When discussing a casino game in expert’s terms or comparing casino games, the ‘house edge’ is the most used technical term. Why is house edge so important when analyzing a casino game?
The house edge is a statistical indicator characterizing every bet that can be placed at a game or the game itself.
From the casinos, operators, and game developers’ perspective, the house edge represents the mathematical guarantee that a game is worth functioning because it ensures a profit for the house over the long run.
From the players’ perspective, the house edge is essential in its responsible-gambling and strategic aspects: It submits to the more general principle that being informed about the mathematical aspects of the game and gambling aligns with a healthy attitude about gambling and prevents the development of problematic gambling behavior.
In addition, most of the objective or optimal strategies in gambling employ house edge as a statistical indicator; even the simplest strategy - that of choosing a game or another or a bet or another within the same game has as its main criterion the value of the house edge of that game or bet.
In this guide, we will explore:
Prof. Catalin Barboianu, PhD will explain house edge and provide numerical values for the most popular casino games. Finally, he will show how inadequate interpretation of this notion may turn into misconceptions considered risk factors in problem gambling.
Let's dive right in!
In statistics, the central notion is that of probability. We cannot define house edge or any other statistical indicator without the notion of probability, and we cannot understand them adequately without knowing what probability is and its role in gambling.
Classical Definition of Probability
The general definition of probability in mathematics is that of a particular measure obeying some axioms (non-negativity, normalization, and countable additivity). It is a function defined on a countable field of events.
Due to the axioms of its definition, probability takes values in the interval [0, 1].
While not providing here the formal definition of a field of events, think of such a field as a set of sets of data that can be operated between them with two operations that have the same properties relative to each other, like addition and multiplication (in arithmetic), operators ‘and’ and ‘or’ (in propositional logic), or union and intersection (in set theory) have.
In such a structure, the probability of a data set will be measured relative to the whole set of data, which is called the sample space.
Each random experiment has a sample space and a field of events attached.
We may say that the probability of an event measures the shreds of evidence for the possibility of that event to happen, provided that all the evidence belongs to a mathematical structure as a field of events.
This is why we can say what the probability is for a red number to occur at a roulette spin.
Still, we cannot say the probability of the Sun not rising tomorrow just because this latter event does not belong to a suitable mathematical structure (so its probability does not make sense mathematically).
For example
In the experiment of rolling a die, the sample space is the set of all possible outcomes of this action, namely Ω = {1, 2, 3, 4, 5, 6}, numbering six elements. Any event can be defined as a subset of Ω (for instance, {1, 2}, {1, 2, 3}, and so on). These events form the field of events of the experiment.
The probability of any event will be the ratio between the number of elements of that event and the total number of elements of the sample space (six). For instance, the probability of the event {2, 4, 6} (even number) is P({2, 4, 6}) = 3/6 = 1/2 = 0.5 = 50%.
In gambling, the sample space of any experiment (game) is finite, and the field of events can be represented as finite sets of equally-possible outcomes (called elementary events).
For such fields, probability can be defined as follows:
‘Probability of an event A is the ratio between the number of situations favorable for A to occur and the total number of equally possible situations.’ It is called classical (or Laplacian) probability.
Such a definition is suitable for and applicable to experiments with finite sample spaces; it cannot measure events of high complexity, but it suffices to be used in any statistical computations in gambling.
Examples
- For computing the probability for a pair of dice to roll a sum of 7, we count first the number of situations/evidences/elementary events favorable for that event to occur – these are the possible pairs of outcomes (one for each die) (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1), numbering 6 – and divide that number by the total number of possibilities for such pairs, which is 6 x 6 = 36; hence the probability is 6/36 = 1/6 = 16.66%.
- What is the probability of a Hold’em player being dealt a pair of kings as their hole cards? Since there are four kings, the situations favorable for the event are the combinations of four taken two, numbering six, namely the 2-card combinations K♠K♣, K♠K♥, K♠K♦, K♥K♦, K♥K♣, K♦K♣. The total number of equal possibilities is the number of combinations of 52 cards taken each 2: C(52,2) = 1,326. The probability is then 6/1,326 = 0.00452 = 0.45%. The exact likelihood stands for any other pair, whether of aces, tens, nines, or whatever.
Keep in mind
Probability is defined in mathematics as a measure function whose definition domain is a mathematical structure called field of events.
For the particular case of gambling experiments, the definition of the probability of an event reverts to a simple ratio – the ratio between the number of situations favorable for that event to occur and the total number of equally possible situations. For computing the probability of a gambling event, we just have to count those situations.
Probability is mathematically defined as a measure of an event, and we saw that classical probability measures finite sets of elementary events.
However, the probability is also associated with the experiment that generates outcomes, which may or may not belong to the estimated event (since the sample space is a characteristic of that experiment).
Therefore, even if we talk about an isolated event that is measured, its probability embeds all the possible (past and future) instances of that experiment (called trials) and has an infinite inner feature.
This seemingly philosophical argument is made concrete by a theorem, which in the simplest possible terms, says that “For any sequence of independent trials of a random experiment, the sequence of relative frequencies of an event A is convergent toward the probability of A.”
It is called Borel’s theorem or The Law of Large Numbers.
For example
Performing successive independent rolls of a die and recording the frequency and relative frequency of a number, say, 6, we get a specific sequence of relative frequencies: Say that we recorded: 0 (occurrences) out of 1 (rolls), 0 out of 2, 1 out of 3, 1 out of 4, 1 out of 5, and so on, 11 out of 70, etc.
The Law of Large Numbers says that the recorded sequence of fractions 0/1, 0/2, 1/3, 1/4, 1/5, …, 11/70 approaches the probability of the event ‘occurrence of 6’, which is 1/6, when the number of rolls increases.
In non-mathematical terms, The Law of Large Numbers says that over the long run, outcomes occur in an experiment the way that theory says they do.
However, in mathematical terms, “long run” means an infinite number of trials.
In experimental terms, for a sufficiently large number of trials n, the relative frequency (sometimes called experimental probability) approaches (or comes acceptably close to) the theoretical probability.
The LLN is the mathematical result that makes a connection between the real-world context of the random experiments and the abstract probability theory and grants probability the status of a limit.
The Law of Large Numbers helps us in interpreting probability not as an absolute measure for certainty regarding an isolated event, but rather as a kind of infinite average specific for the experiment that generates that event.
Be aware
Probability does not indicate whether an event will occur or not at a certain trial, nor over a definite interval of trials, but it reflects a mathematical limit over all possible trials – the limit of the sequence of the relative frequencies of the occurrence of that event, provided that the trials are random and independent to each other.
Keep in mind
Probability is both a measure and a limit, provided that the events are organized in a suitable mathematical structure.
Expected Value
Now that we know basically what probability is, we can move to another important statistical notion in gambling, that of expected value of a bet.
Every time we play a game of chance, we place a bet. This happens not only when we place chips on a casino table, but also when we insert coins in a slot machine, and even when we buy a lottery ticket or a scratch card.
We pay a price for every bet, like for any commercial product, called stake or wager.
In case we win the bet, we get in return our stake multiplied by the payout odds of that bet, according to the game’s rule. As such, a bet is characterized not only by the event on whose occurrence we bet, but also by its stake and payout odds.
Thus, a complete mathematical description of a bet will employ probability of winning, the stake of the bet, and its payout odds.
Given a bet B on an event A, with stake S, the expected value (EV) or mathematical expectation of bet B is the following sum of products:
EV(B) = (probability of winning B) x profit if you win + (probability of losing B) x loss if you lose, where the loss is expressed as a negative number.
For example
A Dozen bet with a $1 stake at European roulette has an expectation 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 dozen will occur, 25/37 is the probability of the contrary event, $2 is the profit in case of a win (since the Dozen bet pays 2 : 1), and $1 is the possible loss, that is, the stake.
It reads as “If placing this bet constantly, you are expected to lose on average 2.7 cents at every dollar bet”.
EV can also be expressed as the ratio of the stake. In our example above, EV(%) is –2.7 cents / $1 = –2.7%
Usually games offer a payout schedule such that each payout applies to a type of outcome or to an outcome (the more likely the outcome, the lower the payout). Therefore any bet is actually a combined bet, where a certain payout out of several is applied when the bet is won.
For example, in slots a payout is applied for every winning combination noted in the paytable.
In general, if we consider:
the winning outcomes (combinations of symbols in the case of slots),
are respectively the probabilities for these combinations to occur, are respectively the payout multipliers, and c is the credit wagered, then the Expected Value (EV) is:
or, as a ratio of the initial credit:
For being able to compute numerically the EV by the above formula, we have to know the probabilities pi.
These probabilities are known or can be picked from expert sources for the classical casino games, with the exception of slots, whose parametric configuration is usually kept secret by the producers and this lack of information prevents one to do the probability computations.
In statistics terms, EV is a mean of a random variable, but this mean should not be understood as an arithmetical mean or average, but as a statistical average, that is, a weighted mean where the weights are probabilities.
This is why any interpretation of the EV should not be as an expected quantity, but as an infinite average or limit (remember that probability is a limit).
If we say that in roulette you are expected to lose 2.7 cents at every dollar bet, this should not be understood literally, nor in a limited context – that is, lose that amount money in a game or two or more, but over the long run.
Keep in mind
Expected value is a statistical indicator characterizing any bet, expressed as a statistical average. It is computed by doing each product between the probability of every possible event related to a bet and its possible profit or loss and adding them up.
EV should not be interpreted as applying over a short session of games, but only over the long run, that is, an indefinite number of games.
Now that we know what expected value of a bet is, we can move to the notion of house edge of a bet, which is defined as the opposite (in sign minus or plus) of the EV, expressed as a ratio of the initial stake:
HE = –EV(%)
So an EV of –2.7% for a roulette bet turns into a 2.7% house edge of that bet.
Since EV of a bet is the average profit that the player is expected to make over the long run with that bet, by definition HE is then the overall profit that the house is expected to make from the players’ bets over the long run, relative to their wagers.
Like for the EV, we have to be careful when interpreting HE in terms of actual gambling.
“Overall” profit does not mean the profit made by the house upon a player’s bets, nor over certain time from all players’ bets, but from all-time all players’ bets, over the long run. Like the EV, the HE is also a statistical average and involves probability.
This is why any interpretation of the HE in terms of actual play should not concern individual plays or limited playing sessions.
Keep in mind
House edge of a bet is a statistical indicator expressed as the opposite of the EV as percentage. For getting the formula of the HE, we have to change every sign (minus/plus) in the formula of the EV. The HE reflects the overall profit that the house makes over the long run from all players’ bets.
Using the definition of the HE in the explicit formula (*) of the EV for a multiple payout schedule, we get:
With formula (**) we can compute the HE of any bet, provided that we know the probabilities of all the events associated with the payout odds.
While the expected value is a notion defined as applying to a bet, the house edge was defined still applying to a bet, but it can also apply to a game.
We can talk about the HE of a certain game if there is only one type of bet that that game consists of, even though the game may have several payouts for the various outcomes.
In other words, if that game is a bet itself.
For instance, in blackjack the bet is that you will beat the dealer and the possible outcomes are: you win, the dealer wins, or a tie occurs; each of these three outcomes has its own payout odds, so the house edge can be computed by the classical formula (**) of the HE of a bet with multiple possible outcomes.
Instead, in baccarat you can bet on the Banker, on the Player, or on a Tie, each such bet having its own payout.
Each of these three bets thus has its own EV and HE. Using these individual house edges we can calculate an averaged HE by doing the product between the HE of each bet and its winning probability and adding them together, for obtaining a kind of house edge of the game of baccarat.
Or, by doing the arithmetical mean of the individual HEs.
Or, by using computer simulations of random betting choices. However, any of these would be just a cumulative statistical indicator; from the players’ perspective, the HE of each individual bet is more relevant as mathematical information.
Therefore, baccarat does not have a HE per the mathematical definition of this notion, but we can assign it a statistical indicator as an averaged HE.
In slots we have a multiple payout schedule and any credit insertion is a bet, whose HE would be computed by the same formula if the probabilities were known. Hence slots do have a HE.
Roulette is the exception to our rule of assigning a HE to a game.
Although the game of roulette allows different bets to be placed, with different payouts, it has a special status among the casino games with respect to house edge: Whatever bet we place (simple or combined), it will have the same EV of –2.7% for the European roulette and –5.26% for the American roulette.
The EV is constant over the bets because the payout schedule of the game of roulette is so conceived to provide this constancy.
Therefore there is a house edge of roulette – the HE of the European roulette is 2.7% and for American roulette it is 5.26%.
The classical casino games that have a HE per its mathematical definition are: roulette, slots, and blackjack. The games that don’t have one (but only a statistically averaged HE) are: baccarat and craps.
Keep in mind
The house edge is theoretically defined for a bet and not for a game. Still, there are casino games having a mathematical house edge (roulette, slots, and blackjack). We can talk about a house edge of baccarat and craps, but it is an averaged house edge.
The house edge is sometimes expressed in terms of what is called the payback percentage or return to player (RTP).
The RTP is the percentage of the total amounts wagered that a game or machine pays back to its players as prizes over the long run. The difference 1 – RTP represents the house’s profit on that machine.
Therefore, we have the following relation between HE and RTP: RTP = 1 – HE or HE = 1 – RTP.
The RTP is a term used mostly in slots, as a statistical indicator that is usually made public by the games’ producers.
RTP in slots range usually between 80% – 98%, which reverts to a range of the HE between 2% – 10%.
However, the RTP applies to any casino game and any game has its own RTP.
Although RTP and HE are in a mathematical sense equivalent as statistical indicators (knowing one directly determines the other and once one increases the other decreases in the same amount), it was ascertained by researchers that players come to perceive differently the two values.
Some players have a “positive” perception about the RTP, which is due to its values which usually ranges higher, while having a negative perception about the house edge.
They take the former to be a kind of gain and the latter to be a loss (what actually is), however the RTP also reflects the same loss expressed by the HE.
A RTP of 97% means a house edge of 3%, higher than the house edge of European roulette and far higher than the house edge of blackjack, for example.
Then, we should take the RTP as a mode to express the HE, that is, what the house takes from our pocket over the long run, rather than any actual return.
Be aware
The RTP is not any gain or bonus for the player. It is a statistical indicator reflecting what the house edge is as opposite to what the game returns to players as prizes. The lower the RTP, the higher the HE. It does not reflect any average prize amount to expect over a definite session of plays, but just a way to express HE as the difference 1 – RTP.
With formula (**) we can compute the HE of any bet or of any game that has HE, provided that we know the parameters involved, that is, the payout schedule and the probabilities of the winning outcomes.
Yet the HE of a game is not always constant. Why is that?
For the most popular casino games, versions of these games evolved and became popular.
Roulette has its own versions, blackjack has, baccarat has, and craps has.
A version of a game may differ from the original one in three aspects:
- The payout odds of one or more winning events are modified,
- One or more winning events are introduced in or removed from the payout scheme,
- The rules of the game are slightly changed (whether we talk about permissions, restrictions, or privileges, number of decks used or numbers on the betting table, etc.).
Let’s take them individually: If the payout odds are modified, then one or more parameters in the HE formula change and thus HE changes. If any winning event is changed (that is, reformulated), then its probability changes and the probabilities of the other events change, so one or more parameters change in the HE formula.
If new winning events are added or events are removed, the entire distribution of the probabilities changes, new payout odds may appear and therefore the parameters in the formula of the HE changes.
This is why we have different house edges for European and American roulette, classical blackjack and Spanish blackjack, or 6-deck baccarat and 8-deck baccarat.
But the game version is not the only element causing the HE to vary.
The HE also changes with player’s strategy. This might seem a bit counterintuitive, as one may see the HE as a parameter characterizing a game itself, not in any relation with the player. Moreover, looking at formula (**), we see just parameters associated with the game’s rules.
However, remember that HE was defined as the opposite of the EV, which characterizes any bet as placed by the player.
When talking about optimal strategies in gambling, their mathematical aim is to maximize either the probability of winning or the expected value of a bet, by a certain kind of play.
Either such change results in changing the HE (actually reducing it, in the favor of the player).
However, not every game of chance allows this effect of an optimal strategy on the HE or allows such a strategy itself.
Among the classical casino games, blackjack and baccarat allow optimal strategies.
It is difficult to see this effect while looking at formula (**). We could rather understand it by following the course of a specific game played with such a strategy.
For example
In general, optimal strategies based on card counting in blackjack or baccarat increase the initial probability of winning a round when playing by them and this increase reflects in the increase of the EV.
But is not only one probability that may change in the formula of the HE with an optimal strategy.
Optimal strategies (even basic) assume an algorithmic play depending on situation or configuration of the game at a certain moment.
The way the game develops with a certain action may change the number of events taken into account in the formula of the HE.
For instance, in blackjack optimal play certain pairs are required to be split against dealer’s certain totals.
With this move, you create a new tree of possibilities that count as events that change the composition of the formula of the HE.
In our odds and house edge guides you may see concretely the various values of the HE for different versions of the most popular casino games and also how the HE changes with the optimal play in blackjack and baccarat.
We will make a brief application of what we have learned in these sections for some popular casino games and provide some numerical values for the HE in these games.
Keep in mind
The house edge is not a constant characterizing a casino game. It changes with the game versions (rules) and it also changes for the players using optimal strategies (for the games that allow them), by being reduced.
Let’s take the simplest bet in roulette, the straight-up bet. In European roulette, its winning probability is 1/37 and payout odds 35 : 1. Its house edge according to formula (**) will be then HE = 1 – (1/37)x(1 + 35) = 1/37 = 0.0270 = 2.70%.
Take another bet, say, the Dozen bet.
Its winning probability is 12/37 and payout odds 2 : 1. Then, its HE is HE = 1 – (12/37)x(1 + 2) = 1/37 = 0.0270 = 2.70%.
Will any bet we take, have the same house edge? In roulette, yes.
Why is that?
Take a look at the next table, and observe the following: on any row, if you add 1 to the payout multiplier then multiply the result with the fraction in the probability column (as indicates the formula of HE), you always get 36/37.
This happened because the payout odds are such chosen to have this property. Then 1 – (36/37) = 1/37 is the HE.
| Simple bet | Winning probability | Payout odds |
| Straight Up | 1/37 | 35 to 1 |
| Split | 2/37 | 17 to 1 |
| Street | 3/37 | 11 to 1 |
| Trio | 3/37 | 11 to 1 |
| Corner | 4/37 | 8 to 1 |
| Four numbers | 4/37 | 8 to 1 |
| Line | 6/37 | 5 to 1 |
| Column | 12/37 | 2 to 1 |
| Dozen | 12/37 | 2 to 1 |
| Red/Black | 18/37 | 1 to 1 |
| Even/Odd | 18/37 | 1 to 1 |
| Low/High | 18/37 | 1 to 1 |
Therefore, any bet in European Roulette has the HE = 2.70%. If we apply formula (**) for any bet in the American Roulette, we get the HE = 5.26%, almost twice than in the European Roulette.
This is why it is better to play in the former (the winning probabilities are higher and the payout odds are the same).
In fact, the HE reflects the extra numbers (0 for the European and 0 and 00 for the American roulette) that fall outside the bets except the straight-up bet.
One may fairly ask now: Ok, the HE of any simple bet in roulette is constant (either 2.70% or 5.26%), but how about the combined bets? Is there any increase or decrease in HE for such bets?
Well, although the probability of winning increases with the coverage of a bet (the more numbers we cover, the higher the probability of winning), we have the same HE for such bet as for the simple bets in roulette.
In roulette theory, this can be proved by using some properties of the equivalence of the bets.
This result is not weird at all: even though the winning probability increases, the possible loss also increases as we choose more numbers to cover with our bet.
Check our Roulette Odds and House Edge Guide to learn more about the probabilities associated with this game and other mathematical facts of roulette.
Keep in mind
The house edge of the game of roulette is 2.70% for the European roulette and 5.26% for the American roulette and the same numbers stand as the house edge of any roulette bet, respectively, whatever complex this bet may be.
The apriori overall probabilities characterizing the game of blackjack are:
You have a probability of 42.43% to win, 8.48% to go into a tie, and 49.09% to lose. These odds do not take into account any other information, such as any strategy used or cards out of play.
Among those 42.43% winning odds, about 4.50% are the odds of winning with a blackjack.
Doing the math for the house edge (incorporating in the HE formula these probabilities and their associated payout odds, that is, 1 : 1 for a win with no blackjack and 3 : 2 for a win with blackjack in the classical variant of the game), we get a standard HE of about 0.5%.
But as we said house edge varies with the game versions and strategy and the game of blackjack has very many versions.
The number of decks used, the payout odds for the blackjack win (3/2, 6/5, or 2/1), the rules variation (in regard to splitting, doubling down, dealer hitting at soft 17, etc.) result is many versions of blackjack, each one with its own house edge, which also changes depending on the strategy that a player uses.
The HE of blackjack ranges between 0.1% and 2.7%, depending on all the factors mentioned above.
For more details about the odds in blackjack and tables of numerical values for the HE of blackjack for the various versions of the game, check out our Blackjack Odds and House Edge Guide.
They say that blackjack offers the best odds among the casino games and this is true in terms of house edge.
You won’t find the standard HE of 0.5% in any other casino game, not mentioning the optimal play, which can reduce it down to 0.1%.
The statement is not true in terms of winning probabilities, where roulette is in the top of the casino games with winning probabilities of over 92% for some combined bets.
Keep in mind
The house edge of blackjack is one of the lowest house edges in casino games. From a standard 0.5%, it can go down to 0.1% if the game is played by an optimal strategy.
Craps seems to be the most complicated casino game, and this is also reflected in the payout schedule of craps.
We have so many payout odds, associated with each type of bet.
Computing house edge in craps is also a sensible problem. That’s because there are some bets (the place bet, for example) for which many rolls may be required to resolve it.
During these rolls, the player can take down the bet at any time.
In these conditions, there are three options to define the HE for a craps bet, namely per bet made, per bet resolved, and per roll.
Here are the HEs of some bets in craps which may take multiple rolls to resolve, in each of the three options of defining HE:
| 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 9 | 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% |
Here are the HEs of some bets in craps that are resolved in a single roll:
| 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% |
Even from these partial results one can see there is a wide spread for HE, from close to 1% to over 16%, depending on the bet.
This spread is also an argument against taking an averaged HE as relevant HE for the game of craps.
Like in other games, the HEs and odds of the craps bets vary with the versions of this game.
Keep in mind
In craps there is a wide spread of the values of the house edge of the bets. The house edge of the bets that may take multiple rolls to resolve is not absolute, as there are three ways of defining it.
In baccarat there are three main bets: on the Banker, on the Player, and on a Tie. There are also side bets associated with every version of the game; in side bets you bet on the occurrence of a particular outcome of the game.
In baccarat there is the HE of each type of bet that is more relevant as statistical information than any averaged HE of the game. A particular reason is that the main three bets are not uniformly distributed in the actual play – players prefer to avoid the Tie bet for instance, as offering the lowest odds of winning.
In baccarat we can compute the HE of each type of bet by using formula (**).
As in other games, the HE differs among the versions of baccarat, because both the probabilities and payout odds change with the versions. For instance, the number of decks used influences the probabilities, while the commission on the Banker bet influences the payout odds for this bet. As an example, let’s take the standard 8-deck game and compute the HE of the Player bet:
The parameters of the Player bet are noted in the next table.
| Event | Payoff | Probability |
| Banker wins | -1 | 0.458597 |
| Player wins | 1 | 0.446247 |
| Tie | 0 | 0.095156 |
Example of calculation
Applying formula (**) for these three events with their payout odds and probabilities, we get:
EV = 1 – 0.458597 x (–1 + 1) – 0.446247 x (1 + 1) – 0.095156 x (0 + 1) = 0.01235 or 1.23% for the Player bet.
Among the three base bets, the Banker bet offers the lowest house edge, and the Tie bet the highest house edge, between 14.35% and 15.76%, depending on the number of decks used.
With a house edge close to 1% for the Banker bet, baccarat is one of the casino games offering the lowest HE, being outranked only by blackjack played with strategy, with its house edge under 1%. The Player bet also has a low HE, under 1.3%.
Baccarat side bets have a far higher HE, ranging between 10% and 30%, depending on type of bet and game version.
Check our Baccarat Odds and House Edge guide for finding all the figures associated with the bets and versions of baccarat.
Keep in mind
Baccarat Banker and Player bets offer some of the lowest house edges among the casino games. Side bets have a far higher house edge.
I have left slots intentionally at the final, as the game of slots has a special status with respect to its available statistical information: It is the only casino game whose mathematical configuration is kept secret by its producers.
The reason for this secrecy is not justified in any way and has raised concerns with respect to the ethical aspects of slots gambling.
The only available on-screen information was for a long time he payout schedule of the machine (the variables in our formula of HE). However this is not enough for a computation for EV or HE, as long as the probabilities of winning are missing.
The parametric configuration of a slot machine – called PAR (Probability Accounting Report) sheet – necessary for the probability computations was retrieved for some games by researchers and experts through legal intervention.
Other slot machines were “deconstructed” by statistical methods using long-time observation and recording.
Over the last decade, gambling regulations in many jurisdictions over the world imposed producers to display on their machine one statistical indicator, namely the payback percentage (or RTP).
This information is of course not enough for retrieving the winning probabilities associated with the prizes, but suffice for knowing the game’s house edge.
As we saw in the theoretical section, the relationship between RTP and HE is RTP = 1 – HE, so HE = 1 – RTP. T
his relationships tells us that the higher the RTP, the lower the HE. Combining this relation with formula (**), we get that
where the pi are the (unknown) winning probabilities associated with the prizes and oi are the payout odds of these prizes respectively.
Many slots players choose a slots game or another by RTP.
Using RTP as the main criterion for choosing a slots game is not wrong, since choosing the games with higher RTP (and thus lower house edge) is favorable for the player. Yet other statistical indicators (if available) may be used as well for informed decisions.
Keep in mind
On many slot machines, their RTP is displayed. The RTP reflects the house edge of the machine even if it is the average payback percentage to players.
We saw when we defined house edge mathematically that it is a statistical indicator of a bet or game in the form of a statistical average, as expected value is.
This means that the payouts show in the mathematical expression of the HE with probabilities as coefficients (multipliers).
By this reason, the HE is not an arithmetical mean, but a statistical kind of mean.
While an arithmetical mean expresses an average of a finite number of values, a statistical average expresses an infinite average despite holding a finite number of terms in its expression.
This might seem counterintuitive, but to understand the nature of the house edge, we have to understand the nature of probability.
Probability is a limit and a limit is a kind of infinite average (of all the terms of a convergent sequence). Any term showing in the explicit formula of the HE expresses such an infinite average.
The payout odds apply in the real game any time the associated winning event occurs, and this happens with probability:
If you ask how many times the game will payout you with those odds over a definite interval of play – say, session, day, week, month, or a certain number of plays – this question has no answer, as the outcomes are unpredictable.
You can only answer this question precisely after you played the game over that interval, but not before.
If you ask before playing the game for an average of these payouts over the same interval, the only answer available is:
That's because probability is the only measure for the uncertain possibility.
But this average concerns not only your play, but everybody’s play at that game, and not only that interval, but any interval of play. It’s a feature of the game itself and not of your play.
This is why that statistical average (either if we talk about the HE or EV) embeds a potential infinity, namely all the plays for that bet or at that game, including those in the past and those possible in the future.
As such, the HE should be interpreted in this cumulative sense and not in a limited sense.
We should interpret the HE as applying to an infinite number of plays, although any gambling experience is finite.
No matter how much we would like to make the statistical average a concrete thing, applicable to the real game and characterizing it, it remains abstract, just because probability is something abstract.
The only compromise we can make to use the abstract and the concrete for a statistical measurement is the wording “over the long run” as an attribute of that measurement.
The longer the interval of play and the more number of players placing a bet, the concrete gain associated with that bet comes to approach it’s abstract value of the EV.
The same holds for the HE.
It is the same mechanism that makes the relative frequencies to approach probability, as stated by the Law of Large Numbers.
The same interpretation should be adopted from the casinos/operators/producers’ perspective.
The HE should be seen as an overall statistical average profit characterizing a bet or a game, regardless of how many gamblers have played it, for how long, or the strategies they used.
A positive HE is actually the mathematical guarantee for these businesses that the game they offer to players won’t make them going bankrupt and as such they are worth functioning.
This means that even if one or more players made a big hit in one day and the house had a loss in that day, the house won’t be concerned with that, knowing that their game has a positive HE, which means an average positive profit over the long run.
The more crowded is the casino and the longer is in business, the house’s profit from any bet or game comes closer to the value of its HE.
Keep in mind
The house edge is an abstract mathematical notion, like probability is. We can interpret it in the real world of gambling, but only in a cumulative statistical sense, as applying “over the long run”.
Misinterpretations of the House Edge and Implications
People that are not math inclined or don’t have a good grasp of the mathematical definition of the HE may be easily subject to misinterpretation of the HE.
In general, probability is hard to digest even for math-educated people; and it is not so much about the formal mathematics behind probability, as about the relationship of this concept with the real world of uncertainty.
A poor understanding of the concept of probability implies a poor understanding of the statistical notions related to it, including expected value and house edge.
Research in problem gambling has dealt with the issues of misunderstanding house edge. Empirical studies have detected various modes of misinterpreting house edge among gamblers.
The most spread misinterpretation is to take house edge to be an indicator applicable to definite intervals of plays or to one player.
A player may fallaciously believe that a game with a house edge of, say, 2% will retain 2% as the house’s profit from their wagers during a whatever long session of play, and consequently will return them 98% of their wagers as prizes over that session.
This misinterpretation implies a misinterpretation of the RTP as well, which comes to be perceived as a gain, although it just reflects the house edge, that is, a loss. This misinterpretation of the RTP is common among the slot players.
Empirical studies have shown that slot players come to misinterpret also the RTP messages displayed on the slot machines.
EV, HE, or RTP are statistical averages that cannot be applied to definite intervals of plays. They just characterize a bet or a game statistically and their values do not mean or entail any prediction about the real gain or loss during a definite playing interval.
The misinterpretation above is widespread and may affect even those players who are informed about the definition of the HE and have a basic mathematical understanding of it.
But some misinterpretations reflect a total lack of knowledge of the definition of the HE.
For instance
There are players equating house edge with probability: some may take the house edge to be an overall probability of losing – that is, a 10% house edge means for them that they will lose with a 10% probability and win with a 90% probability, which is entirely wrong since the HE consists of both probability and payout and the percentage that the HE expresses actually applies to stakes.
Other came to interpret it as a combined probability-incidence measure, concretely that 9 out of 10 gamblers playing that game or bet will win.
Or that the game will payout a prize 9 times out of 10. All such detected misinterpretations may seem weird and rare for a person knowing the basic statistics behind casino games; but studies has shown that the phenomenon of misinterpreting house edge is not marginal and should be given utmost attention.
There is also a strategic-type of misunderstanding of the house edge.
Among those players knowing that playing by optimal strategies one can lower the house edge of a game, there are players that take this decrease of the HE to be equivalent to a guaranteed win.
They believe that reducing the HE in blackjack down to 0.1% by using a high-low counting strategy it’s a success in itself, which must reflect in the monetary gain.
Again, these players should be aware that the house edge is not probability and any positive HE means an overall loss over the long run, even if this might not be the concrete case for lucky plays in some sessions. In gambling slang, the house edge is many times referred as ‘odds’, which is not entirely correct, although it is acceptable.
Odds are more associated with probability, but house edge is a statistical indicator concerning profit and loss.
Another type of strategic misunderstanding of the house edge may occur when it is used as the only criterion of choosing a game or another.
Like in the case of reducing the HE with optimal strategies, choosing a game with a lower house edge does not necessarily entail higher chances of winning.
If this latter aim is essential for a player, then they should check on the probability of the winning events as the main criterion for choosing between several games.
This latter criterion is not applicable to most of the games of slots, where only the information about RTP is usually available.
Be aware
Both the formal mathematical definition of house edge (or RTP) and its application in the real world of gambling may be easily misinterpreted by persons who not have a good grasp of the probability-related concepts.
For preventing misinterpretation, you should stick with the mathematical definition of the HE and also bear in mind that the HE and EV are abstract concepts, which do not apply literally or limitedly in the real world of gambling.
Problem-gambling research has indentified these misinterpretations of the house edge as risk factors for the development of a problematic gambling behavior.
They submit to the more general case of math-related gambling cognitive distortions, in the form of misconceptions, fallacies, and irrational beliefs caused by poor understanding of the math concepts related to gambling and their application in the real world of gambling.
Indeed, having an erroneous belief about a statistical indicator like house edge – for example, taking it as a prediction for outcomes rather than a parameter characterizing the game as a whole – may result in an increase of the gambling activity and irrational choices causing money loss.
Correction of such cognitive distortions is not an easy task for the experts, as it mostly depends on the cognitive profile of the subject and their education.
Besides, concepts of probability theory and statistics are hard to understand even by scientists, especially when applied to the real world.
There is a philosophy of probability in which sensible issues are in ongoing debate since the birth of this theory, and neither mathematicians nor philosophers came to a decisive answer regarding the true nature of probability.
All the more so these aspects are difficult to understand for the average player.
Being informed is not enough in this matter.
Having a mathematical sense or inclination may not be sufficient either, researchers said.
However, getting informed about the mathematical facts of gambling, perhaps with the help of an expert, is a prerequisite of any endeavor of self-correcting gambling-related cognitive distortions.
Responsible gambling also means getting informed about the mathematical aspects of gambling, just because inadequate understanding may be concretely harmful when translated into gambling decisions.
Reading this guide is a first step in having an adequate understanding of the house edge and of other gambling-math related concepts.
Make sure you stay up-to-date by reading other Academy Online Gambling guides and getting in line with all the information before you commit any real money.
Conclusion
House edge is a statistical indicator of a bet defined as the opposite (in sign) of the EV. It expresses the overall profit of the house over the long run, as a statistical average.
HE can also be defined for a game, if that game can be considered as a single bet, even with multiple payouts depending on the outcome.
In that case, the HE is a parameter characterizing the game as a whole.
The HE varies between the casino games and also within the same game with the versions of the games and the strategies used by players.
The HE often stands as a criterion of choosing a game or another, but other statistical criteria are available as well for the choice.
The house edge is subject to many misinterpretations, in regard to both its mathematical definition and how it applies in the real world of gambling.
Problem gambling deals with such misinterpretations and other math-related cognitive distortions, which can be corrected only if the subject gets informed about the mathematical facts of gambling from expert sources and comes to an adequate understanding of the concepts involved.
References
- Bărboianu, C. (2022). Understanding Your Game: A Mathematician’s Advice for Rational and Safe Gambling. Târgu Jiu: PhilScience Press.
- Bărboianu, C. (2014). Is the secrecy of the parametric configuration of slot machines rationally justified? The exposure of the mathematical facts of games of chance as an ethical obligation. Journal of Gambling Issues, Vol. 29, 1-23.
- Beresford, K., & Blaszczynski, A. (2020). Return-to-player percentage in gaming machines: Impact of informative materials on player understanding. Journal of Gambling Studies, 36(1), 51-67.
- Leonard, C. A., & Williams, R. J. (2016). The relationship between gambling fallacies and problem gambling. Psychology of Addictive Behaviors, 30(6), 694.
- Newall WS, P., Walasek, L., Ludvig, E. A., & Rockloff, M. J. (2020). House-edge information yields lower subjective chances of winning than equivalent return-to-player percentages: New evidence from support forum participants. Journal of Gambling Issues, Vol. 45, 166-172.
- Newall, P. W., Byrne, C. A., Russell, A. M., & Rockloff, M. J. (2022). House-edge information and a volatility warning lead to reduced gambling expenditure: Potential improvements to return-to-player percentages. Addictive Behaviors, Vol. 130, 107308.
- Newall, P. W., James, R. J., & Maynard, O. M. (2023). How does the phrasing of house edge information affect gamblers’ perceptions and level of understanding? A Registered Report. Addiction Research & Theory, 1-9.
- Pennington, C. (2022). Does relaying ‘house edge’information influence gambler’s perceived chances of winning and their factual understanding of the statistical outcomes?. Peer Community in Registered Reports, Vol. 1.
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