Baccarat Odds Demystified: Understanding the Math Behind House Edge

Master the baccarat odds and house edge using  Dr. Catalin Barbioanu's expert insights. In this guide, he demystifies the intricacies of the maths behind Baccarat. 

Here is how this article will help you optimize your gameplay and increase their winning potential:

• What is Baccarat
• Baccarat Bets and their Payout Odds
• Basic Probabilities and House Edge in Baccarat
• Probabilities and House Edge for Side Bets
• Enhancing the Baccarat Winning Odds
• How the House Edge Changes with Different Baccarat Variants
• The Odds of Streaks in Baccarat

Delving into the mathematics behind the popular casino game, Dr. Barbioanu expertly breaks down the probabilities of each outcome, be it Player, Banker, or Tie.  

With a clear explanation of the house edge for each bet and how it affects your chances of winning, this article serves as a valuable resource for Baccarat enthusiasts and newcomers alike. 

Keep reading to learn more.

What is Baccarat?

The game of baccarat is in many respects similar to blackjack, but only in what concerns the movement of the cards and the goal of reaching a total value higher than that of the other side. 

Baccarat is usually played with one, six, or eight decks of cards dealt from a shoe, usually with eight decks. 

The big difference is that in baccarat, you don’t bet on your hand but on either an imaginary Banker’s or the Player’s hand, which the dealer actually plays. 

Hence, the baccarat game is a side bet in itself and thus simpler than blackjack. 

This feature makes any strategic play in baccarat less complex than in blackjack. On the other hand, unlike blackjack, where exceeding 21 points takes you out of the game, in baccarat, there is no bust due to the rule of removing the first digit of a total higher than 10. 

This rule renders in some respects the mathematics of baccarat more complex than that of blackjack in what concerns the odds computations.

Let’s explore the odds, probabilities, and house edges associated with the game of baccarat and its versions.

Baccarat Bets and Their Payout Odds

Three types of bets are available in baccarat: on the Banker, on the Player or on a Tie. 

• The Player bet pays off evenly (1 to 1)
• The Banker pays evenly, but the house retains an usual 5% commission for this bet, so the actual payout odds for it are 19 to 20. 
• The Tie bet usually pays off 8 to 1 (or 9 to 1 in some casinos), and a bet on the

For the Banker bet, the payout odds are displayed in the table below as 1: 1; you should be aware that they are lower due to the retained commission.

Some casinos also have a pair bet in the schedule of the base game (other have it as a side bet), which wins if either the Player or the Banker (as nominated by the bettor) have a pair of cards of the same value (usually paying 5 to 1).

Bet	Payout odds
Player	1 to 1
Banker	19 to 20
Tie	8 to 1 (9 to 1)

These payout odds approximate the likelihoods for the respective events to happen, as ‘true odds’ or probabilities – odds for the Banker to win are slightly higher than for the Player, while the odds for a Tie are the lowest. 

The reason is that the Banker benefits from a positional advantage, with more information available to decide to draw a third card. The difference between the payout odds and true odds gives the house an advantage or edge. 

Some players may consider baccarat a boring game, due to its simplicity and the very limited number of base bets (three). 

This is why casinos have enhanced the game with the so-called side bets, that is, bets placed on the configuration and features of the game's final outcome, other than on who is winning or on the tie. 

As in many other games of chance, in baccarat, players have the chance to place side bets, which are optional and differ from one casino to another and also vary with the version of the game. 

Such side bets include those called:

• Dragon Bonus: Commonly offered in Mini Baccarat. Betting on either hand achieving the maximum of 9 points. Pays 30 to 1;
• Panda 8: Commonly offered in EZ Baccarat. Betting on the Banker’s hand achieving 8 points with three cards. Pays 25 to 1;
• Dragon 7: Commonly offered in EZ Baccarat. Betting on the Banker’s hand achieving 7 points with three cards. Pays 40 to 1;
• Pair: Betting that the two cards dealt to the Player or the Banker (as nominated by the bettor) are paired. Pays 5 to 1;
• Pairs: Betting that the two cards dealt to the Player and the Banker are paired. Pays 11 to 1 (or 10 to 1);
• Perfect Pair: Offered in the online baccarat offered by Playtech. Betting that either the Player’s or Banker’s opening hand will be a pair, with different payout odds for suited and non-suited pair. Some platforms require for this bet the pair to be of the same suit, with unique payout odds. It has different payouts: 5 to 1, if the pair is not of the same suit, 25 to 1, if the pair is of the same suit, and 12 to 1 unique payout odds if the pair is of the same suit)
• All Red/All Black: Betting on the Player’s hand consisting only on red (diamonds and hearts) or black (spades and clubs) cards. Pays 22 to 1 (all red)24 to 1 (all black)
• Egalité: Betting on the exact value of the tie result. The payouts are different: 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: Betting on the Banker or Player having a natural total of 8. Pays 9 to 1 (or 8 to 1);
• Natural 9: Betting on the Banker or Player having a natural total of 9. Pays 9 to 1 (or 8 to 1)
• Natural 8 or 9: Betting on both numbers. Pays 4 to 1.

 

There are a few other side bets, but they're not as popular. The bets mentioned above are the most common ones you will encounter.

Basic Probabilities and House Edge in Baccarat

Probabilities of the various events in baccarat are combinatorial probabilities and are computed either by compact formulas, recursive methods, or computer simulations, depending on the complexity of the event. 

A probability of an event in baccarat is the ratio between the number of card combinations for both the Banker’s and Player’s hands (the card configuration of the table when the game is over) favorable for that event to happen and the total number of possible card combinations for both hands. 

House edge of a baccarat bet is the opposite (as sign +/–) of its expected value relative to the initial stake. 

It reflects the share of the stakes that goes to the house as a profit from your bets over the long run. In the equations of the expected value and house edge are employed the probabilities of winning and loosing the bet and the payout odds of the bet.

The winning probabilities (for the base bets and side bets) vary with the number of decks used. In the tables below are probabilities of the base bets along with their house edge for eight, six, and one deck used, for a standard commission of 5% on the Banker bet.

Banker Bet — 8 Decks

Player Bet — 8 Decks

Tie Bet — 8 Decks

Banker Bet — 6 Decks

Player Bet — 6 Decks

Tie Bet — 6 Decks

Banker Bet — 1 Deck

Player Bet — 1 Deck

Tie Bet — 1 Deck

As one may see, there are slight differences in the probability of the same event for the three cases, and also in what concerns house edge.

Among the three base bets, the Banker bet offers the highest return and implicitly the lowest house edge, and the Tie bet (the most risky) has the highest house edge, between 14.35% and 15.76%.

We can also estimate an overall house edge by doing the statistical average of the house edges of each bet. 

Such an indicator is only relevant under the assumption that the types of bets are distributed uniformly over the long run (which of course does not happen in practice, since the Banker bet is the most frequent). For instance, in case of an 8-deck, the overall house edge is 2.40%.

With just 1.06% house edge for the Banker bet, baccarat is one of the casino games offering the highest return, being outranked only by blackjack played with strategy, with its house edge under 1%. 

The most popular strategic advice in baccarat is to bet only on the Banker due to that slight difference in winning probability compared to the Player bet. 

Such advice can be extended with the recommendation to place the Banker bet at a 1-deck baccarat game due to another slight increase in probability in the favor of this latter game. Yet the effects of such choices may be visible only over a very long period of plays or at all as well.

Probabilities and House Edge for Side Bets

Baccarat side bets have their own probabilities and house edge, which depend on the number of decks used (as in the case of base bets). The number of possible side bets is big and many of them are associated with a certain version of baccarat.

In this section, we only provide the figures associated with the Pair bet, Panda 8 bet, Egalité bet, Natural 8 bet, Natural 9 bet, and Natural 8 or 9 bet.

Pair Bet Odds — 8 Decks

Pair Bet Odds — 6 Decks

Pair Bet Odds — 1 Deck

The Panda 8 is a side bet in EZ Baccarat (where is no commission on winning Banker bets and a winning Banker bet with a 3-card seven will push; EZ baccarat is played with 6 or 8 decks, usually 8). Panda 8 bet pays 25 to 1 for a 3-card winning Player total of 8.

Panda 8 Odds (8 decks)

Egalité Odds (8 decks)

Natural 8 — Win Pays 9 to 1 (8 decks)

Natural 8 — Win Pays 8 to 1 (8 decks)

Natural 9 — Win Pays 9 to 1 (8 decks)

Natural 9 — Win Pays 8 to 1 (8 decks)

Natural 8 or 9

Unlike the base bets, in side bets there are generally big differences in house edge between the game versions played with 8, 6 or 1 deck. 

We may see that in the Pair bet, whose house edge increases from 10.36% for 8 decks to 29.41% for 1 deck. Yet the winning probability does not follow the same rate of increase: from 7.46% to 5.88% respectively.

Third Card Rule Explained – Hows and Whys

Baccarat is known as one of the simplest casino games with respect to playing and betting. 

Yet the only thing that makes it seemingly complicated sometimes is the so-called third card rule, used to determine whether the Player or Banker hand ends the round with two or three cards.

The rule is quite straightforward when it comes to the Player hand, which consists of the first two cards to be revealed on the table in every round. The Banker being dealt a third card however depends on all currently revealed cards on the table.

The Player is dealt a third card only depending on the total of Player’s first two cards:

• (p1) If the Player’s first two cards have a total of 0-5, the Player is dealt a third card.
• (p2) If the Player’s first two cards total 6 or 7, Player stands and no third card is dealt.
• (p3) If the Player’s first two cards total 8 or 9, both Player and Banker stay with two cards.
• (b0) If the Player stands pat with just two cards, the same third card rules apply to the Banker side. (The Banker draws a third card if the total of the first two cards is 0-5).

If the Player has three cards on the table, things get a bit complicated with a new set of rules:

• (b1) If the Banker’s first two cards total 0, 1, or 2, Banker always draws a third card.
• (b2) If the Banker’s first two cards total 3, a third card is drawn in all cases except if the Player’s third card is an 8.
• (b3) If the Banker’s first two cards total 4, a third card is drawn if the Player’s third card is 2, 3, 4, 5, 6, or 7.
• (b4) If the Banker’s first two cards total 5, a third card is drawn if the Player’s third card is 4, 5, 6, or 7.
• (b5) If the Banker’s total is 6, a third card is drawn if the Player’s third card is 6 or 7.
• (b6) If the Banker totals 7, 8, or 9 with the first two cards, no third card is dealt.

This third card set of rules for the Banker in the situation Player has a three-card hand is usually comprised in a chart that you will see in almost any baccarat guide.

But what’s the reason behind this “bushy” third card rule?

First, remember that the Banker - Player match is just illusory; there is actually the dealer playing both their roles and the two hands that the Banker and Player end up the game with are in fact parts of a single outcome of the game (a four-, five-, or six-size combination of cards). 

However, since the dealer acts as two “players”, it would be unfair to favor one or another side to a significant extent and this would also be counterproductive for the game, as the bettor would constantly bet on that side that has greater chances to win; house edge would become negative and the game would no longer be profitable. 

By keeping that difference low, bettors are encouraged to bet uniformly. 

Moreover, the Player as a real player would have the option to choose whether to stand or hit; since this is not the case, the third card rule removes this option.

Like in blackjack, where players are not forced by any rule to stay at a certain total, in baccarat they are only allowed to take one additional card per the third card rules. 

As we saw in the previous sections, the difference in probability of winning (estimated before any card is dealt) is not big between the Banker and the Player and this is a specific feature of the game of baccarat. 

From the Player’s standpoint and given that Player is dealt before the Banker, by the third card rule the Player has the chance to improve their total, as they do not “know” what the Banker’s hand will be. 

In the table below are noted the probabilities of winning for the two sides conditioned on the first two cards dealt to the Player.

Probabilities of winning after two cards dealt to the Player (8 decks)

Observe that for a total 0 – 5 the probability that the Player wins is far lower than the probability that the Banker wins, not mentioning the Tie odds, which add against the bettor who bets on the Player. 

This explains the rule for the Player to be dealt a third card in this situation (p1). With a total of 6, probabilities become almost equal, regardless the tie, while for a total of 7 they become seriously reversed. This explains why the Player is forced to stay for a total equal to or above 6 (rules p2 and p3).

Moreover, 0 is the most likely value coming from a single card. That’s because 10s, Jacks, Queens, and Kings all count as 0, so it’s four times more probable than any other value. In other words, the Player will probably end up right back where it started after drawing the third card. 

Even if the Player only draws the third card with weak hands, the hand is likely to be weaker after drawing than before. 

For instance, if a hand has a total of 5, the only cards that will actually make that hand stronger are 1, 2, 3, and 4. Everything else either reduces the value of the hand or keeps it.

Now move to the Banker. The rationale behind the third card rule relative to the Banker follows the principle that the Banker only draws the third card when it suits them. If they were a real player, this would be the natural action. 

Another principle is that the Banker’s small advantage over the Player, reflected in the odds and house edge of a Banker bet, should be maintained. Per this principle, the Banker only draws the third card if it has an unfavorable hand compared to the Player, that is, only if it’s not about to win the current hand. 

Probability for the Player to win after four cards dealt (two for the Player and two for the Banker) (8 decks)

If the Player has a total of 6 or 7 and the Banker has 0 – 5, obviously a third card dealt favors the Banker, even though the probability for the Player to win is high (highlighted in the above table with dark red). Observe in the table that for a total of 0 – 5 for the Banker and 0 – 5 for the Player, the probability for the Player to win if the Banker draws a third card is more than 50% in only nine of the cases (highlighted in blue in the table).

In the rest of the cases, it is more likely for the Banker or a Tie to win. This expresses Banker’s overall advantage if it draws a third card and as such justifies the rule b0. Despite the noted exceptions, the rule makes justice for the two sides in regard to the basic rule of drawing a third card at a total of 0 – 5.

Probability for the Player to win after five cards dealt (three for the Player and two for the Banker) (8 decks)

If the Banker has a total of 0 – 2 and the Player has a total of 4 – 9, drawing a third card favors the Banker, even though the Player has a high probability of winning the hand. 

If the Player’s total is 0 – 4, the overall probability for the Banker or Tie to win if the Banker draws a third card is higher than the probability for the Player to win. 

This justifies the rule b1.

Looking at the whole table, we can see that it is in the Banker’s favor to draw a third card in almost every situation, except when the Banker holds a total over 6 less than the 

Player’s total, which prevents it to draw per rules b5 and b6. However, the situations in which the probability for the Player to win if the Banker draws the third card is over 50% are almost half of the total of situations. 

This means that, although the Banker is in advantage by having the chance to draw the third card, that advantage is not statistical (it could be materialized only in about half of the situations).

The rules b3 to b5 contribute to keeping the Banker’s advantage in statistical terms. When the Banker’s total is between 3 and 6, drawing a third is likely to downgrade it. 

Constraining the draw by the Player’s third card balances back the Banker’s overall advantage. 

The conditions in the rules b3 to b5 are actually arbitrary, since it is not a specific card that counts toward the probability of winning, but all the cards in the Player’s hand. 

Yet as a whole they are chosen still taking into account specific probabilities associated with the entire hand, which are very difficult to compute manually.

Rule b6 again consolidates the Banker’s advantage, since a total of 7, 8, or 9 gives it the highest odds of winnings.

The whole set of third card rules is devised to maintain the Banker’s advantage over the Player, but the rules are such chosen for this advantage to remain minimal, as reflected in the probability balance 45.86% (for the Banker) – 9.51% (for a Tie) – 44.62% (for the Player), for a 8-deck baccarat game, and the house edge of 1.05% for a bet on the Banker.

Enhancing the Baccarat Winning Odds

The resemblance between blackjack and baccarat is also reflected in the answer to the question of whether the baccarat winning odds can be enhanced. 

Theoretically, these odds may be enhanced, but only in certain conditions and through a card counting strategy, as in blackjack.

The only possible such information is available when the multiple hands are played before the shoe is shuffled. With every card dealt, the odds of one side or another winning (or a tie) change. 

This means that by observing the cards dealt and using a counting algorithm, you may find when the odds favor a side or another.

Blackjack card counting is known as effective (although hard to put into practice) by reducing the house edge and providing positive expectation in certain situations.

The question is to what extent a card counting strategy is effective in baccarat. First, let’s see what the effect is of removing a card from the shoe on the probabilities. 

Probability after card removal (8 decks)

We can see that the initial probabilities did not change much for one card removed.

House edge after card removal (8 decks)

We can see that house edge did not change much either for one card removed, but there are situations in which it increased and others where it decreased. The effect of card removal on the house edge is given in the next table, where minus means a decrease and plus an increase:

Effect of card removal on house edge (8 decks)

The situations with decrease (minus) indicate an advantage for the bet on Player, while those with increase (plus) an advantage for the bet on Banker. Any new card dealt may increase or decrease this advantage. 

Like in blackjack, an objective measure for this advantage is obtained by card counting, which is an algorithm that operates values associated with each card for providing a final overall value (the running count). 

Then the running count is divided by the number of cards left in the shoe to get the final measure that will indicate which bet is advantageous (the true count).

There are several card counting systems for baccarat (as are for blackjack), which you may read about in most of the baccarat strategy guides. 

There are also several strategies of using the running or true count. For certain values of the running count and strategies, the house edge of your bet in baccarat can be reduced relative to its standard value (associated with the non-strategic play).

The table above shows the effect on house edge of only one card removal, but the cumulated effect increases with every card dealt. An optimal strategy is that in which you exploit the situations associated with the favorable values of the house edge, that is, bet on either the Banker, the Player or Tie, or holding your bet and wait for the next favorable value.

The true count can indicate situations in which the house edge passes the zero point, that is, the expectation of that bet is positive. 

This does not mean that you will certainly win that bet in that particular instance, but that you may expect a positive profit over the long run if playing that way every time that situation occurs.

The next table shows the ratio of hands played, based on a simulation of 100 million, in which the true count passes the zero house edge point. The left-hand column indicates the ratio of cards dealt before the cards are shuffled.

Positive Expectation

If talking in terms of profit, the values of the above table applied for 100 bets and a $1,000 stake every time a positive expected value occurred gives the following table: 

Expected profit for the optimal play based on positive expectation

With no further calculation, one can see how the low profit rate is in this strategic play.

Yet the strategic play based on card counting lower the house edge, but not that much. 

For instance, with a basic strategy, the house edge of the bet on the Banker comes to 0.99% compared to the house edge of 1.06% for that bet when not counting cards.

The simplistic general strategic recommendation to always bet on the Banker stands as long as card counting is not employed; yet even a simple observation that many large cards were dealt (so the shoe remains rich in low cards) indicates that the Player is favored and the recommendation changes.

Now, leaving the figures aside, remember that a strategic optimal play assumes that the player is able to keep a perfect count and apply it and both the player and the casino is not going to mind the player only making a bet once every tens or hundreds of hands. 

For instance, if following the positive-expectation-only strategy, a bet should be placed at 475 hands or less. 

Adding to this the low profit rate it follows clearly that the card counting play, even mathematically optimal, is not effective in practice, as it is inconvenient, in the sense that the cumulated results do not worth the effort to achieve them; besides, it would simply cancel the joy of playing the game.

Effectiveness of card counting is yet different in blackjack, where the values of the positive expectations (when occurred) are much higher, and the overall reduction of the house edge is higher than in baccarat.

How the House Edge Changes with Different Baccarat Variants

Like in every game of chance, any modification in the rules or payout schedule of the game affects either the probabilities of the winning events or the statistical indicators of the game, as expectation and house edge (or both).

There are several versions of baccarat with respect to the basic game (not taking into account the side bets). A category of versions is given by the commission on the Banker bet, which may vary with values less than 5%.

The next table shows the house edge on the Banker bet depending on the commission and number of decks used.

House edge of the Banker bet

Some casinos (mostly in UK) have a Red 8 rule, where the winning Banker bets pay no commission on certain wins. Some of them have a Red 8 rule where the Banker bets pay 1 to 1 if:

The Banker has two-card total of three points.

The Player has two-card total of five points or less.

The Player’s third card is an 8.

Assuming eight decks, this rule reduces the house edge on the Banker bet from 1.06% to 0.81%.

Probabilities and house edge for the Banker bet under the Red 8 rule (8 decks)

There are several versions of commission free baccarat. The most popular version is the EZ Baccarat. Its rules follows standard baccarat except on the Banker bet: 

1. all wins pay even money, 
2. if the Banker wins with a three-card total of 7, then the banker bet will push.

The following table shows the house edge of the bets on the Banker in EZ Baccarat. The overall house edge is 1.02%.

Probabilities and house edge for the Banker bet in EZ Baccarat (8 decks)

Another version of commission free baccarat is Monkey Baccarat. As in EZ Baccarat, the Banker bet pays 1 to 1, except on a winning three-card total of 7, when the Banker bet is a push.

The next table shows the probabilities and house edge for the Banker bet in Monkey Baccarat. The overall house edge of this bet is 1.02%.

Probabilities and house edge for the Banker bet in Monkey Baccarat (8 decks)

Another version of commission free baccarat is Nepal Baccarat. This variant of baccarat follows the conventional rules, with the following exception: a winning Banker bet pays even money, except on a winning total of six, when they pay 1 to 2. The house edge on the Player and Tie bets are the same as in conventional baccarat. The house edge on the Banker bet is 1.46%. 

The next table shows the probabilities and house edge for the Banker bet in Nepal Baccarat.

Probabilities and house edge for the Banker bet in Nepal Baccarat (8 decks)

Changing the payout odds for the Tie bet gives a new version of baccarat. The Tie pays 9 to 1 at some casinos. The house edge of the Tie bet paying 9 to 1 is:

• For 8 decks: 4.844%
• For 6 decks: 4.931%
• For 1 deck: 6.385%

The Odds of Streaks in Baccarat

With only three base bets to choose from, a regular bettor or just an observer of the game will of course see sometimes the same bet winning for several times in a row (event called streak in the gambling jargon). 

When this happens, one may ask of course what the odds of that are. For instance, what are the odds of the Banker winning four times in a row? 

Such information has no mathematical or strategic relevance, since the games are independent to each other. 

Moreover, for those playing progressive systems like the martingale knowing these odds might fuel the cognitive distortion known as the Gambler’s Fallacy. 

Yet for the sake of completion I present the odds of the baccarat streaks in the next table.

Probability of streaks in baccarat (6 decks)

The figures apply to any starting point for the streak (no matter it is about the first, second, third game, and so on). Tie was not included.

Conclusion

Baccarat is a game similar in many respects to blackjack, but also very different.

Baccarat is together with blackjack in the category of games of chance having the lowest house edge. In regard to betting, baccarat is a very simple game, offering three base bets, whose probabilities and house edges can be easily memorized. 

Yet these parameters change with the variant of the game, including for the various numbers of card decks used. This simplicity is counterbalanced by the side bets that many variants offer. 

The baccarat odds associated with the side bets are much different from those of the base bets (winning probabilities are much lower and the payout odds are much higher accordingly).

The only complicated thing of baccarat is the third-card rule, which is justified by precise probability computations and its role is to maintain the advantage of the Banker over the Player at a minimal edge.

Strategic play is possible in baccarat and is based on card counting, like in blackjack, however it is far less effective than in blackjack, given the very low profit rates and the long sessions of holding the bets. 

While the mathematical information expressed by odds and house edge is of no practical strategic help in baccarat, getting informed on these figures is important for those who play regularly to make a profit rather than for fun. 

For these regular gamblers, these mathematical indicators are objective criteria for choosing a game or another or choosing between several variants of the same game.