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1. Probability Of Winning Roulette
2. Roulette Wheel Probability
3. American Roulette Probability
4. Roulette Probability Statistics Problem
5. Roulette Wheel Probability Calculator
6. Blackjack Probability Example

You hear people talk about “odds” and “probabilities” all the time. You might have wondered if these words mean the same thing or if they have different meanings.

The answer to your questions is “yes” (and “no”).

American English is often described as a “lazy” language because no matter how precise a meaning you intend for any newly devised word, sooner or later someone will use that word in an unexpected way.

The English language has developed a huge selection of synonyms, words with the same or nearly same meaning, for many different concepts. “Odds” and “probabilities” can be synonymous and in common usage often are.

Probabilities for such are calculated by dividing the number of elements in E by the number of elements in Ω. Let’s define some events in Roulette, namely the events of a red number, a non-zero. For example, why is it that no one comes to me saying this? 'I was playing roulette the other day, and I played six times. The first time it got a 15. Then it got 7, then 19, 1, 5, and finally 27. What are the odds of that happening?' Assuming you have an American roulette wheel (with 38 slots, not 37), the probability of that event is 1/38^6. If, for example, the number is located at the corner of the inside part of the roulette table layout (like numbers 34 and 36), the maximum bet is only 18 chips. The smallest maximum bet is on 0 and consists of only 17 chips in total. A great example of this margin that the house hopes to exploit is in a standard roulette wheel. The numbers on a roulette wheel range from 0–36. For example, the probability of Paul Pogba. For example, why is it that no one comes to me saying this? 'I was playing roulette the other day, and I played six times. The first time it got a 15. Then it got 7, then 19, 1, 5, and finally 27. What are the odds of that happening?' Assuming you have an American roulette wheel (with 38 slots, not 37), the probability of that event is 1/38^6.

But even in gambling circles you’ll find people using either word with a precise meaning that is not intended to be synonymous with the use of the other word.

In other words, it’s okay to say either “odds” or “probabilities” in most situations. Use whichever word you prefer; most people will understand your meaning.

But as you learn more about gambling and statistics you’ll recognize those special cases where each word means something different from the other one.

This post explains the differences for you in (hopefully) plain, simple language.

Let’s begin with simple definitions for these words:

What Are the Definitions for “Odds” and “Probability?”

Odds – The primary definition given for this word is about the “ratio between two amounts staked by parties to a bet”. This technical definition can be and often is clarified. Here’s an example:

The odds in roulette are typically different from the odds in craps. The reason for these differences is that each game has different expected probabilities. A bet on a single number in roulette pays off at 35 to 1 odds, but the odds of winning are 37 to 1. A bet on a specific number in craps has different odds depending on which number you’re choosing. The payoff odds for those bets vary, too.

Probability – The usual definitions given for this word are confusing because they use the word “probable”. An example definition might read: “the extent to which something is probable; the likelihood of something happening or being the case.”

That’s okay when you’re speaking casually.

But there’s a more precise meaning for probability in the field of statistics. It’s important to understand the statistics definition for this word because many gambling tutorials and experts draw upon statistics to explain probabilities in games of chance.

How Is Probability Computed in Statistics?

In statistics a probability is derived from the distribution of a set of scores. The distribution is the set of all the values that have been collected from some type of operation. The set could consist of all possible values resulting from the operation OR the set could consist of all recorded values resulting from execution of the operation more than once. That might be a little confusing.

So let’s look at a classic statistics example: a simple coin toss.

A coin has two sides (we ignore the edge of the coin). When you toss a coin it will land either HEADS facing up or TAILS facing up (again, we ignore any case where it lands on the edge). This gives us a probability set of 2 items: HEADS and TAILS. We say, based on the number of items in this 2-item set, that both HEADS and TAILS have a 0.50 (or 50%) probability of turning up as a result of the coin toss. Both outcomes are equally likely.

But suppose you want to test that probability. You might toss the coin in the air 100 times. You could also write a computer program to simulate a coin toss and use it to run 1,000,000 simulations. In this example, you record the outcomes of the simulations or real coin tosses and tally up how many times the coin lands HEADS up and TAILS up. In practice you might not find an exact 50% split between the two possible values. These recorded outcomes are the SCORES in your PROBABILITY DISTRIBUTION.

Probability Of Winning Roulette

In short, “probability” could refer to either the EXPECTED OUTCOME based on the number of possible outcomes.

OR it could refer to the subset of outcomes of a specific score from a set of real outcomes collected from an experiment or data source.

This distinction is important because it will help you understand the limitations of calculating probabilities for gambling games.

How Are Odds Computed for Gambling Games?

Casinos rely on “odds” to determine how much they will pay players who win their games. The odds are calculated so that, over a long period of time, the casinos will retain a certain expected percentage of the players’ wagers. This is where casinos get their income. The players’ winnings are paid out of the total sum of the wagers they place.

In other words, if a gambling game has a probability of 1% of producing a specific result, the casino may pay a player $99 when that result occurs. The $99 is retained from other player wagers. The player who wagered the winning $1 loses his wager but is awarded $99. Or you could say the player wins $98 and keeps his $1 wager. The casino’s $1 profit is called the “house edge”.

Of course, the money really passes back and forth continuously as players make bets. Players win some bets, and they lose some bets. All the casino cares about is that it retains a small percentage of the bets over time.

But each game is designed to retain a different expected percentage. The “odds” for a game are computed to ensure the casino gets its percentage. Here’s an example using roulette:

You’ll find 3 styles of roulette available in the United States today:

1. European Roulette
2. American Roulette
3. Sands Roulette.

The chief difference between these games is found on the roulette wheel:

The European wheel has exactly 18 black slots, 18 red slots, and 1 green slot.

The American wheel has 18 black slots, 18 red slots, and 2 green slots.

And the Sands wheel has 18 black slots, 18 red slots, and 3 green slots.

Roulette Wheel Probability

All things being considered equal, the probability of the roulette ball landing on any number is 1 divided by ( 18 + 18 + number of green slots). In American Roulette the probability works out to 1 / 38 (or about 0.026, aka “2.6%”).

Notice, too, that the probability would be different for the other 2 versions of the game. Each game has a different total number of slots. So the probability on a European wheel is 1 /37, and the probability on the Sands wheel is 1 /39.

But if you place a bet on any specific number and the ball lands on that number’s corresponding slot, you’ll only be paid a prize of 35 times your bet. That’s true regardless of which game you play. The difference between the payoff odds and the odds of the bet winning is where the house edge stems from.

In other words, the casino has shaved off a little bit of the probability so that it makes a slight profit on the odds. The probability is what is likely to happen but the odds are what you are paid.

How Reliable Are Probability Simulations?

If you research gambling games you’ll find that people have calculated some impressive probability tables for virtually all the major games. These probability tables may be based on millions, even billions of iterations (repeated simulations). They’re considered reliable in establishing expected outcomes, but they don’t predict what’s going to happen in a specific game. We use these probability tables to estimate what is safe to bet on a specific situation, such as the cards in a hand of blackjack or poker, or the available numbers on a roulette table.

But there are two inherent flaws in relying on probability tables.

First, these results are based on computer random number generators. While random number generators are good they’re not perfect. We don’t yet have true random number generation.

In real life events are random, so there is a slight bias in all probability tables.

Second, any probability distribution–regardless of whether it is based on simulations or real game play–only represents the past. In other words, because future events are truly randomized no probability table can accurately predict the percentage of possible outcomes for any length of play.

In other words, not only is the casino risking NOT retaining as much money as it expects from player wagers, you might play 100 losing hands in a row, or make 1,000 unrewarding slot spins in a row. However unlikely these events may seem to be, they can happen. Probability tables won’t tell you when these least likely series of outcomes will happen. At best a sophisticated probability table will tell you that these unfortunate losing runs are mathematically possible.

Nonetheless, even though a probability table won’t reveal the future to you, it’s better to consult these tables when deciding how you’re going to bet your money. Experienced gamblers have learned to fold their cards in certain situations because the chances of winning are far outweighed by the chances of losing. Experienced roulette players may make more conservative wagers than inexperienced players because they know their chances of winning are better with the conservative wagers.

Odds and Probabilities Tell Us One Important Thing about Gambling

Both odds and probability tell us something important about casino gambling:

The house always wins.

The longer you play, the more of your money the house will keep.That’s just how the math works.

So why do people keep gambling?

For one thing, taking any kind of risk can be fun. People enjoy taking risks. And given that only about 10% of the population are prone to become addicted to gambling, most people can safely gamble for a few hours a night and still use good judgment. Of course, if you’re drinking or doing drugs and gambling your judgment will be impaired.

But we gamble because it’s fun–not because we’re probably going to win.

That being said, professional gamblers not only enjoy the game but count on it for an income.These pros find ways to take advantage of inherent conditions in the game which can provide them with a mathematical edge over the house. The classic example is someone who counts cards in blackjack, but poker pros and sharp sports bettors can also tilt the odds in their favor.

If you can stay in the game long enough, even a slot machine game, there is always a chance you’ll win a big prize or pot and get your money back. You may even come out ahead. This happens often enough that people remain interested in gambling. But remember that the probabilities are against you. The players who win are being paid from your wagers. When you win you are being paid from other players’ wagers. That is where the money comes from.

Conclusion

This is why gambling experts caution people not to expect too much from their tips and teachings. It’s great when you win big money but you should look at gambling as a form of entertainment. It’s a way to pass an evening, especially if you’re with friends. The odds will never be in your favor but maybe, just maybe, once in a while you’ll have a really good evening.

That’s what we all hope for.

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.

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.

American Roulette Probability

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.

Roulette Probability Statistics Problem

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.

Roulette Wheel Probability Calculator

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.

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.

Blackjack Probability Example

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.