playing.  And, I would have to say, without 100% written records--no more than 10% in a haphazard accumulation of shoes within that 25,500 even came close to any statistics in most ways.


I think any casino would be seriously worried if after so many shoes only 10% "came close" to the predicted statistics. Of course it depends what you mean by "close", but as a counter-example, here are the results of the Wizard of Odds 1000 6 and 8 deck shoes in terms of B, P, and T percentages:

6 Deck Shoes:
Banker Wins = 28708  (45.8418)
Player Wins = 28038  (44.772%)
Tie Wins =    5878   (9.38618%)

8 Deck Shoes:
Total player wins = 36193, ratio = 0.447496
Total banker wins = 36887, ratio = 0.456076
Total tie wins = 7799, ratio = 0.096428

By any standard, these stats are close to the theoretical prediction, and after "only" 1000 shoes. Sure, you can point to this or that shoe and say it's far off the theoretical stats -- say 35% B and 65% P -- but these by definition can't be typical shoes, otherwise the averages would not be what they are. By the Normal distribution approximately 68% of shoes will be "close" to the predicted averages; a long way from 10%.

Mike, I was generalizing.  I do from time to time.  The nature of my work, 'Hazardous Materials Remediation' I do that daily, back-fill quantities, topography materials, just about all areas of my work in fact except highway signage which the D.O.T. has pretty much down to a science.   Estimation, but estimation based on something.  And, that something is my experience. 

I seriously think I am pretty close.  However, I think you are taking it to another level for argument/challenge.  That's fine.  I am not going there, cannot prove anything or help anything even if we know know the exact figure.  The game will not change, 99% of all players will fade and a new group of gun-ho newbies will overcome the tables.  The game is powerful to the means of 'talk ability' and 'how easy it is to make a fortune' with anyone that gets a taste of it.  They watch, they see it, even if they did not realize profit, their mind did at the tables and they ran back to their own neighborhood and told everyone.  Boom!  It grows and grows like a strip r with pretty good food coming into a white rural suburban neighborhood where they have never ever been.  ( I been there and owned one in New Jersey back in the 1990's) it was like free cocaine and top shelf whisky to the suburban upper middle class to wealthy people in central and western New Jersey that could never get to sneak out on their wives to NYC or Phila.  OMG, what a gold mine!  What headaches, heartburn, family violence, police action and more came with it all, almost every night, but not at fit, it took a good couple of years and then it all set in.)

As far as coming close, I meant almost spot on, real close.  But I did compound my estimation at 15 shoes a week for 35 years.  Actually amounts in total should be higher.  I remember lots of my play, I don't remember lots of it as well.  But I do I have an overall gist of the percentages of results in clusters.  Nothing scientific by far.  It is also from all over the country, noting to do with one property in one jurisdiction, saying they are 90% off of statistics.  Again, the shoes I sit in on, are going to be different the majority of the times from the shoes others play.  I guess if I further broke it down, it might be something like the following:

10%                  ---           20-40%           ---                  20-40%               ---               10%
(As I said, about              (Lower/Off)                         (Higher/Off)                          (Way outside/Extremely
spot-on)                                                                                                                   high or unusual and notable)

The game makes every single casino money, if it DID NOT--it would be removed.  The few bac tables that are removed from a few properties are usually due to the casino hosts and their staff unable to promote or compete with another close by property, or the casino moved them and their personnel to another co-property for marketing purposes, etc.  The game tricks people and deceives people, playing on the players sheer greed and self ignorance to the max.  The players lose themselves more money than the game every genuinely takes away from them as their opponent, etc. 

Baccarat is no different than Wall Street, NONE!  I have played on tables where there was an easy 1.5 Million Dollars of bank chips with more readily available.  In one pit there would have been 20 million dollars on the tables of the bank's capital with 100 times more in reserve.  As far as the player's money the streets money let's say--there could have been an easy 1 million to a few million on the tables outright in cash.  As far as credit, markers, reserve, stashed, etc., all untold--no where to estimate whatsoever.

However, with the above, just compare it to Raj Rajaratnam from Goldman Sachs in Manhattan.  No different.  NONE! If you do not know about Raj and what happened on Wall Street  the mid 2000's and the federal indictment and imprisonment and the snitching out (ratting) of most all of the suits) with federal indictments and federal prison sentences for the multi billionaires, read about it.  It resembles baccarat to the tee!  Actually lots of Wall Street people were regular at the baccarat tables in Atlantic City in the 1980's and all of the 1990's.  Another name that comes it mind is the west coast wanna bee oller, Danny (the fake everything) Peng.  Look him up, he is a fake from his educations claims, to the claim and molding of a distant share holder of the Orange County Family that founded and owns the majority of Orange County, California.  Peng is the Taipei roller that was a huge gambler in Vegas as well.  Another story, not now.

Anyway, Wall Street built the casinos of this country back with Trump and junk bonds, the rest is history.  Anywhere there is cash and lots of it, there are dreams, cons, lies and greed---greed--greed--greed, did I say greed?  Players of baccarat have a huge problem that I have been harping on for years and that is that, they are their own worst enemies.  The same as Raj Rajaratnam. 

https://en.wikipedia.org/wiki/Raj_Rajaratnam

https://www.cnbc.com/id/100991730

His own greed was his total undoing, the same as it is for almost anyone, ALMOST ANYONE AND EVERYONE, at the baccarat tables.  I saw a new player win $60,000.00 the other week.  That is huge!  Think about it.  $60,000.00.  Gave it all back and lost thousands of his own cash.  Still losing, wins---but sill gives it all back and chases and chases and chases.  It is the same on Wall Street.  Lives in a modest suburban $1,500.00 a month apartment, goes through school, graduates and gets a job on Wall Street.  Works hard, makes money and moves into NYC on the east side and gets a $5,000.00 a month or better apartment.  Then moves uptown and even spend more and greater a month.  Then get a house out on Long Island, NY in the 1 million dollar range.  Then moves up to the 5-10 millions range further out on the Island, Connecticut or the Jersey shore.  The story goes on, the work gets tough and the person gets greedy.  Happens more than it doesn't happen. 

Greed, knoweldge and a game.  Men want money, Raj got money.  He also got a federal penitentiary term.  Money will get you, money will change you, money will over rule you in almost every case.  Not just Wall Street, the Miami drug trade of the 1970's and on.  And, almost every other industry as well.   

You guys talk like it is science to gambling.  It is not.  There are so many variables and mind plays, you have no idea.  You sit here and claim this and that and claim how you have everything all figured out.  You claim how you live off gambling and you claim how much you can make and do make.  You claim you have the mathematical part down and it is beatable.  If it was truly 'beatable' by answering a riddle or figuring out the answer to a mathematical question---the game would not be in a casino, period.  However, there are games and ways to capitalize on them without being able to truly have the claim, 'you can beat them with definable attack'.  But none of you show anything or relate to anyone to prove 1/100th of your claims.  You can't period.   Every time you guys explain how mathematical things work, somehow your answers, they always stop far shy of the actual explanation and proof, usually resulting in something like, "Well I can beat it" or "I have the definitive answers and the casino is always watching me" or some rubbish similar to those.   Sure there are players out there struggling to earn a couple hundred a day or every other day.  But, reality is the same as Wall Street, with any kind of success, earned, given to, fell upon, in the right place at the right time, earned, cheated, stole, criminally took, whatever--Greed and more greed.  Greed ti the prosecutors mass weapon the catches almost every single one that is committing a crime without much ado. 

When some player wins money, and let's say it is because  of his or her knowledge and mathematical skill, nothing else.  The highest majority of them, like in the upper 99.9 percentile will give it back and sink themselves.  Even if you know what you claim you do, which is against what the game is designed to produce and how it is produced, will not settle for what you haphazardly obtain from time to time.  The story goes on.  Been there and done that.

Don't get excited about coincidences.  You checkmate yourself that way!

[Mike]

You guys talk like it is science to gambling.  It is not.  There are so many variables and mind plays, you have no idea.  You sit here and claim this and that and claim how you have everything all figured out.  You claim how you live off gambling and you claim how much you can make and do make.  You claim you have the mathematical part down and it is beatable.  If it was truly 'beatable' by answering a riddle or figuring out the answer to a mathematical question---the game would not be in a casino, period.  However, there are games and ways to capitalize on them without being able to truly have the claim, 'you can beat them with definable attack'.  But none of you show anything or relate to anyone to prove 1/100th of your claims. 

Glen,

Since this post was addressed to me, I'll respond. In the first place, I make no claim that I make a living playing any casino game. Blue_Angel does, but I'm not Blue_Angel. Neither have I claimed that such games are beatable. In fact, I'm known on this and other forums for denying that they are, at least by the systems that are proposed. Second, the "claims" I made are with regard to statistical facts, verifiable by anyone empirically if they make the effort. What you seem to forget (or fail to understand) is that if the empirical stats did not conform to the predicted probabilities then the casinos would not have the edge that they do, that was why I said that they would be worried if only 10% of shoes conformed to the stats (your claim).

The payouts are calculated on the presumption that the theoretical outcomes will hold good over time; if "anything can happen" then it wouldn't be possible for the casinos to create an edge in the first place. So I'm not suggesting that these stats in any way give an advantage to the player, or that they manifest in every shoe. It's rather that precisely BECAUSE they are so reliable that the games are so hard to beat. No matter how "clever" your bet selection is, no matter how many hands you skip or patterns you track, no matter if you "attack the game from all sides" (whatever that means), the outcomes will always tend towards 0.45865 for Banker, 0.44627 for Player, and 0.09506 for Tie. That's good for the casino, not so good for the player.

You say

'none of you show anything or relate to anyone to prove 1/100th of your claims.

Right back at you, Glen. Have you ever proved any of your claims that Baccarat is beatable by 'attacking from all sides'? You could start by explaining what this actually means. You spend a lot of time ridiculing others' claims and systems but imply that you're a winner.

[alrelax]

Glen,

Since this post was addressed to me, I'll respond. In the first place, I make no claim that I make a living playing any casino game. Blue_Angel does, but I'm not Blue_Angel. Neither have I claimed that such games are beatable. In fact, I'm known on this and other forums for denying that they are, at least by the systems that are proposed. Second, the "claims" I made are with regard to statistical facts, verifiable by anyone empirically if they make the effort. What you seem to forget (or fail to understand) is that if the empirical stats did not conform to the predicted probabilities then the casinos would not have the edge that they do, that was why I said that they would be worried if only 10% of shoes conformed to the stats (your claim).

The payouts are calculated on the presumption that the theoretical outcomes will hold good over time; if "anything can happen" then it wouldn't be possible for the casinos to create an edge in the first place. So I'm not suggesting that these stats in any way give an advantage to the player, or that they manifest in every shoe. It's rather that precisely BECAUSE they are so reliable that the games are so hard to beat. No matter how "clever" your bet selection is, no matter how many hands you skip or patterns you track, no matter if you "attack the game from all sides" (whatever that means), the outcomes will always tend towards 0.45865 for Banker, 0.44627 for Player, and 0.09506 for Tie. That's good for the casino, not so good for the player.

You say


Right back at you, Glen. Have you ever proved any of your claims that Baccarat is beatable by 'attacking from all sides'? You could start by explaining what this actually means. You spend a lot of time ridiculing others' claims and systems but imply that you're a winner.

First, I never said the game was 'beatable', you have clearly inserted that word into my posts or on my behalf attempting to make others say, "Gee--Alrelax talks rubbish and knows zip" or something real close to that anyway.  I rather not get involved in a war or a spat with you, seriously.  I do feel that is what you are looking for.

Second, if you do not like, enjoy or appreciate things I write about, skip them---saves your valuable time as well as lowering yourself to read my writing. 

Third, I do very well overall at baccarat with my experience that dictates my beliefs, my wagering, my protocols and yes--employing 'directions' rather than limited sight and attempt to catch a few winning hands, here and there and make $100 or $300, etc., on a few hands and claim I conquered baccarat in the long term, shoe after shoe after shoe, endless play at a time, etc., etc.  The latter is very prevalent on the board here as well as other boards. 

Fourth, I look at 10 shoes at a whack.  And my own 'stats' would resemble the rough 4 section graph I made in the above post in this thread with the 10%, 20-40%, 20-40%, 10%.  The shoes I play or you play or another plays, is so trivial as compared to the hundreds of thousands if not, millions of shoes where all the industry statistics do come from, that is what I was trying to say in the first place.  You can sit there and claim, reclaim and stand fast on, I am wrong and do not know what I am talking about.  I don't care Mike.  But, that is merely your interpretation of baccarat.  When I sit there and I see 3 shoes and one shoe is dead even at the end, the other one is in the favor of the Bankers by 20+ hands and the 3rd one was in favor of the Players by say 15 hands or so, I add up 3 shoes.  One being about what national industry 'accepted' statistics prove.  The other two don't go away, as much as you think they do or as much as you want to write them off as a rarity and won't happen again for quite sometime.  Wrong.  False.  The common players mistake to ignoring what baccarat is really about, IMO!

So, like I said, that you challenge and you say is not true, is the results of shoes.  They are not what the national industry statistics dictate.  I am not saying those statistics are wrong.  I am saying those statistics are not applicable to the table I sit down to play at.  3 things can happen.  The shoes produced will be spot on those statistics.  The 3 shoes I play might be all or partially under those statistical results in resemblance or 3rd, one or all of them can also be greater over those statistical numbers. 

I am sure I just further aggravated you and you will not side with or understand where I am coming from, right or wrong that is a fact.  When I sit down at the table tonight to play and I encounter a shoe that is dead even, meaning say 14 Bankers and 14 Players and maybe 3 ties.  When I see something happen where the following 6 hands are all naturals and go into a chop.  I might feel based on other things, that the Banker will prevail and get extremely strong for a section because  of a turning point I clearly discovered.  Then 9 Bankers come out followed by an immediate 8 to 10 Players right next to it---while every single person lost all their money and most of their mouths are open in awe--I smacked the casino and caught every Banker except the last and then caught every Player that was produced and lost the last Player as well.  No matter how you look at that, evening/equally out or a rarity and a stroke of luck, doesn't matter.  That is baccarat and happens all the time, just in different presentments while each of us is trying to wager from some 'direction' or another.  It is all tied in together, oops--probably the wrong thing I could have said I can only imagine.

[Mike]

First, I never said the game was 'beatable', you have clearly inserted that word into my posts or on my behalf attempting to make others say, "Gee--Alrelax talks rubbish and knows zip" or something real close to that anyway.  I rather not get involved in a war or a spat with you, seriously.  I do feel that is what you are looking for.

Second, if you do not like, enjoy or appreciate things I write about, skip them---saves your valuable time as well as lowering yourself to read my writing. 

No I'm not looking for any war with you, but I did feel compelled to reply to your post as it was specifically directed at me. You put words in my mouth by saying that I had made various claims which I hadn't. I admit that annoyed me.  I think it's a matter of semantics whether you say baccarat is 'beatable' or that you 'do well overall' at the game. It amounts to the same thing, but anyway I'm not going to argue about your claim, it would be pointless without knowing more details and the actual numbers involved.

The shoes I play or you play or another plays, is so trivial as compared to the hundreds of thousands if not, millions of shoes where all the industry statistics do come from, that is what I was trying to say in the first place

And I posted the numbers for just 1000 shoes which come very close to the 'official' stats, so it's not true that they are only meaningful after hundreds of thousands or millions of shoes. In fact even 100 shoes would come pretty close to the theoretical probabilities. Take it or leave it, those are the facts.

[alrelax]

And I posted the numbers for just 1000 shoes which come very close to the 'official' stats, so it's not true that they are only meaningful after hundreds of thousands or millions of shoes. In fact even 100 shoes would come pretty close to the theoretical probabilities. Take it or leave it, those are the facts.

Players of all types attempt to apply the statistics repeatedly and with great devotion and verbal/physical explanation and course of wagering action---most of it fails when attempted on much more than a hand or two or a few hands within most shoes.  If you understood what I just said, there is no way on a consistent basis, say greater than 60-70% of a shoe will wagering and statistical percentages match up to benefit the player.  If it does, it will certainly be where the player eventually rode out a deficit of either the B or the P to 'catch up' and equal out.  But the player will usually be lucky if he recoups the money he lost doing the exact same thing prior to the deficit coming out in the first place.  Once again, that is what I am referring to within 'statistics really don't apply' and 'directions'. 

[Mike]

Yes, many players do that, but this irrelevant to my previous posts about the stats. You seem to be assuming that I'm recommending that it's a good idea to choose your bets this way, presumably because I cast doubt on your 10% figure. But I only commented on that because it would be miraculous if after so many thousands of shoes the stats didn't conform to the 'official' statistics. But actually I'm in full agreement with you regarding the merit of betting by following the stats. There is no merit it in because although the outcomes do approach the theoretical probabilities, they don't necessarily do so in the present shoe, or from shoe to shoe or even the next few shoes. You may actually be correct quite often in terms of the NUMBERS or ratio of B to P, or the numbers of streaks of 1, 2, etc, but generally not in the ORDER in which they arrive in a shoe, except sometimes by luck. If you try to do this you're just committing the gambler's fallacy. Trying to apply the stats to the next few hands, or the next few shoes doesn't work because as you rightly point out, the stats are not derived from just a few hands or shoes; at least a few dozen shoes are needed before the stats converge to the 'true' probabilities, and the smaller the probabilities are, the more shoes you need.

[Blue_Angel]

Yes, many players do that, but this irrelevant to my previous posts about the stats. You seem to be assuming that I'm recommending that it's a good idea to choose your bets this way, presumably because I cast doubt on your 10% figure. But I only commented on that because it would be miraculous if after so many thousands of shoes the stats didn't conform to the 'official' statistics. But actually I'm in full agreement with you regarding the merit of betting by following the stats. There is no merit it in because although the outcomes do approach the theoretical probabilities, they don't necessarily do so in the present shoe, or from shoe to shoe or even the next few shoes. You may actually be correct quite often in terms of the NUMBERS or ratio of B to P, or the numbers of streaks of 1, 2, etc, but generally not in the ORDER in which they arrive in a shoe, except sometimes by luck. If you try to do this you're just committing the gambler's fallacy. Trying to apply the stats to the next few hands, or the next few shoes doesn't work because as you rightly point out, the stats are not derived from just a few hands or shoes; at least a few dozen shoes are needed before the stats converge to the 'true' probabilities, and the smaller the probabilities are, the more shoes you need.

Hello Mike, I'd like to listen to your opinion regarding my hypothesis;

1) In order the house edge to take its toll events have to balance, or at least come very near equilibrium state.
Do we agree so far?

2) If the statistical certainty (?) of large numbers it's only matter of time to occur then why this equates to doom for the players?

Let me provide an example, let's say that a player by knowing these statistics had developed a strategy being based on the eventual and unavoidable (?) equilibrium state.
No, I'm not talking about an idiotic Martingale progression on what is "due"! 
As casinos are so certain for their built in edge based on lesser payouts (according probability), so could be the player who aims for long term profit instead of short term.

To become more specific, let's say that we were sure (?) that after 5000 spins black and red would be very close, like I said the aim is in long term and we could divide 5,000 into 10 checkpoints of 500 bets each.

So we pick 1 of the 2 sides and stick with it always, if we are ahead then we made nice profit by flat betting, when we are down we would increase by 1 unit for the next 500 bets, in short, the application of d'Alembert (+1 on loss, -1 on win) in a much greater scale, like 1:500 or even 1:1000.

Of course events won't balance perfectly but ain't necessary in order d'Alembert equilibrium principle to come out on top of the house edge.
As a matter of fact we would need 40% to 41% wins out of the total, from 48.65% there is from 7.65% up to 8.65% difference, in which if being translated in units for many thousands of bets it would be very rewarding, especially when the unit value is green, black or higher.

All come down to the wrong mentality of the majority, I call it snatch mentality which is short-lived.
I have to admit that such long term approach might seem unattractive on first glance, but what if it was the only viable solution?

Since we cannot dispute the facts the next sensible conclusion is that with a strong BR and the patience of Job one could turn winning into formality!

Why we won't see gamblers doing so, like I said it begins from the mentality, gamblers have not the investors mindset and are reaping what they sow.

The bottom line is that the coin has 2 sides, if casinos can be certain when dealing with variance, the same could be for the players!


Yet you see all those Cassandras preaching doom from forum to forum like after Jesus Christ prophets!
Sorry to be blunt, but we don't need anymore parrots and monkeys, we need more out of the box thinkers!

[alrelax]

Yes, many players do that, but this irrelevant to my previous posts about the stats. You seem to be assuming that I'm recommending that it's a good idea to choose your bets this way, presumably because I cast doubt on your 10% figure. But I only commented on that because it would be miraculous if after so many thousands of shoes the stats didn't conform to the 'official' statistics. But actually I'm in full agreement with you regarding the merit of betting by following the stats. There is no merit it in because although the outcomes do approach the theoretical probabilities, they don't necessarily do so in the present shoe, or from shoe to shoe or even the next few shoes. You may actually be correct quite often in terms of the NUMBERS or ratio of B to P, or the numbers of streaks of 1, 2, etc, but generally not in the ORDER in which they arrive in a shoe, except sometimes by luck. If you try to do this you're just committing the gambler's fallacy. Trying to apply the stats to the next few hands, or the next few shoes doesn't work because as you rightly point out, the stats are not derived from just a few hands or shoes; at least a few dozen shoes are needed before the stats converge to the 'true' probabilities, and the smaller the probabilities are, the more shoes you need.

Mike, you inspired me to write this just now.

http://betselection.cc/alrelax's-blog/baccarat-ideas-executions-of-those-ideas/msg60673/?PHPSESSID=6nbcra2qmeretvt6dcei5qegt4#msg60673

"It is only a matter of time, however--no one really has the correct answer as to if it is the time or not".