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.'
Casino (Blu-ray) (1995) No One Stays At The Top Forever. Director: Martin Scorsese OVERALL: Draw. Click a link to jump to that release. Aspect Ratio: 2.35:1. Picture Format: 1080p 24fps VC-1 Soundtrack(s): English DTS-HD Master Audio 5.1 French (Canadian) DTS 5.1 German DTS 5.1. Different techniques to measure or determine aspect ratios were reported in the literature. A vast number of studies (Baudet et al., 1993, Jennings and Parslow, 1988, Lohmander, 2000, Pabst et al., 2000, Pabst et al., 2001, Pabst et al., 2006a, Pabst et al., 2006b, Pabst et al., 2007, Pabst and Berthold, 2007, Slepetys and Cleland, 1993) described how laser ensemble diffraction and single. 'Casino': The Story' (8 min.) traces Scorsese's late-breaking interest in the material. He 'owed' Universal one more movie after 'Cape Fear' and 'Age of Innocence,' and only jumped into 'Casino' after deciding at the last minute to pass on directing 'Clockers,' handing that project over to Spike Lee.
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%)
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.
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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. |
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Note: This summary is the intellectual property of the author and the University of Nevada, Las Vegas. Do not use or reproduce without proper citation and permission.
This page is part of © FOTW Flags Of The World websiteLast modified: 2014-02-16 by rob raeside
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Summary of Ratios of Flags
My 'holy book' (Whitney Smith, Flags Through the Ages and Across the World) quotes for the United States' flags a 10:19 ratio, really closer to the 1:2 traditional ratio of the British flags, from which the 'Star Spangled Banner' comes.
In fact there are three main 'threads' in the world of flags:
British flags have a 1:2 ratio (United Kingdom,Australia,Bahamas,Canada,Ireland and with the little correction of 10:19 United States and of course Liberia);
French flags have a 2:3 ratio (France,Italy,Cameroon,Ivory Coast,Algeria,Spain and the most of Latin-American flags);
German flags have a 3:5 ratio;
Moreover some nations have unusual ratios, as Denmark (28:37) or Belgium (13:15).
Alessio Bragadini
The British ensigns (including the Union Jack) ratio varied with the standard breadth of the textile industry, but always retaining a length of 18:
Year | Ratio of Length to Breadth |
---|---|
1687-17xx | 11:18 |
17xx-1837 | 10:18 (5:9) |
1837-present | 9:18 (1:2) |
On the other hand, I'm not sure how strict this regulations would be followed in civilian rebellions taking place in faraway Australia, nor how fast they were enforced throughout the empire..
I don't know at what point after 1837 the proportions were actually regulated. Possibly not until the reorganisation of Squadron colours in 1864.
David Prothero, 03 June 1999
The Golden Section
Many flags, picture frames, book covers, etc., are proportioned in accordance with what artists and mathematicians call 'the golden section.' This relationship exists when the length and width of a rectangle are divided into extreme and mean ratio, or when the parts follow (or approximate) the formula:
Another way to look at it is if the length and width roughly equal 62 and 38 percent of their sum respectively.
Lou Stewart, 1998 January 30
If you solve the equation Lou Stewart gave analytically,
you'll find a solution:
where:
Mathematically, there's another solution to this equation, namely
but I don't think we're looking for a flag with a negative length.
So, the ratio is 1.618…:1.
This ratio was already known to the Greeks, and the Acropolis reflects this ratio in many ways (correct me if I'm wrong).
Filip Van Laenan, 1998 January 30
Let's try it this way:
the first format to think of is 1:1 (A:B), then we put the B as a new A, and A+B as a new B. So next we'll get 1:2, and next 2:3 and 3:5 and 5:8 and 8:13 and 13:21 and 21:34 and 34:55 and 55:89 and so on.. We'll get closer and closer to 1.618 or something like that, the golden section. It has been used a lot in art and Kepler spoke of 'divina proportio'. It is mostly a proportion that 'looks nice'. Many mathematicians and physicists have written about it.
Ole Andersen, 1998 January 30
1.1618… is phi the golden ratio and is, like pi, irrational. However, if we look at the Fibonacci series we'll see that the difference between each number gets closer and closer to the ratio first over then under. The series is 1 1 2 3 5 8 13 21 .. where each number is the sum of the previous two. You will also note that the numbers are not far from many flag ratios 5:8 8:13 13:21 etc.
Rich Hansen, 1998 January 30
A more interesting approach uses the Golden Ratio's connection to the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13,…, in which each new member of the sequence is the sum of the preceding two. Then, you can generate successive (and closer) approximations to the Golden Ratio as 1/1, 2/1, 3/2, 5/3, 8/5, 13/8,.. where the numerator is just the member of the sequence that is one ahead of the denominator. (The two.. approximations above are derived from this method, just a bit further along in the sequence).
An awful lot has been written about the Golden Ratio (also called 'The Divine Proportion'). It crops up in nature: shapes of shells, arrangements of sunflower seeds; in architecture (the most aesthetic shapes are ones having proportions equal to the Golden Ratio), and flags! It was known to the Ancient Greeks (look at a picture of the Parthenon), and probably earlier. Interesting it should show up in flag design too!
anonymous,Aspect Ratio Tires
1999 February 03At least one flag used the 'golden proportions' as an integral part of its design: Saarland
Dave Martucci, 1998 January 30
No. There wasn't such flag manufactured in Saarland. I don't know where the document which mentioned this was to be found, but all the projects of laws of 1947 mentioned a flag with the proportions 1 x 1,5, not 1 x 1,61803398875. If such flag was proposed in Saarland, this was really absurd and ridiculous: how can you draw precisely such a flag, and above all how can you manufacture such a flag: it is impossible and not practical! If the flag existed it was only a proposition, not a real flag.
Pascal Vagnat, 1999 February 05
New Brunswick's isn't a national flag, but its 5:8 ratio is the closest approximation you can get to the golden ratio with one-digit numbers. The designer probably considered this when choosing the ratio. Any other flags with this ratio were probably designed with the golden ratio in mind.
Dean Tiegs, 1999 February 05
Artists Bruno Tuukkanen and Eero Snellman had the Golden Ratio in their mind when they designed the Finnish flag. Ratio 11:18 = 0.6111… which differs very little from 0.6180…
Ossi Raivio, 1999 February 06
Chuvash Republic - the designer of this flag artist Mr. E. Jurjev has specially made ratio of width and lengths of a flag as golden ratio.
Mikhail Revnivtsev, 12 August 2005
Vertical Proportions of Flags
The proportions of vertical stripes on French naval flag are 30:33:37, to enable good visual effect of flag when flying. Portugal, too, obviously, has an off-centered pattern and I suppose the Scandinavian cross flags have the same reason for the vertical bar shifted right.
Željko Heimer, 23 September 23 1995
Bangladesh, North Korea, Nauru, Turkey and the Japanese Ensign all shift their designs to the hoist. Whitney Smith's book mentions that Bangladesh does this so that the flag will look proper while flying. There is no reason given for the others and in the case of Nauru especially I suspect that the star is toward the hoist for some other reason.
Nathan Augustine, 27 September 1995
Why keep the right proportions?
Since we are talking about flag proportions, I was wondering if the proportions are ever symbolic in and of themselves, or always more or less arbitrary. (Let's leave oddballs like Nepal and Qatar out for the moment.) This question arises from the question of why it's so important to keep the proportions right. For instance, Ron pointed out that many of the errors are caused by standardizations of the flag manufacturing process. Earlier, someone said that all the flags of the former Soviet Union kept to the proportions of the old Hammer and Sickle. Similarly, looking at my flag chart, all the flags of the former Yugoslavia seem to be more or less the same proportions. I'm willing to bet that this is a result less of nostalgia for the old days and more of the fact that it was easier to leave the settings on the flag-making machines as is..
Thus, I ask again, why is it important to keep proportions straight? Colors and symbols have meanings which it would wrong to alter, but if proportions are chosen arbitrarily..
Josh Fruhlinger, 29 January 1996
One pair of flags that differ only in their proportions are those of Indonesia (2:3) and Monaco (4:5). Of course, I don't know whether the proportions have significance in themselves, but they have significance in relation to each other in that they are the only way to distinguish the two flags.
I'm having a hard time thinking of a real-world situation in which these two countries' flags could be confused, though. (Shipwrecked sailors wash up on an unfamiliar shore; 'What country are we in?' 'Must be Indonesia -- look at that flag.' 'Yes, and that big building up there must be the famous Djakarta Casino!')
Bruce Tindall, 29 January 1996
The UJ family is shaped 1:2 and I suppose that with 'commercial reasons' a flagmaker has to provide the correct ratio for those and other flags, otherwise he'll keep sitting on his stuff (standardization of production would suggest a general 2:3 ratio and I think there is quite a bunch of 2:3 flags just for this reason).
Aspect Ratio Pune
I place my questions earlier: at the time before mass production, when design and measurements of a flag are layed down. I'd like to look over the shoulders of those in past and present who are in charge of creating a flag or who elaborate the demands for the designers. Why did they choose 1:2 or 2:3? I tried to give some reasons for the 2:3 / 3:5 choice but very probably there are more. Seafaring nations may have old maritime flag traditions for example. Maybe some documents could tell more about it.
Martin Karner, 13 January 2005
It seems that manufacturers like to pick a ratio, and make all their flags 2:3, or all 1:2 or whatever. Remember the comments about the latest Georgia State flag: they said if it was 1:2, it would be longer than the US flag, since the Stars & Stripes is de facto made in 2:3. Canadian (or at least British Columbia) city flags are primarily made in 1:2, even if the Heralds' illustration shows a 2:3 flag.
2:3 Canadian flag
image by Martin Grieve, 4 December 2005
modified by Dean McGee, 13 January 2006
While a display like the UN's may look better with the flags to a uniform size, many designs look 'wrong' when they're in the wrong proportion. British Columbia's flag is often made in 1:2, which stretches out the setting sun to look like a banana with a crown, and Canadian flags in 2:3 have too much white above and below the Maple Leaf.
Casino 1995 Aspect Ratio Ratios
The original message centred around a Bahamian-flagged cruise ship which belonged to an American-run company. They probably ordered their flags from a company which uses 2:3 as its standard, or maybe they order their flags 4' x 6' without thinking about proportions. I have to say that to my eye, horizontally striped flags like Bahamas, Malaysia and US are not as noticably 'wrong' when the proportions are wrong.
Dean McGee, 13 January 2006
Casino 1995 Aspect Ratio
According to 'major' flag makers I have spoken with about this very subject, the issue is 'automation' for lack of a better word.
It is rare in the west nowadays for a flag maker to have large numbers of folks who are genuinely skilled in the craft of making fully sewn flags with the exception of perhaps said company's national flag. When flag makers do have such folk on staff, those capable of making other things, they pay a premium price to keep them around. The cost is passed down to the customer.
Automation means certain standards are set for all flags whether proportionally correct or not.
I have seen (probably) the same afore mentioned Bahamas ensign while I was in Venice. When I got up close to the cruise ship, I noticed that the canton was printed and the rest of the parts were joined together. This was a much cheaper way of making the ensign a opposed to having everything custom sewn.
Most likely, the Bahamian ensign in Venice was American made, meaning it was fairly inexpensive compared to European flags.
It has been a while since I was in the Bahamas, but when I was there, I rarely saw any kind of flag or ensign proportioned 1:2. Almost everything with the exception of a very few government flags were 2:3 or 3:5 and Annin or Dettra made. Come to think of it, I don't recall ever seeing a 1:2 Bahamian red ensign in Bahamian waters.
Clay Moss, 13 January 2006
I can tell some ideas on the flag size question for Austria-Hungary, and I think that flag ratio is somewhat related to this.
Austro-Hungarian maritime flags originally were made of stripes of 48 cm height, called 'Kleid' (plural 'Kleider'), so the height of the flag was a multiple of 48 cm (approx.) [bmg77]
This is most probably typical for other countries as well, that one unit was based on a more or less standardized measure of cloth available. The other side would be any measure convenient for manufacture, either determined by the length of available cloth, or by the size of the manufacturing facilities, or simply by some easily measurable proportion (1:2 or 2:3 rather than say 4:7).
Marcus E.V. Schmöger, 5 February 2006
Casino 1995 Aspect Ratio Formula
There was a question why flags are lowered during the night and I guessed that this has to do primarily with the preservation of the flag material. Since the winds on the sea and in coastal regions are harder and more constant, flags are lowered when they are not seen, i.e. Slots free 2021. on the high seas and during the night. In landlocked countries the flags are not lowered at night because the winds are less hard. This corresponds perfectly to your remarks. In addition to Marcus' production points we thus have three climatic factors which may have been in favour of longer flags in maritime regions: the rough winds which consume the cloth from the fly towards the hoist, the easiness to repair a long flag and the absence of the problem that a long flag flies less easily than a short one.
Martin Karner, 6 February 2006
Longer flags (e.g., 1:2) may have been more useful for seafaring nations because of the old method of repairing weather-damaged flags on board ship (which was IIRC to cut off a strip from the fly to repair any damage). A long flag could therefore be repaired more readily. On land, this became less of a problem because new flags could be more readily obtained, and the dynamics of flag-flying (which ISTR say that a shorter flag will fly more easily) may have come into play in the design. For that reason, it would make sense if countries with naval traditions favoured longer flags, and landlocked countries favoured shorter flags. Of course, this doesn't explain why countries like the Netherlands prefer shorter flags, but it might explain the UK and its possessions, and also countries like Switzerland.
James Dignan, 5 February 2006
It is thought that British naval flags attained a ratio of 1 : 2 through carelessness. 17th century English naval ensigns were made from material that was about eleven inches wide. It was stipulated that the length of a flag should be eighteen times the number of widths of material used to make the flag. The ratio at that time was therefore 11 : 18. Over the years, for reasons that I have never seen explained, the width of the material used to make flags was reduced, but no corresponding adjustment was made to the stipulated length. The length of the flags thus increased relative to their width. By about 1840 the width of the material had been reduced to nine inches, giving a ratio of 8 : 18. Standard sizes were now introduced, in which the length was twice the width.
David Prothero, 6 February 2006
According to Pepys writing in the last half of the 17th Century 'It is in general to be noted that the bewper (bunting) from which colours are made being 22 inches (approx 56cm) in breadth and half of that breadth or 11 inches in ordinary discourse by the name of a breadth being wrought into colours, every such breadth is allowed half a yard (18 inches or approx 46cm) for its fly'.
If the flag sizes given for 1742 may be cited as evidence the 'breadth' had decreased to 10 inches by that date, and a surviving 20 foot x 40 foot White Ensign of 1787 (not counting an Establishment of 1822) seems to indicate that the breadth had reduced yet again to its modern width of 9 inches by the later 18th Century?
Christopher Southworth, 6 February 2006