The meanings and importance of odds
In the context of gambling, the term ‘odds’ may have any of the following main meanings: probability, payout rate, and sometimes house edge or player’s return. In the gambling jargon, ‘odds’ is also meant sometimes as a format in which a numerical probability can be expressed. We will take further each of these possible meanings to define and clarify, but let’s first note that in the absence of a standard of definition for them, which to be adopted by the wide majority of the users of this term, one could use it with a personal subjective meaning that might not match the meaning that their hearer or reader will assign to the term. This is the traditional recipe of a semantic conflict, applicable in any communication, including beyond gambling. Using ‘odds’ several times in the same text without a clear context or clarification for each usage also submits to the meaning mismatch, as it may create confusion for the reader or mislead them.
Clarification of the meaning of the term ‘odds’ is important not only for communication, but also because odds (in any meaning) have associated information that characterizes the game in question. Such information is often used by the players as criteria of choice and decision in gambling and thus taking the right meaning in a given context may prevent the misapplication of those criteria and decisions that could be wrong even from their setup.
In problem gambling – the academic area dealing with problematic gambling behavior, including pathological gambling and addiction – the cognitive distortions (in the form of misconceptions, fallacies, or irrational beliefs) are considered risk factors. Since many of these cognitive distortions are related to the understanding of the concept of odds, distinguishing between the various meanings of this term and having a good grasp of them add to the importance of this knowledge.
Odds as probability
When someone asks “What are the odds of that?” as referring to an uncertain event, they assign to ‘odds’ the meaning of measurable likelihood or chances for that event to happen. Since an answer to the previous question would be in a numerical form, the likelihood becomes a measure for that possibility and such a qualification is mathematical. This is why the meaning of ‘odds’ in that context is mathematical, namely the notion of probability.
What is probability?
In mathematics, the notion of probability is defined as a particular measure, in terms of measure theory. This measure function measures sets that belong to a certain structure (of measurable sets, called Boolean structure), takes values in the interval [0, 1], and has certain properties, among which the most important is additivity: the probability of a union of two or more mutually exclusive sets is the sum of their probabilities.
When interpreted in the real life, those measurable sets represent random events, and are sets of evidences or possibilities. When estimating the probability of a random event, we actually count the evidences, instances, or situations that are in the favor of the occurrence of that event, relative to the total number of possibilities, which form what is called the sample space of that experiment.
Example:
The event ‘The die will roll a five or a six’, associated with the experiment of rolling a die, is represented as the set of evidences {5, 6} and its probability is 2/6 (or 1/3 = 0.3333 = 33.33%), because there are 6 possibilities in the sample space for that event. We can say that the odds for the die to roll a five or a six are ‘one in three’ or express that as a percentage.
The simplest definition of probability is that in which it is expressed as a ratio:
The probability of a random event E is the ratio between the number of situations (evidences or instances) favorable for E to occur and the total number of possible situations.
It is called classical probability. Although the general mathematical definition of probability is far more complex and involves infinity, the classical probability suffices to be applied in gambling, where any gaming event can be expressed as a finite set of evidences and the sample space of any experiment is finite.
- The event ‘The ball will land on a red number’ at European roulette has the probability 18/37 = 0.4864 = 48.64%, because there are 18 red numbers on the roulette wheel out of 37 possible numbers.
- At craps, the probability for the two dice to roll a sum of 7 is 3/18 = 1/6 = 0.1666 = 16.66%. Here the possibilities are expressed as 2-size combinations (of numbers on the two dice, regardless order). There are 3 favorable combinations for that event: (1, 6), (2, 5), and (3, 4), out of 18 possible combinations (of numbers from 1 to 6).
- If you have a blackjack hand consisting of a queen and a seven, dealer’s face-up card is a three, in a two-deck game where you are the only player at the table, the probability of not going bust with the next card if hit is 31/101 = 0.3069 = 30.69%. This is as such because: The cards favorable for that event are A2, 2s, 3s, and 4s, in total number of 32, of which one is in the dealer’s hand: 32 – 1 = 31. The total number of possibilities is 104 (the total number of cards in the two decks) minus 3 (the cards showing in your hand and the dealer’s hand): 104 – 3 = 101.
Now that you know in brief what probability is and have an insightful image of how it applies to gambling events, we can talk about the usage of the term ‘odds’ with the probability meaning. For identifying this meaning in a given text, we first have to get assured that the context in appropriate for that meaning. Such a context should refer directly or indirectly to chances or likelihood and necessarily mention an event as associated with ‘odds’. The reference to an event could be in the form of “the odds for [that event] to occur/happen” or as a question, like “What are the odds of [that event]?” or “Which of [two or more events] has better odds?”. If reference to an event as associated with the odds is missing, there might not be probability that the author of that text had in mind as meaning for their usage of ‘odds’.
It is important to note that ‘odds’ as probability is associated with a gaming event, but not with a game. If someone says “the odds of roulette” or “the odds of blackjack” as referring to certain numbers, they might not think of probabilities, as there are several possible events in these games, each having its own probability. It is possible that in this case the intended meaning was ‘house edge’ or ‘player’s return’.
How probabilities are computed in gambling
In gambling, the events that can be measured in probability range from very simple (as in our previous examples) to very complex and so does range the complexity of the computations necessary for obtaining the probabilities of those events. The easiest to compute probabilities are those associated with the game of roulette and dice games, while the most difficult are those in card games, like blackjack, baccarat, or poker.
The difficulty of the probability computations depends on both the event in question and the game it belongs to. For computing a gambling probability, one could use the definition of classical probability, but also other mathematical properties of probability, depending on the event to be measured.
We can compute the probability of an immediate event (following to occur with the next stage of the game) or a long-shot event (occurring in a further stage or as a final outcome of the game). The probabilities associated with these categories of events are called immediate odds (for the former events) and long-shot odds (for the matter events). The long-shot odds are the most difficult to compute, as often they are conditional probabilities (depending on intermediary uncertain events whose probabilities have to be employed in the equation).
For example:
The odds of being dealt certain card or cards in blackjack when hit are immediate odds, while the odds of winning the hand, estimated when you received the first two cards, are long-shot odds, because they depend on the probabilities of the dealer’s hand.
The odds for a certain card configuration to occur at the flop are immediate odds, while the odds for any of your opponents to beat your hand by river (estimated before the turn) are long-shot odds in Hold’em poker; the latter involves the probabilities that your opponents will beat your hand (if reaching a showdown) conditional on the probabilities for you to acquire the various draws.
Even when computing the probabilities by their classical definition, as a simple ratio, things are not always simple. Most of the difficulties arise in combinatorial games, like card games, where the count of the favorable situations for the measured event involves combinatorial calculus. This is why such probabilities are also called combinatorial probabilities.
In case of very difficult computations that cannot be done by applying compact formulas, mathematicians may use various techniques of approximation for the sought probabilities. But there are situations (games, stages, and/or events) where a direct computation is actually impossible. When mathematics fails in computing odds due to complexity and difficulty, the experts can approximate them through computer simulations, by various statistical methods, as it is the Monte Carlo simulation.
In such simulations, the game artificially runs for millions of rounds (or even billions, if required), the measured event is tracked and its occurrences are counted, then its relative frequency is taken as an approximation for its probability (as the ratio between the number of occurrences and the total number of rounds). The longer the run, the more accurate is this approximation.
Even when computing the probabilities by their classical definition, as a simple ratio, things are not always simple. Most of the difficulties arise in combinatorial games, like card games, where the count of the favorable situations for the measured event involves combinatorial calculus. This is why such probabilities are also called combinatorial probabilities.
In case of very difficult computations that cannot be done by applying compact formulas, mathematicians may use various techniques of approximation for the sought probabilities. But there are situations (games, stages, and/or events) where a direct computation is actually impossible. When mathematics fails in computing odds due to complexity and difficulty, the experts can approximate them through computer simulations, by various statistical methods, as it is the Monte Carlo simulation.
In such simulations, the game artificially runs for millions of rounds (or even billions, if required), the measured event is tracked and its occurrences are counted, then its relative frequency is taken as an approximation for its probability (as the ratio between the number of occurrences and the total number of rounds). The longer the run, the more accurate is this approximation.
Example:
A probability of 1/4, which reads as ‘1 to 4’, can be expressed as the result of the division 1 : 4 = 0.25 = 25%.
In the gambling language, in the appropriate context, ‘odds’ (as probability) and ‘probability’ can be used interchangeably, but sometimes ‘odds’ is meant as a certain format in which a probability can be expressed. In the customary usage of such distinction, a probability can be expressed as a fraction (ratio), percentage, or “odds”.
The odds format represents a way of expressing the probability of an event relative to the probability of the contrary event. The following formula is used to convert a probability into odds:
odds = probability / (1 – probability)
For example:
A probability of 1/3 of event A is converted into the odds format as follows:
odds of A = (1/3)/(1 – 1/3) = (1/3)/(2/3) = 1/2
This is denoted by 2 : 1 (the traditional denotation for odds) and reads as ‘2 against 1’ or ‘2 to 1’. It means ‘the chances for event A to occur are 2 against 1, meaning there are two chances in three for non-A to occur and one chance in three for A to occur.’
The expression of probability in odds format usually holds only natural numbers, so we must make an approximation if necessary. For example, if the above formula returns 3/7 = 1/2.333..., we can approximate it as 3 : 1 odds.
Conversely, to convert the odds format into a fractional probability, the following formula is used:
probability = odds / (1 + odds)
For example:
3 : 2 odds are converted into fractional probability as follows:
(2/3) / (1 + 2/3) = (2/3) / (5/3) = 2/5 or 40% as percentage.
Even though the words probability and odds as format (in a probability context) have different definitions attached, they express the same quantity and quality, so their undifferentiated usage along with the associated figures is not quite an error, but rather a deviation from consistent terminology.
Where the probabilities are displayed
The odds (as probability) associated with a game are not visible information. No producer or operator of a game of chance displays this information along with the presentation of that game or in its descriptive materials published along with the game’s rules, as no gambling regulation requires them to do so. Therefore, if you want to know more about a game’s probabilities, you have to dig for that information, by retrieving it from specific resources, if not computing the odds by yourself.
First, it worth noting that the probabilities associated with the events of a game can be in the order of hundreds, thousands, or even millions. Some of the odds are simple to compute, like in roulette or dice games, but computing them for games like blackjack or poker requires extensive math knowledge and skills, not mentioning that not all such odds can be actually calculated. This is why the available expert resources (websites, books, or math sheets dedicated to gambling mathematics) remain the effective means by which one can retrieve such mathematical information in a user-ready form.
Regarding odds computation, the very first requirement for it to be possible is that the person doing it to know how the game is configured with respect to producing its outcomes, that is, the parametric configuration of that game.
For instance, we can compute the odds of roulette because we know how many numbers are on the wheel and on the table, and how many numbered are covered by each type of bet. Or, we can compute odds in blackjack as long as we know the number of decks used and all the game’s rules. This basic information is transparent, as it is in other casino games, too.
The exception to this transparency occurs in the games of slots. The parametric configuration of these games (essentially the weighting of the reels) is kept secret by their producers for various reasons. The lack of information regarding the inner design of a slot game prevents one for computing the probabilities of winning each of the prizes mentioned in the payout schedule of the game, as well as other statistical indicators of that game. The only statistical indicator frequently made public with a slot machine is its RTP (Return to Player).
The parametric configuration of a slot game is enclosed in the so-called PAR sheet (Probability Accounting Report) of the game, which is an internal document for the producer of the game. In such sheets, the probabilities are displayed under the label ‘hit frequency’ or ‘hit rate’, for each possible combination of symbols, including the winning ones.
PAR sheets are retrievable only by legal intervention or by special requests for research purposes. You can find PAR sheets of some slot games made public on the internet:
Probability tables
In what follows we provide some winning odds comprised in tables for the most popular casino games. These odds are just illustrative and do not cover all the events associated with the respective game.
| Simple bet | European | French | American |
|---|---|---|---|
| Straight Up | 1/37 = 2.70% (36:1) | 1/37 = 2.70% (36:1) | 1/38 = 2.63% (37:1) |
| Split | 2/37 = 5.40% (17.5:1) | 2/37 = 5.40% (17.5:1) | 2/38 = 5.26% (18:1) |
| Street | 3/37 = 8.10% (11.3:1) | 3/37 = 8.10% (11.3:1) | 3/38 = 7.89% (11.1:1) |
| Trio | 3/37 = 8.10% (11.3:1) | 3/37 = 8.10% (11.3:1) | 4/38 = 7.89% (11.6:1) |
| Corner | 4/37 = 10.81% (8.2:1) | 4/37 = 10.81% (8.2:1) | 4/38 = 10.52% (8.5:1) |
| Four numbers | 4/37 = 10.81% (8.2:1) | 4/37 = 10.81% (8.2:1) | 4/38 = 10.52% (8.5:1) |
| Line | 6/37 = 16.21% (5:1) | 6/37 = 16.21% (5:1) | 6/38 = 15.78% (5.2:1) |
| Column | 12/37 = 32.43% (2:1) | 12/37 = 32.43% (2:1) | 12/38 = 31.57% (2:1) |
| Dozen | 12/37 = 32.43% (2:1) | 12/37 = 32.43% (2:1) | 12/38 = 31.57% (2:1) |
| Red/Black | 18/37 = 48.64% (1.0:1) | 18/37 = 48.64% (1.0:1) | 18/38 = 47.36% (1.1:1) |
| Even/Odd | 18/37 = 48.64% (1.0:1) | 18/37 = 48.64% (1.0:1) | 18/38 = 47.36% (1.1:1) |
| Low/High | 18/37 = 48.64% (1.0:1) | 18/37 = 48.64% (1.0:1) | 18/38 = 47.36% (1.1:1) |
| Total | 17 | 18 | 19 | 20 | 21 | blackjack | bust |
| Probability | 0.1451 | 0.1395 | 0.1335 | 0.1803 | 0.0727 | 0.0473 | 0.2816 |
| Dealer’s first card Dealer’s result | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | A |
|---|---|---|---|---|---|---|---|---|---|---|
| 17 | 0.1398 | 0.1350 | 0.1305 | 0.1223 | 0.1654 | 0.3686 | 0.1286 | 0.1200 | 0.1114 | 0.1308 |
| 18 | 0.1349 | 0.1305 | 0.1259 | 0.1223 | 0.1654 | 0.1063 | 0.1378 | 0.3593 | 0.1114 | 0.1308 |
| 19 | 0.1297 | 0.1256 | 0.1214 | 0.1177 | 0.1063 | 0.0786 | 0.1286 | 0.3508 | 0.1114 | 0.1308 |
| 20 | 0.1240 | 0.1203 | 0.1165 | 0.1131 | 0.1017 | 0.0786 | 0.0694 | 0.1200 | 0.3422 | 0.1308 |
| 21 | 0.1180 | 0.1147 | 0.1112 | 0.1082 | 0.0972 | 0.0741 | 0.0694 | 0.0608 | 0.0345 | 0.0539 |
| BJ | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0769 | 0.3077 |
| Bust | 0.3536 | 0.3739 | 0.3945 | 0.4164 | 0.4232 | 0.2623 | 0.2447 | 0.2284 | 0.2121 | 0.1153 |
| Dealer’s first card Player’s resul | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | A |
|---|---|---|---|---|---|---|---|---|---|---|
| BJ | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 0.9231 | 0.692 |
| 21 | 0.8820 | 0.8853 | 0.8880 | 0.8918 | 0.9028 | 0.9259 | 0.9306 | 0.9392 | 0.8886 | 0.6384 |
| 20 | 0.7580 | 0.7650 | 0.7723 | 0.7787 | 0.8011 | 0.8473 | 0.8612 | 0.8192 | 0.5464 | 0.5076 |
| 19 | 0.6283 | 0.6394 | 0.6509 | 0.6610 | 0.6948 | 0.7687 | 0.7326 | 0.4684 | 0.4350 | 0.3768 |
| 18 | 0.4934 | 0.5089 | 0.5250 | 0.5387 | 0.5885 | 0.6309 | 0.3733 | 0.3484 | 0.3236 | 0.2460 |
| ≤ 17 | 0.3536 | 0.3739 | 0.3945 | 0.4164 | 0.4231 | 0.2623 | 0.2447 | 0.2284 | 0.2121 | 0.1153 |
| EVENT | PROBABILITY |
|---|---|
| Banker wins | 0.458597 |
| Player wins | 0.446247 |
| Tie | 0.095156 |
| EVENT | PROBABILITY |
|---|---|
| Banker wins | 0.458597 |
| Player wins | 0.446247 |
| Tie | 0.095156 |
| EVENT | PROBABILITY |
|---|---|
| Banker wins | 0.458597 |
| Player wins | 0.446247 |
| Tie | 0.095156 |
Importance of probabilities
The probabilities associated with a game are crucial information for evaluating that game or a certain bet of that game. First, it is about playing informed, which includes assessing the risks and expectations, and probability is the most relevant measure for such tasks.
Imagine someone propose you to bet that you will jump down from a high place and land on your feet. Of course, it would be at least useful for you to know in advance or measure the height from which you will jump, as for some figures you may decline the bet or propose another; this means informed decision. The same happens in casino gambling – playing informed means not only to know the characteristics of the game, but also to know the probability of winning (and losing) for any bet you intend to place in that game. Otherwise, it is like shooting at a target blindfolded.
Probabilities also have strategic roles in some games. All optimal strategies employ criteria based on evaluation or comparison of probabilities. Making the right move is driven by probabilistic reasoning and this principle also applies to simple choices between games and bets.
Yet probability is not the only criterion of choosing and moving strategically in gambling. Other factor also counts in this equation, namely the payout rates offered in the games. There is always a balance between the probabilities of winning and the payout rates (also called “odds” often), such conceived to maintain the house’s advantage, as we will see further.
Odds as payout rate
Any bet in whatever game is characterized mathematically by three parameters: its stake, the probability of winning it, and the payout rate. The payout rate is defined as the multiplier (as integer or decimal number) applied to the stake for determining the return to the player as profit in the case the bet is won. In gambling jargon, the payout rate is often termed as ‘payout’, however the payout is actually defined as the amount received by the player when they win a bet (also called payback), that is, the stake multiplied by the payout rate of that bet.
A Split bet in roulette has a payout rate of 17. If you win a Split bet, you receive 17 times your stake as profit and your stake back. If you wagered $5, the payback is $5 x 17 = $85.
A Tie bet in classical baccarat has a payout rate of 8. If you win a Tie bet, you receive 8 times your stake as profit and your stake back. If you wagered $3, the payback is $3 x 8 = $24.
The payout rate is usually denoted in one of the following forms: x (the multiplier sign) followed by the multiplier (example: x2), using ‘to 1’ after the multiplier (example: 2 to 1), or using the division sign between the multiplier and 1 (example: 2 : 1).
The last two forms induced in the gambling jargon the habit of calling the payout rate ‘odds’, just because the probability in odds format is denoted in the same way. Still, the two notions remain different – the odds of winning a bet have nothing to do with the odds as payout offered by the casino operator or bookie. Actually, it is this difference that ensures the house’s advantage in the long run for any bet.
Using the same term (odds) in a given context for referring to probability of winning and payout rate may create confusion and is definitely an error to equate the two notions. For avoiding such confusion, while maintaining the term, the right wording for the payout rate would be ‘payout odds’.
Payout rate in sports betting
The gambling field in whose jargon the usage of ‘odds’ for naming the payout rate is widely accepted and taken as a standard is sports betting. When one talks about the odds of a bet in sports betting, everybody knows that they refer to the payout rate offered by the bookie for that bet. The danger of confusion is actually very low in this case also because probabilities of the outcomes of sports events do not make a mathematical sense (there are too many physical factors determining the outcome, which cannot be quantified in a probability field). In sports betting we can talk about likelihoods (as subjective measures for the possibility), but not mathematical probabilities.
In sports betting, there are three specific formats for expressing the payout odds: decimal (European), fractional (British), or moneyline (American).
The decimal odds
The decimal odds represent the gross payout rate of the bet (the return), which includes your stake ( as x1 multiplier). They represent the payback of a won bet with a stake of one unit, written as a decimal number higher than 1 with two decimal places.
Example: 1.58 odds means that you are paid back 1.58 times your stake if you win the bet, so if you bet $1 you will get back $1.58 and as such make a profit of $0.58.
The fractional odd
The fractional odds are written as a fraction in terms of the net profit, namely the ratio of the possible profit won to the stake.
Example: 5/3 odds means that you can make $5 as a profit from a $3 stake; this means a gross return of $3 + $5 = $8.
The moneyline odds
The moneyline odds do not express rates or multipliers, but amounts of either the stake or profit of a won bet. The odds for the favorites are in the form of a negative integer, representing the amount you need to stake to make a profit of $100. The odds for the underdogs are in the form of a positive integer, representing the profit made for $100 staked.
Example: –175 odds means that you need to bet $175 to make a profit of $100.
+210 means that you can make a $210 profit if you bet $100.
In sports betting terminology, there is the notion of implied probability. Implied probability is just another way of expressing the payout odds, namely as percentage. As percentage, it looks like a probability but actually it is not.
Converting payout odds into implied probability assumes running a simple algorithm of arithmetical operations. Let’s see how this conversion works for the moneyline format, with an example.
Example:
A NFL match between Bills and Patriots has the following moneyline:
Bills: –200
Patriots: +185
For the Bills:
1. Remove the minus sign from the odds and add 100: 200 + 100 = 300.
2. Divide the odds as a positive number to the number obtained at step 1 and write the result as a decimal number rounded at maximum four decimal places: 200 : 300 = 0.6666.
3. Multiply the result obtained at step 2 by 100 and apply the percentage to find the implied probability: 0.6666 x 100% = 66.66%.
For the Patriots:
1. Add 100 to the odds: 185 + 100 = 285.
2. Divide 100 by the number obtained at step 1 and write the result as a decimal number rounded at maximum four decimal places: 100 : 285 = 0.3508.
3. Multiply the result obtained at step 2 by 100 and apply the percentage to find the implied probability: 0.3508 x 100% = 35.08%.
These percentages in the above example reflect the likelihood of the victory of every team from the bookie’s perspective, but they are just another form of expressing the payout odds. Hence we should not equate the odds as probability with the bookie’s odds. One additional argument for this distinction is that the bookie’s commission (called the vigorish) is actually included in these odds: If you add the odds offered for every possible outcome of a given event to bet on, you will see that their sum exceeds one. If they were probabilities, their sum should have been 1. What exceeds 1 is actually the vigorish.
Therefore, in sports betting we should be careful with the meaning of odds as probability – although we know that ‘odds’ is meant ‘payout odds’ in communication, in our reasoning we should not think of it or of implied probability as a mathematical probability.
Payout odds tables
In what follows we provide in tables the payout rates of a few popular casino games. The payout rates are part of the rules of a game and such information should be visible in any description of that game, including on the game itself (on screen or betting table).
| Inside bet | Payout odds |
|---|---|
| Straight Up | 35 to 1 |
| Spli | 17 to 1 |
| Street | 11 to 1 |
| Trio | 11 to 1 |
| Corner (or Square) | 8 to 1 |
| Four numbers | 8 to 1 |
| Line | 5 to 1 |
| Column | 2 to 1 |
| Dozen | 2 to 1 |
| Red/Black (or color) | 1 to 1 |
| Even/Odd | 1 to 1 |
| Low/High | 1 to 1 |
| Payout odds | Rule of application |
|---|---|
| –1 | When the dealer achieves a better hand than you or a blackjack they take your wager. |
| 0 | No money will exchange if you and the dealer achieve the same total value (a push); they return your wager. |
| 1 (or 1 : 1) | If you beat the dealer without a blackjack, you are returned your wager and paid the same amount. |
| 3/2 (or 3 : 2) | When you achieve a blackjack and beat the dealer, you are returned your wager and paid 1.5 times your wager. |
| 6/5 (or 6 : 5) | When you achieve a blackjack and beat the dealer, you are returned your wager and paid 1.2 times your wager. |
| Bet | Payout odds |
|---|---|
| Player | 1 to 1 |
| Banker | 19 to 20 |
| Tie | 8 to 1 (or 9 to 1) |
| Side bet | Payout odds |
|---|---|
| Dragon Bonus | 30 to 1 |
| Panda 8 | 25 to 1 |
| Dragon 7 | 40 to 1 |
| Pair | 5 to 1 |
| Pairs | 11 to 1 (or 10 to 1) |
| Perfect Pair | 5 to 1 (if the pair is not of the same suit) 25 to 1 (if the pair is of the same suit) 12 to 1 (unique payout odds if the pair is of the same suit) |
| All Red/All Black | 22 to 1 (all red) 24 to 1 (all black) |
| Egalité | 80 to 1 (tied value 9 or 8) 45 to 1 (tied value 7 or 6) 110 to 1 (tied value 5) 120 to 1 (tied value 4) 200 to 1 (tied value 3) 225 to 1 (tied value 2) 215 to 1 (tied value 1) 150 to 1 (tied value 0) |
| Natural 8 | 9 to 1 (or 8 to 1) |
| Natural 9 | 9 to 1 (or 8 to 1) |
| Natural 8 or 9 | 4 to 1 |
| Craps Bet | Payout Odds |
|---|---|
| Pass Line | 1 to 1 |
| Come | 1 to 1 |
| Don't Pass | 1 to 1 |
| Don't Come (Bar 12) | 1 to 1 |
| Pass / Come
Free Odds | 2 to 1 on 4 or 10
3 to 2 on 5 or 9 6 to 5 on 6 or 8 |
| Don't Pass / Don't Come
Free Odds | 1 to 2 on 4 or 10
2 to 3 on 5 or 9 5 to 6 on 6 or 8 |
| 7 Out | 4 to 1 |
| Yo 11 | 15 to 1 |
| 3 | 15 to 1 |
| 2 Snake Eyes | 30 to 1 |
| 12 Box Cars | 30 to 1 |
| Hi-Lo | 15 to 1 |
| Craps | 7 to 1 |
| C and E | 3 to 1 with craps
7 to 1 with 11 |
| Field | 1 to 1 with 3, 4, 9, 10 or 11
2 to 1 on 2 or 12 |
| Field (alternate) | 1 to 1 with 3, 4, 9, 10 or 11
2 to 1 on 2 3 to 1 on 12 |
| Horn | 27 to 4 with 2 or 12
3 to 1 with 3 or 11 |
| Whirl / World | 26 to 5 with 2 or 12
11 to 5 with 3 or 11 0 to 1 on 7 (push) |
| Hard Way 4 or 10 | 7 to 1 |
| Hard Way 6 or 8 | 9 to 1 |
| Big 6 or 8 | 1 to 1 |
| Place 4 or 10 | 9 to 5 |
| Place 5 or 9 | 7 to 5 |
| Place 6 or 8 | 7 to 6 |
| Buy 4 or 10 | 2 to 1 plus commission (5%) |
| Buy 5 or 9 | 3 to 2 plus commission (5%) |
| Buy 6 or 8 | 6 to 5 plus commission (5%) |
| Lay 4 or 10 | 1 to 2 plus commission (5%) |
| Lay 5 or 9 | 2 to 3 plus commission (5%) |
| Lay 6 or 8 | 5 to 6 plus commission (5%) |
In slots, the payout schedule of each game is displayed on the front panel of the machine or the main interface in online slots. The payout odds for the winning combinations in slots differ from game to game and could range in order from hundreds to one or thousands to 1 for the highest prize (base jackpot) for the most of the games.
Odds as house edge
Odds are sometimes meant as house edge in gambling texts or speech. The context of such a usage is usually that of a general talk about chances in casino gambling and in particular chances for profit in each game, including comparisons between games in that respect. The so popular question “Which casino game offers the best odds?” is many times answered in terms of house edge. One reason for that is that house edge is a statistical indicator associated with almost every game and as such answering the question in terms of that indicator be the simplest answer. But such usage could also be the result of a poor knowledge about what odds mean and the distinctions between the various meanings of this word.
Even though using ‘odds’ with the meaning of house edge may be a marginal or accidental phenomenon in some communities, we have to clarify also this meaning for having the complete picture of the palette of meanings of this term and correcting inadequate usages, depending on contexts.
What is house edge
House edge or house advantage is a statistical indicator in the form of a statistical average, associated with a bet or a game operated by a casino. It reflects the average gain of the house over the long run, as a percentage of all amounts wagered at that bet or game.
For understanding the mathematical nature of the house edge and how it can be computed, we must first define the expected value of a bet (EV):
Expected value of a bet
For a bet B on an event A, with stake S, the expected value (EV) or mathematical expectation of bet B is the following expression:
EV(B) = (probability of winning B) x profit if you win + (probability of loosing B) x loss if you loose,
where the loss is expressed as a negative number.
EV can also be expressed as the ratio of the stake into a percentage form:
EV(B)(%) = EV(B)/S
A Column bet with a $1 stake at European roulette has an expected value of (12/37) × $2 – (25/37) × $1 = –2.7 cents
In this expression, 12/37 is the probability of the event that a number in that column will occur, 25/37 is the probability of the contrary event, $2 is the profit in case of a win (since the Column bet pays 2 : 1), and $1 is the possible loss.
EV(%) = –2.7 cents / $1 = –2.7%
The EV of a bet is a statistical average and reflects the overall profit that the player would make if placing that bet in identical conditions (including with the same stake) indefinitely or, else said, their average profit over the long run, where the ‘long run’’s mathematical counterpart is infinity.
Hence the average in EV has to be meant in a statistical sense, not arithmetical: The average is not a mean over a definite number of plays or time interval, but it is a special weighted mean, where the weights are probabilities; we could also meant it as a limit.
EV (%) = –2.7%, as in the previous example, reads as “If placing this bet constantly, you are expected to loose on average 2.7 cents at every dollar bet”.
The fact that EV of a bet has negative values should not be interpreted as we will experience a loss every time we place that bet or over whatever number of such bets, but that the more bets, the more our overall profit will approach that negative value.
House edge as opposite to expected value
Getting back to the house edge of a bet, this is just the opposite in sign of the expected value of that bet: HE(B) = – EV(B).
Therefore, if the EV is negative, the HE should be positive, or, interpreted in real gambling terms, what you loose become the house’s gain. Having a positive HE for every bet in a game is the mathematical guarantee of the operator that that game will bring them profit over the long run whatever stakes and strategies the players will run.
We can also define the house edge of a game, as the statistical average of the house edges of all the possible bets or payouts available in that game.
In slots, the house edge of a game can be expressed as HE = 1 – RTP, where the RTP denotes the return to player as the average ratio between the players’ total amount of payouts and their total amount of wagers.
In some games, like roulette, the house edge of any bet is the same and such is the house edge of the game. In other games, like craps for instance, each bet has its own house edge.
It is important to know that, even though the house edge is an indicator associated with a game, for some games it is not constant. First, it varies with any version of the game, as rules, including payout rates. It also changes in games allowing optimal strategies of playing, like blackjack. For a player using such a strategy, the house edge can decrease from its standard value. For instance, a player using a card-counting strategy in blackjack can reduce the house edge of the game from a standard 0.5% in classical blackjack to about 0.1%.
House edge is one of the most important statistical indicators, as it stands as an objective criterion for choosing between bets and games, especially for those playing longer sessions and constantly. Obviously, choosing the games or bets with a lower house edge is in the player’s advantage as long as such style of playing is in effect.
House edge tables
In this section we provide the values of the house edges associated with a few popular casino games.
| Bet | House edge |
|---|---|
| Any bet in classical European roulette | 2.70% |
| Any bet in classical American roulette | 5.26% |
| Non even-money bet in French roulette | 2.70% |
| Even-money bet in French roulette with La Partage or En Prison | 1.35% |
| Roulette version | House edge |
|---|---|
| European roulette | 2.70% |
| American roulette | 5.26% |
| French roulette | 2.66% |
| Game | House edge |
|---|---|
| Atlantic City Blackjack | 0.36% |
| American Blackjack | 0.35% |
| Big Five Blackjack | 0.47% |
| Classic Blackjack | 0.13% |
| Double Exposure | 0.69% |
| European Blackjack | 0.42% |
| Pontoon | 0.41% |
| Premier Blackjack | 0.42% |
| Spanish Blackjack | 0.38% |
| Super Fun 21 | 0.94% |
| Vegas Downtown Blackjack | 0.39% |
| Vegas Single Deck Blackjack | 0.35% |
| Vegas Strip Blackjack | 0.35% |
| No. of decks | House edge |
|---|---|
| 2 | +0.412% |
| 4 | +0.556% |
| 5 | +0.584% |
| 6 | +0.603% |
| 8 | +0.627% |
| BET | HOUSE EDGE PER BET MADE | HOUSE EDGE PER BET RESOLVED | HOUSE EDGE PER ROLL |
|---|---|---|---|
| Pass | 1.41% | 1.41% | 0.42% |
| Don't Pass | 1.36% | 1.40% | 0.40% |
| Place 6 and 8 | 0.46% | 1.52% | 0.46% |
| Place 5 and | 1.11% | 4.00% | 1.11% |
| Place 4 and 10 | 1.67% | 6.67% | 1.67% |
| Big 6 and 8 | 2.78% | 9.09% | 2.78% |
| Don't Place 6 and 8 | 0.56% | 1.82% | 0.56% |
| Don't Place 5 and 9 | 0.69% | 2.50% | 0.69% |
| Don't Place 4 and 10 | 0.76% | 3.03% | 0.76% |
| Hard 6 and 8 (US) | 2.78% | 9.09% | 2.78% |
| Hard 6 and 8 (AU) | 1.39% | 4.55% | 1.26% |
| Hard 4 and 10 (US) | 2.78% | 11.11% | 2.78% |
| Hard 4 and 10 (AU) | 1.39% | 5.56% | 1.39% |
| BET | HOUSE EDGE |
|---|---|
| Any craps (2, 3, or 12) | 11.11% |
| Any craps (2, 3, or 12) | 5.56% |
| Any seven (US) | 16.67% |
| Any seven (AU) | 8.33% |
The casino games offering the best odds
Now that you know exactly what the possible meanings of ‘odds’ are in the gambling language, you saw that they are distinct – thought related to each other – and explored their mathematical descriptions, we can turn back to the popular question “Which casino game offers the best odds?” to try to answer it objectively.
Since we have three possible meanings for ‘odds’, the answer will not be straightforward, because it depends on the chosen meaning and hence we actually have three answers to that question.
If we refer to odds as probability of winning a bet, the casino game offering the best odds of winning for its bets is roulette, with probabilities of winning of over 90% for specific types of large-coverage bets. These are combined bets covering many numbers on the roulette table, chosen by certain criteria. However, there is a cost of such a high probability of winning, namely a low profit rate and a big possible loss. Probability of winning is not the only factor is the equation of gambling profit, just because it relates to the payout rates and the house edge of the game. Then, we have baccarat, with about 46% probability of winning the Banker bet, and blackjack, with an apriori winning probability of about 42%.
In the house edge meaning, the casino game offering players the highest odds (meaning the lowest house edge) is blackjack, with a house edge ranging from 0.1% to 0.94% over all blackjack versions, but under 0.5% for an optimal play. It is followed by baccarat, having a house edge of about 1% for the Banker bet, roulette (1.35% as the lowest one in French roulette), and craps (1.36% as the lowest one).
As for the payout rate meaning, the figures range from x1 to x10,000 or more in slots, but making a top 3 would actually be irrelevant, as the higher a payout rate the lower the probability of having that payout, so all that count as criteria of assessing a bet is its EV (or HE) and the probability of winning it.
Probability, expected value, house edge and payout rate are related to each other mathematically and such relationships are so balanced for the house to conserve its statistical advantage. Thus, any apparent advantage in terms of one factor is counterbalanced by another factor.
Roulette and blackjack are the most popular casino games and this popularity seems to relate to these rankings in terms of odds (as probability and as house edge). However, the criteria by which players can choose the games they intend to play are not based only on such ranking, but also on other subjective factors, as their particular style of playing, goals, strategies, and bankroll.
Conclusions
The term ‘odds’ have three popular meanings in the gambling jargon, but not all people is aware of this distinction and the undifferentiated usage can cause semantic conflicts, misunderstandings and misinterpretations in communication, and influence strategic decisions.
For making a sharp distinction in terms, which to prevent such issues, the optimal usage is with ‘odds’ meaning probability, ‘payout odds’ meaning payout rates, and ‘house edge’ for the meaning of the statistical notion of house advantage. The exception is in sports betting, where ‘odds’ is unanimously accepted as payout rate. Yet, in this market we should not equate odds (meant as payout rate) with probability.
Having a good grasp of the definitions of these terms, including their mathematical description, is a step ahead not only for avoiding the communication issues, but for playing informed and using them adequately as factors influencing your playing decisions.
