If you plan to play online roulette, stick with the bets whose odds almost mirror their payouts. These 'even money' bets include betting on Even, Odd, Low (numbers 1 through 18), High (numbers 19 through 36), Red, or Black. All of these wagers pay out at 1 to 1. Game Odds Articles. The payout for a winning even bet is the same in American roulette as in European roulette but the true odds are 1.11-to-1. As you can see European roulette gives better chances of winning the even bet so if you have the choice between the two types of roulette, we recommend you opt for European. The “En Prison” Rule.

By Ion Saliu, Founder of Roulette Mathematics

This question was posted in mathematical newsgroups (alt.math.recreational, alt.math.undergrad, alt.sci.math.probability): 'Winning and Quitting on Red/Black in Roulette'.

'Obviously in roulette betting on in the long run you are going to lose your money but at some point chances are you'll be in profit. To take an extreme example if you had $1000 you could reasonably expect to be up $1 at some point. Is it possible to generalize this? I want to win W dollars at which point I will quit. How much cash C would I need to have probability P of succeeding? Let's say I'm betting on a 37 number roulette wheel (18 red 18 black and one green 0)?'

On the surface, the best probability for the roulette player to be ahead is in one trial (spin): 48.6% to win (versus 51.4% to lose), as far as even-money betting is concerned. I don't agree that it is the best strategy (betting all your bankroll on one spin).

Theoretically, no bankroll will put a player ahead guaranteed, IF flat-betting and playing very long consecutive sessions. There are moments, however, when the roulette player can be ahead by at least one betting unit. Even in even-money bets, the player has a good chance to be ahead by at least one unit after 5, or 10, or even 100 spins. But more than 20 spins are NOT recommended; the probability (odds) to lose go(es) above 50%! Think about it!

The main thing, mathematically, is the number of player's wins in N trials. To be ahead, means the player has won at least one more roulette spin (number of successes) than the number of losses in N trials. The question then becomes:

'What are the probabilities for the player to be ahead in various numbers of trials?'

Everybody can use my probability software SuperFormula, option L: At Least M successes in N trials.
Winning probability: p = 18/37; M must be at least (N/2) + 1.
Here is a number of cases from the player's perspective.

The figures are applicable to all even-money roulette bets: black or red; even or odd; low or high (1-18 or 19-36).

1 trial (spin)
- probability (odds) to win: 48.6%; odds = 1 in 2.05
- probability (odds) to lose: 51.4%; odds = 1 in 1.95
(the probability to lose is 19/37; adding zero to unfavorable cases).

2 trials (spins)
- probability (odds) to win 2 of 2: 23.7% (1 of 2 doesn't mean 'being ahead')
- probability (odds) to lose 1 of 2: 76.3%

3 spins
- probability (odds) to win at least 2 of 3: 48%
- probability (odds) to lose at least 2 of 3: 52%

10 spins
- probability (odds) to win at least 6 of 10: 34.4%; odds = 1 in 2.91
- probability (odds) to lose at least 6 of 10: 41.1%; odds = 1 in 2.43

20 spins
- probability (odds) to win at least 11 of 20: 36.5%
- probability (odds) to lose at least 11 of 20: 46.2%

100 spins
- probability (odds) to win at least 51 of 100: 35.5%; odds = 1 in 2.82
- probability (odds) to lose at least 51 of 100: 56.8%; odds = 1 in 1.76.
It's getting worse for the player...

The roulette strategy (or system) is a totally different ball game! But there are professional gamblers out there, including roulette players! They must have strategies, some roulette systems deduced from some figures like the ones above! The player can be ahead at any point in the game. If so, maybe it's time to move to another (or casino) table: It improves the odds of winning!

Always keep track of the losing and winning streaks. Be strong and put an end to a winning streak. You are ahead, you quit the roulette table. Go to another table and wait until you are ahead. The bankroll is of the essence: It must assure going through long losing streaks. Divide the streaks in 10 spins or 20 spins. Never fight aggressively short or mid-term losing streaks. This is the best approach for those who do not know Ion Saliu's casino gambling systems. A good approach to gambling is the next best thing to a good gambling system! Applicable to blackjack and baccarat, too!

Axiomatic one, everybody knows that the casinos have an edge or house advantage (HA) in all the games they offer, roulette including. The house advantage is created by the payouts in rapport to total possibilities for the respective bet. We can apply this simple formula based on units paid UP over total possibilities TP:

HA = 1 – (UP / TP)
(always expressed as a percentage.)

For example, in single-zero roulette, the one-number (straight-up) bet has payout of 35 to 1. The to qualifier is very important: the casino pays you 35 units and they give you back the unit you bet; thus, you get 36 units. There are 37 possibilities in single-zero roulette: 36 numbers from 1 to 36 plus the 0 number. Therefore, HA = 1 – (UP / TP) = 1 – (36 / 37) = 1 – 0.973 = 0.027 = 2.7%.

Let's calculate HA for the 1 to 1 bets: black/red, even/odd, low/high. HA = 1 – (UP / TP) = 1 – (2 / 2.055) = 1 – 0.973 = 0.027 = 2.7%. There are little differences among bets depending on how many decimal points we work with in our calculations.

The point is, the casinos have an advantage, or the players have a disadvantage. Nonetheless, the players' disadvantage is far better than what they face in state-run lotteries. Yet, most casino gamblers lose big, including at roulette tables. They do not have sufficient bankrolls to withstand long losing streaks.

However, around 45% of the roulette numbers lead the gamblers to profits in a few thousand spins. That is, with a sufficient bankroll, a player has a pretty good chance to make a profit, even if playing a random roulette number, or a favorite number. I analyzed about 8000 roulette spins from Hamburg Spielbank (casino). Quite a few numbers ended up making a profit: roulette systems, magic numbers.

By contrast, the more lottery drawings a player plays, the higher the degree of certainty of a loss. Let's make a comparative analysis to the roulette long series above (spins: total roulette numbers, 37, multiplied by 200). If playing the pick-3 lottery for some 100,000 drawings, it is guaranteed that all pick-3 straight sets will be losers. Some numbers will hit up to 3% to 5% above the norm — but that is not nearly enough to assure a profit. A frequency of 3% to 5% above the norm leads to profits in roulette, however.

Ion Saliu's Paradox and Roulette

Ion Saliu's Paradox of N Trials is presented in detail at saliu.com, especially the probability theory page and the mathematics of gambling formula. If p = 1 / N, we can discover an interesting relation between the degree of certainty DC and the number of trials N. The degree of certainty has a limit, when N tends to infinity. That limit is 1 — 1/e, or approximately 0.632....

If you play 1 roulette number for the next 38 spins, common belief was that you expected to win once. Not! Non! Only if you play 38 numbers in 1 spin, your chance to hit the winning number is 100%. Here is an interesting table, which includes also The Free Roulette System #1 presented at the main roulette site.

The maximum gain comes when playing 38 numbers in one spin: 36.3%. Obviously, it makes no sense to play that way because of the house advantage. On the other hand, a so-called wise gambler is more than happy to play one number at a time. What he does is simply losing slowly! Not only that, but losing slowly is accompanied by losing more. That cautious type of gambling is like a placebo. A roulette system such as Free System #1 scares most gamblers. 'Play 34 or 33 numbers in one shot? I'll have a heart attack!' In reality, the Free Roulette System #1 offers a 28.8% advantage over playing singular numbers in long sessions. That's mathematics, and there is no heart to worry about, axiomatic one.

You can also use SuperFormula to calculate all kinds of probabilities and advantage percentages. The option L — At least M successes in N trials is a very useful gambling instrument. If you play 19 numbers in one spin, the probability to win is 50%. If you play 19 numbers in 2 consecutive spins, the probability to win at least once is 75%.

Editor's note
• In an apparent change of heart, the Hamburg casino (Spielbank) offers online roulette results for free. The new link is (for the time being!):

www.spielbank-hamburg.de/spielsaal/permanenzen.php4

• • Real-life roulette spins are also available from the Wiesbaden, Germany, Casino (Spielbank)

www.spielbank-wiesbaden.de/DE/621/Permanenzen2.php: Wiesbaden Spielbank Permanenzen

Roulette: Software, Systems, Super Strategy

See a comprehensive directory of the pages and materials on the subject of roulette, software, systems, and the Super Strategy.

• Theory, Mathematics of Roulette Systems, Strategies, Software.
• The Best-Ever Roulette Strategy, Systems based on mathematics of progressions and free-for-all.
• An inventory of free and outrageously pricedroulette systems.
• Software, Systems: Roulette Wheel Positioning, Sectors, Birthday Paradox.
• 'Roulette System that Won Millions!'
  ~ Ion Saliu's roulette theory, systems, and software were pirated by an Australian gambling group.
• James Bond Roulette System for Dozens, Columns.
• Roulette Systems, Threats from Casino Chairman.
• Anti-Gambler Advice from John Patrick, Casino Moleextraordinaire and conspirator.
• Wizard of Odds Had High Praise For Ion Saliu's Gambling Theory.
• Casinos pay troubled individuals to intimidate intelligent gambling system authors.
• Download roulette, blackjack, casino gambling, systems, software.

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Roulette is one of oldest casino games and its origins could easily be traced to more than a millennium ago when various simplified versions of it existed as a form of entertainment. Today, gamblers have access to dozens of betting systems, strategies, and even cheating devices that promise long-term success to those who dare use them. Still, there are only two simple concepts they need to grasp in order to improve their play and their chances of winning and that is the odds and the casino advantage that is built-in in every roulette game.

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Knowing the odds is essential in playing roulette or any other game of chance where players’ decisions have no impact on the outcome of the round. Games of skill, in contrast, typically require at least basic knowledge of the rules and give skilled players the opportunity to improve their chances of winning. Unlike classic casino games such as blackjack or poker where individual actions can significantly change the outcome, roulette is a game of pure chance where the outcome is completely random.

Once players recognize this fact, they can truly understand the nature of roulette and learn how to optimize their play to avoid significant losses on the roulette table. Those who want to be successful in the long term need to learn the basic probabilities in this game and determine how likely to win or lose any given bet is. Moreover, all roulette fans should bear in mind that every bet they place is more likely to lose than to win – this is how casinos manage to always make a profit even with the occasional big wins they have to pay out.

So, what is probability and is there a difference between the true odds in roulette and the casino odds offered to players? These are fundamental questions that need to be answered honestly before gamblers sit around the roulette table.

Probability and True Odds

Often, the terms “probability” and “odds” can be confused by even professional casino players. Although they are, indeed, very similar in terms of their usefulness and relevance to evaluating different roulette bets, the two are quite different – at least theoretically.

First of all, players should know that if fair and unbiased, roulette wheels produce random results with every spin. Each of the numbers is equally likely to win or, rather, equally likely to lose. At the same time, for each number from 0 to 36, there are only two possible outcomes – a win or a loss.

Roulette Odds One Number

Probability

The probability is the likelihood of any given outcome compared to all outcomes that are possible. Expressed as a number, the probability always exists between 0 and 1 – 0 would indicate that something is impossible to occur, while 1 would mean absolute certainty for the outcome. Even chances, for example, would be expressed as a 0.50 probability. In roulette, we have either 37 or 38 numbers in total, so these are all the possible outcomes.

Let us take the single-zero roulette wheel where for each number, there is only 1 way to win and 36 ways to lose. So what is the probability that the ball will fall on 17 in the next spin, for example? We can calculate it by dividing the number of ways to win by all possible ways – 1/37, which is 0.027. For most people, the probability is easier to understand as a percentage, so we simply multiply the decimal by 100 and we get 2.70%.

True Odds in Roulette

Now that we have established what probability is and how it is determined in roulette, we can compare it to the notion of “odds”. Odds are also used to describe the chance of an event occurring but they compare the number of ways it can occur to the number of ways it cannot occur. The odds of any particular number winning in roulette could be simply displayed as 1:36 or 1/36 where 36 is, once again, the number of ways to lose.

Sometimes, when it comes to expressing the odds of a particular bet in roulette, they would be in reverse, indicating the odds against winning. Using the same example, the odds of the number 17 against winning would be 36:1, or 36 to 1. These are known as “true odds” as compared to what some players refer to as “casino odds”. There is a huge difference between the two terms, but it will be explained in detail in the following section.

To make this even clearer, let us see the odds for probably the most commonly placed bet in roulette – red/black. If we place our chips on red, the probability would be 18 out of 37, while the odds would be 19:18 against us because there would be 19 ways to lose against 18 ways to win. In percentages, the probability would be 18/37 = 0.48648648648, 48.65%.

Casino Odds

When determining whether a roulette bet has good or bad odds, experienced players take into account not only their probability of winning and the true odds but also the potential reward they could bring. Each bet pays out differently, depending on its likelihood of winning – the less likely a given outcome is to occur, the more its potential payout would be. There is a sound logic behind this and in a perfect world where casinos would not have to make a profit, the payout of every bet would be equal to the odds against winning.

Let us take a look at the straight bet in European roulette – the odds against the player here are 36 to 1 (36:1) because as we have already shown above, there are 36 ways for this bet to lose and only 1 way to win. In order to offer a reward that would match the risk, the casino would be expected to pay players 36 to 1. In other words, it would be expected to return the original stake and to pay out winnings that are worth 36 times the amount of the bet.

Casino Odds Additional TipsIn reality, this does not happen, however, and the house gives a slightly lower payout of 35:1 on winning straight-up bets. The idea is that on every winning wager, players pay a small fee to the casino and in this case, it is one unit. The difference does not seem significant but it is how casinos gain their advantage over players – by paying every winner just slightly less than they should have. Over time, this tiny difference makes a large profit for the house and provides a guaranteed income in any possible scenario.

These payouts offered by casinos are sometimes referred to as “casino odds” due to their visible similarity. And while both are expressed as ratios of two numbers, they are never equal – the odds against winning are always slightly higher than the casino odds. The closer the casino odds get to the true odds of any given bet, the lower the advantage of the casino.

Roulette House Edge Explained

Clearly, the payout does not correspond to the true odds of roulette bets. This advantage of the casino over its patrons is called house advantage and can be easily demonstrated with the following example – we bet $1 on the number 17 and win. If given the true odds for this bet (36:1), we would receive our initial $1 stake back plus $36 as a payout. Instead, casinos pay us $35 and the original $1 bet but they keep the $1 difference.

The same concept applies to all bets in the game, which results in an average house edge of 2.70% for single-zero roulette and 5.26% for double-zero roulette. There are several formulas for calculating the house edge but probably the simplest one is the following – we need to subtract the casino odds from the true odds against success and then, to multiply that by the probability of success. So, the formula will look like this:

House Edge = (True Odds – Casino Odds) x Probability

When we substitute with the numbers we have already used for the straight bet, we get – (36/1 – 35/1) x 1/37 = 1×1/37 = 0.02702702702, or 0.027. To express it as a percentage, we multiply by 100 and get 2.70%. Several other formulas exist, but they all eventually come to the same conclusion and percentage for the house edge.

Roulette House Edge Additional TipsWhat this means for the players is that they can expect to lose 2.70% of their wager when playing a single-zero roulette. For every $100 they bet, they would lose on $2.70 on average. Of course, this is a theoretical ratio between the stake and the expected loss but things could be very different in real life. If we place a $100 chip on red, we can either win or lose its entire value. But we cannot win the bet and lose $2.70 at the same time.

So, the house edge can only get closer to reality with hundreds or even hundreds of thousands spins of the roulette wheel. It is a theoretical concept and while this may sound too vague to players, it is a good indicator of how much money they can expect to lose over time. Of course, they might win hundreds of dollars by the end of one gaming session or lose hundreds more in the next one. Still, they are less likely to lose their bankroll if they stick to specific games and bets where the casino has the lowest advantage.

Beating the Odds in Roulette

The idea that by using a complex strategy one could beat the odds in roulette is very popular but equally misleading. There are countless guides, books, and websites dedicated to convincing people that there is a guaranteed method of winning in this game. In fact, various betting systems, often inaccurately called “strategies”, have been developed over the years, as well as ways that would supposedly help players exploit the imperfections of physical roulette wheels and overcome the house edge.

Unfortunately, these methodologies have been repeatedly proven to be inefficient in securing winnings over the long term. Moreover, their short-term usefulness is doubtful and the reason is obvious to most expert-level players – roulette is a game with fixed odds that cannot be changed even with the best strategy. As explained above, the outcome of every spin of the roulette is random and based on probability. The house edge, therefore, also remains the same almost as a proof for the popular saying that the house always wins.

Roulette Strategies

The so-called roulette strategies are, in their essence, betting systems based on a progression where the amount of the stake changes after a certain outcome. Famous methods such as the Martingale or the D’Alembert suggest that you increase your bet after every loss, hoping that, in the end, one winning bet will compensate for all the losses. There are also systems where the bet is decreased, while in others, it remains the same throughout the entire game session. Overall, the idea is to help players minimize their losses or even generate some decent winnings over the long term.

While none of these betting progressions provides a fool-proof way to win, another type of strategies comes with exactly this claim. These strategies are based on the notion that you can increase your chances of winning by covering a large part of the table. Indeed, this sounds logical at first and certain methods even suggest that you should cover much more than 50% of the table. This strategy, however, will be too costly for most players, especially after a few losing spins.

If not used carefully, both betting systems and roulette strategies may “eat up” players’ entire bankroll within a short period of time. Placing neighbor bets or any other type of announced bets could be detrimental to those who are not well prepared to suffer serious losses. In conclusion, even the best methods and combinations of bets cannot aid you in beating the odds and overcoming the built-in casino advantage.

Advantage Play

Some roulette players rely on quite different methods for securing winnings. Known as advantage play methods, these include any strategies that give players either a statistical or a mathematical advantage over the casino. If employed successfully, they can beat the standard roulette odds and even if it is just by a little, it should be enough to provide players with long-term winnings. Unlike the betting strategies and systems described above, advantage play does not revolve around the betting layout but rather, around the wheel.

It can be used in both online and land-based casinos and online, advantage players would try to use special software that records and analyzes the results of hundreds of spins. The aim is to find a pattern in the winning numbers (such as repeating sequences of winning numbers) and then, to use it to make predictions. However, most online casinos nowadays offer roulette games based on RNG (random number generator) so finding patterns would be impossible as the outcome of every spin is random.

When it comes to playing a roulette game with a physical roulette wheel, however, advantage play is much more different. Players who try to use this technique would typically stand by the roulette table for at least 40-50 spins and write down all the winning numbers in the hope that they would be able to spot numbers that come out more frequently than others. In fact, sometimes they observe the wheel for hundreds of spins before they can notice repeating numbers, patterns or some irregularities.

Advantage Play Additional TipsThis method was mostly used in the past when casinos did not have such strict maintenance rules and protocols, while the wheels and other gaming equipment were manufactured in a manner that today would be deemed unacceptable. Players who wish to turn the odds in their favor these days need to be extremely discreet if they plan to observe the wheels before the casino security become aware of them. Exploiting roulette wheels’ bias and imperfections for one’s profit is not usually met with understanding from casinos.

How to Increase Player’s Chances of Winning

There are no reliable ways to actually beat the odds when playing roulette and even if such methods exist, they are neither simple nor acceptable by casino standards. Yet, players can increase their chances of winning by following several basic principles that can be applied to both online and brick-and-mortar casinos.

First of all, choosing a good roulette table is essential and obviously, single-zero roulette games are a much better option than double-zero games, which are mostly offered in casinos across the United States. The house edge in the American-style roulette is twice as high due to the additional sector on the wheel, the green 00. But picking French or European-style roulette variations is just the first step in learning how to maximize players’ expected value.

Roulette Odds Event

Second, players should always base their play around wagers with the lowest possible house edge. Of course, the best bets in roulette are the outside bets, which cover large portions of the wheel and require a single chip to be wagered. These include black/red, even/odd, and low/high, where the advantage of the casino is 2.70% while the player’s odds of winning are the highest. It is true that the payouts are not particularly attractive but these bets are less risky options in the game.

Roulette Odds Of Black

When choosing a bet, players need to compare the casino odds to the true odds and find the type of bet where these two as close to each other as possible. Often, the most attractive payouts are offered for bets which the casino pays much less than their actual worth. As a general rule of thumb, players should remember that the less risk they take, the less they can expect to win. All the good payouts, however, come with exceptionally bad odds and require a great tolerance for risk.