Introduction

I've been getting asked a lot about the odds in baccarat for unusual numbers of decks. To hopefully cover all my bases, this page shall show the number of combinations, probability of winning, and expected return, for every number of decks from 1 to 12.

Combinations

The following table shows the number of combinations for all three possible outcomes by the number of decks. The combinations in this page get pretty huge. Unfortunately, Excel displays only 15 significant digits, so please don't write to me that I'm wrong after 15 digits. Instead, please blame Microsoft.

Combinations

Decks	Player	Banker	Tie	Total
1	6,548,674,432	6,737,232,640	1,372,227,328	14,658,134,400
2	487,816,735,232	501,543,728,128	103,154,274,240	1,092,514,737,600
3	5,837,305,562,496	6,000,349,195,008	1,238,768,922,816	13,076,423,680,320
4	33,608,344,225,792	34,543,624,867,840	7,145,601,996,928	75,297,571,090,560
5	130,087,780,016,000	133,700,037,548,800	27,690,387,926,400	291,478,205,491,200
6	392,220,492,728,832	403,095,751,234,560	83,552,962,932,288	878,869,206,895,680
7	995,884,732,700,032	1,023,469,376,328,440	212,268,385,833,280	2,231,622,494,861,750
8	2,230,518,282,592,250	2,292,252,566,437,880	475,627,426,473,216	4,998,398,275,503,350
9	4,540,095,442,933,630	4,665,676,262,757,120	968,435,421,769,728	10,174,207,127,460,500
10	8,570,454,841,408,000	8,807,402,586,035,200	1,828,629,499,384,000	19,206,486,926,827,200
11	15,222,978,065,896,800	15,643,682,376,129,200	3,248,754,169,011,640	34,115,414,611,037,600
12	25,714,619,121,272,800	26,425,038,379,143,100	5,488,795,281,294,720	57,628,452,781,710,600

The next table shows the probability of each outcome by number of decks.

Probabilities

Decks	Player	Banker	Tie
1	0.446760430304146	0.459624155172162	0.093615414523693
2	0.446508150822404	0.459072734551641	0.094419114625955
3	0.446399237681564	0.458867756329930	0.094733005988506
4	0.446340349881558	0.458761476200800	0.094898173917642
5	0.446303626018198	0.458696516686345	0.094999857295457
6	0.446278569838877	0.458652718825324	0.095068711335799
7	0.446260393499810	0.458621195423935	0.095118411076256
8	0.446246609343597	0.458597422632763	0.095155968023641
9	0.446235798628453	0.458578855758138	0.095185345613408
10	0.446227093692859	0.458563953917737	0.095208952389404
11	0.446219934286584	0.458551729606354	0.095228336107061
12	0.446213942593194	0.458541520787272	0.095244536619534

The next table shows the expected return of each outcome by number of decks. Note how as the number of decks increase, the expected value of the Player and Tie bets increase and the Banker bet decreases.

Similar to well-paying odds, the free Baccarat games comprise the same type of games that one handles during paid online games. In as much as players may end up losing, this serves as an eye-opener. This is because one gets to know what to expect while playing the paid games as. Online Baccarat Odds Calculator. Unlike poker, baccarat is a game of complete chance. That's one of the things that attract new players to baccarat tables in droves. It combines many of the aspects of card games with the adventure of fast and fun table games such as roulette.

Expected Returns

Decks	Player	Banker	Tie
1	-0.012863724868016	-0.010117482890592	-0.157461269286765
2	-0.012564583729237	-0.010389052998346	-0.150227968366401
3	-0.012468518648366	-0.010474869168131	-0.147402946103443
4	-0.012421126319243	-0.010516947490797	-0.145916434741219
5	-0.012392890668147	-0.010541935166171	-0.145001284340884
6	-0.012374148986447	-0.010558486954819	-0.144381597977809
7	-0.012360801924125	-0.010570257847072	-0.143934300313697
8	-0.012350813289166	-0.010579057842472	-0.143596287787236
9	-0.012343057129685	-0.010585885658222	-0.143331889479325
10	-0.012336860224877	-0.010591337471010	-0.143119428495365
11	-0.012331795319770	-0.010595791160548	-0.142944975036449
12	-0.012327578194079	-0.010599497845285	-0.142799170424194

Pair Bets

The next set of tables are for the Player Pair and Banker Pair bets. The expected value table is based on the usual win of 11 to 1. This next table shows the number of combination of winning and losing by number of decks.

Pair Bets Combinations

Decks	Win	Loss	Total
1	78	1,248	1,326
2	364	4,992	5,356
3	858	11,232	12,090
4	1,560	19,968	21,528
5	2,470	31,200	33,670
6	3,588	44,928	48,516
7	4,914	61,152	66,066
8	6,448	79,872	86,320
9	8,190	101,088	109,278
10	10,140	124,800	134,940
11	12,298	151,008	163,306
12	14,664	179,712	194,376

The next table shows the probability of winning and losing by number of decks.

Pair Bet Probabilities

Decks	Prob. Win	Prob. Loss
1	0.058823529411765	0.941176470588235
2	0.067961165048544	0.932038834951456
3	0.070967741935484	0.929032258064516
4	0.072463768115942	0.927536231884058
5	0.073359073359073	0.926640926640927
6	0.073954983922830	0.926045016077170
7	0.074380165289256	0.925619834710744
8	0.074698795180723	0.925301204819277
9	0.074946466809422	0.925053533190578
10	0.075144508670520	0.924855491329480
11	0.075306479859895	0.924693520140105
12	0.075441412520064	0.924558587479936

The next table shows the expected return by number of decks. As you can see, as the number of decks increase, so does the expected value. This should be obvious, because as the number of decks increases, the effect of removal becomes less significant.

Pair Bet Expected Returns

Decks	Expected Return
1	-0.294117647058823
2	-0.184466019417476
3	-0.148387096774194
4	-0.130434782608696
5	-0.119691119691120
6	-0.112540192926045
7	-0.107438016528926
8	-0.103614457831325
9	-0.100642398286938
10	-0.098265895953757
11	-0.096322241681261
12	-0.094703049759230

Written by: Michael Shackleford

Introduction

Baccarat, also known as punto banco, is one of the oldest and most popular games in casinos all over the world. It is especially popular among high-rollers and Asian gamblers. In Macau, baccarat is extremely dominant. Although the game seems serious and elegant, it is really as simple as betting on the flip of a coin.

Baccarat Basics - How to Play with Wizard's Free Demo

Format

There are three types of baccarat tables:

Big Table Baccarat is played in a snooty roped off area to separate itself from the other games. The players are usually very well dressed and the table minimums are often high. The regular baccarat table is about the size of a craps table with 3 casino dealers and up to 12 or 14 players. Each player, including the player dealing, may still bet on either the player or the banker but it is customary for the dealer to bet on the banker.

The deal will rotate around the table, much like the dice rotate around the craps table. If a player does not want to deal they may pass the shoe to the next player. The same person will keep dealing as long as the banker keeps winning. The person dealing will put two cards, face down, tucked under the shoe, and give the player with the greatest bet on the player the other two cards, face down. This player then looks at the cards and then gives them back to the player who is dealing. Then the player who is dealing will turn over the cards and one of the casino dealers will announce the totals. Depending on the totals, the dealer may then instruct the person dealing the cards to deal a third card. Finally the dealers will pay winning wagers and collect losing ones out of the dealer's tray. The player who actually deals is not assuming any financial responsibility of the other players bets, unlike pai gow, and is just turning over cards.

Mini baccarat has the same rules are big table baccarat. However, unlike the game at the big table, the dealer turns over all the cards, making for a much faster game. The odds are exactly the same as those of the big table, assuming the same number of decks. Mini baccarat is sometimes dealt from a six-deck shoe, changing the odds slightly. It can usually be found in the main casino areas.

Midi baccarat is the same as mini baccarat, except the size of the table is larger, and it is usually found in the high-limit rooms, as opposed to the main casino floor.

Rules

Following are the rules of baccarat. The terminology can be a little confusing. To try to minimize this, when referring to a particular bet or hand, I will use capital letters. In particular Player and Banker refer to bets as well as particular sets of two or three cards each.

1. Usually eight decks of cards are used.
2. Cards are given point values as follows: Ace = 1, 2-9 = pip value, 10 and face cards = 0.
3. Play begins by all players betting either on the Player or Banker. There is also a side bet on a Tie. At some tables there are also side bets on a Player Pair and/or Banker Pair. There are lots of other newer side bets, which I go into in my baccarat appendix 5.
4. After all bets are down, the dealer gives two cards each to the Player and Banker hands.
5. The maximum number of points in both the Player and Banker hands is 9. The way to arrive at the points per hand is to take the total points of each individual card in the hand. If the sum is more than 9, then drop the first digit. For example, if either hand had a 9 and 7, then you would drop the 1 from the total of 16, for a 6-point hand.
6. A third card may or may not be dealt to either the Player or Banker hands, depending on the following rules.
7. If either the Player or Banker have 8 or 9 points, it is referred to as a 'natural.' If there is at least one natural, then both hands will stand.
8. Otherwise, if the Player's total is 5 or less, then the Player hand will draw one more card, otherwise, with 6 or 7 points, the Player hand stands.
9. If the Player hand stands with 6 or 7 points, then the Banker hand will draw a third card on a total of 5 or less. Otherwise, with 6 or 7 points, the Banker will stand.
10. If the Player does draw a third card, then use the Banker will use his positional advantage to decide whether to take a third card according to his total and the third card drawn to the Player, according to the following table.

    Banker Drawing Rules

    Banker's Score	Player's Third Card
    	0	1	2	3	4	5	6	7	8	9
    7	S	S	S	S	S	S	S	S	S	S
    6	S	S	S	S	S	S	D	D	S	S
    5	S	S	S	S	D	D	D	D	S	S
    4	S	S	D	D	D	D	D	D	S	S
    3	D	D	D	D	D	D	D	D	S	D
    2	D	D	D	D	D	D	D	D	D	D
    1	D	D	D	D	D	D	D	D	D	D
    0	D	D	D	D	D	D	D	D	D	D

    D = Draw third card
    S = Stand on two cards
11. The score of the Player and Banker hands are compared; the winner is the one that is greater. In an event of a tie, the Player and Banker bets push.
12. The Tie bet wins if the Player and Banker hands tie. All other outcomes lose.
13. The Player Pair bet wins if the first two cards in the Player hand are of the same rank. All other outcomes lose.
14. The Banker Pair bet wins if the first two cards in the Banker hand are of the same rank. All other outcomes lose.
15. Following is how to pay each bet:
    
    • Player: Wins pay even money.
    • Banker: Wins pay 19 to 20, which is equivalent to even money less a 5% commission.
    • Tie: Almost all casinos pay 8 to 1 on a tie, which is often expressed as the equivalent to 9 for 1.
    • Player Pair and Banker pairs: Wins pay 10 to 1.

The reason the Banker gets paid less than the Player is the Banker enjoys a positional advantage, with more information available to make the decision on drawing a third card.

Following are some rules on dealing procedures. The beginner can skip this part.

• At the start of a new shoe, the dealer will turn over one card. This will determine how many cards the dealer will burn, according to the baccarat value, except a 10 or face card will result in 10 cards burned.
• The cut card will be placed 16 cards from the bottom of the shoe. When the cut card appears, the dealer will finish that hand, play one more hand, and then start a new shoe. If the cut card comes out instead of the first card, the dealer will finish that hand, and then start a new shoe.
• To speed up the game, the dealer will keep track of commissions owed on winning Banker bets using small laminated markers. He will collect all commissions owed at the end of the shoe.

Eight-Deck Analysis

The following return tables show the possible outcomes of the Player, Banker, and Tie bets for an 8-deck game. As the lower right cells in each table show, the house edge is 1.06% on the banker bet, 1.24% on the player bet, and 14.36% on the tie.

Banker Bet — 8 Decks

Event	Pays	Combinations	Probability	Return
Banker wins	0.95	2,292,252,566,437,888	0.458597	0.435668
Player wins	-1	2,230,518,282,592,256	0.446247	-0.446247
Tie	0	475,627,426,473,216	0.095156	0
Total	4,998,398,275,503,360	1	-0.010579

Player Bet — 8 Decks

Event	Pays	Combinations	Probability	Return
Banker wins	-1	2,292,252,566,437,888	0.458597	-0.458597
Player wins	1	2,230,518,282,592,256	0.446247	0.446247
Tie	0	475,627,426,473,216	0.095156	0
Total	4,998,398,275,503,360	1	-0.012351

Tie Bet — 8 Decks

Event	Pays	Combinations	Probability	Return
Banker wins	-1	2,292,252,566,437,888	0.458597	-0.458597
Player wins	-1	2,230,518,282,592,256	0.446247	-0.446247
Tie	8	475,627,426,473,216	0.095156	0.761248
Total	4,998,398,275,503,360	1	-0.143596

Pair Bets — 8 Decks

Event	Pays	Combinations	Probability	Return
Pair	11	6448	0.074699	0.821687
No pair	-1	79872	0.925301	-0.925301
Total	86320	1	-0.103614

Six-Deck Analysis

The following return tables show the possible outcomes of the Player, Banker, and Tie bets for a 6-deck game. As the lower right cells in each table show, the house edge is 1.06% on the Banker bet, 1.24% on the Player bet, and 14.44% on the Tie.

Banker Bet — 6 Decks

Event	Pays	Combinations	Probability	Return
Banker wins	0.95	403095751234560	0.458653	0.43572
Player wins	-1	392220492728832	0.446279	-0.446279
Tie	0	83552962932288	0.095069	0
Total	878869206895680	1	-0.010558

Player Bet — 6 Decks

Baccarat Odds Chart

Event	Pays	Combinations	Probability	Return
Banker wins	-1	403095751234560	0.458653	-0.458653
Player wins	1	392220492728832	0.446279	0.446279
Tie	0	83552962932288	0.095069	0
Total	878869206895680	1	-0.012374

Tie Bet — 6 Decks

Event	Pays	Combinations	Probability	Return
Banker wins	-1	403095751234560	0.458653	-0.458653
Player wins	-1	392220492728832	0.446279	-0.446279
Tie	8	83552962932288	0.095069	0.76055
Total	878869206895680	1	-0.144382

Pair Bets — 6 Decks

Event	Pays	Combinations	Probability	Return
Pair	11	3588	0.073955	0.813505
No pair	-1	44928	0.926045	-0.926045
Total	48516	1	-0.11254

One-Deck Analysis

The following return tables show the possible outcomes of the Player, Banker, and Tie bets for a single deck game. As the lower right cells in each table show, the house edge is 1.01% on the Banker bet, 1.29% on the Player bet, and 15.75% on the Tie.

Banker Bet — 1 Deck

Event	Pays	Combinations	Probability	Return
Banker wins	0.95	6737232640	0.459624	0.436643
Player wins	-1	6548674432	0.44676	-0.44676
Tie	0	1372227328	0.093615	0
Total	14658134400	1	-0.010117

Player Bet — 1 Deck

Event	Pays	Combinations	Probability	Return
Banker wins	-1	6737232640	0.459624	-0.459624
Player wins	1	6548674432	0.44676	0.44676
Tie	0	1372227328	0.093615	0
Total	14658134400	1	-0.012864

Tie Bet — 1 Deck

Event	Pays	Combinations	Probability	Return
Banker wins	-1	6737232640	0.459624	-0.459624
Player wins	-1	6548674432	0.44676	-0.44676
Tie	8	1372227328	0.093615	0.748923
Total	14658134400	1	-0.157461

Pair Bets — 1 Deck

Event	Pays	Combinations	Probability	Return
Pair	11	78	0.058824	0.647059
No pair	-1	1248	0.941176	-0.941176
Total	1326	1	-0.294118

Baccarat Odds Calculator

Infinite Decks

Some Internet casinos claim to use an infinite number of decks in baccarat. The following four tables show the analysis of the Banker, Player, Tie, and Pair bets assuming an infinite deck shoe.

Banker — Infinite Decks

Event	Pays	Combinations	Probability	Return
Win	0.95	2,212,744	0.458428	0.435507
Tie	0	460,601	0.095426	0.000000
Loss	-1	2,153,464	0.446147	-0.446147
Total	4,826,809	1.000000	-0.010640

Player — Infinite Decks

Event	Pays	Combinations	Probability	Return
Win	1	2,153,464	0.446147	0.446147
Tie	0	460,601	0.095426	0.000000
Loss	-1	2,212,744	0.458428	-0.458428
Total	4,826,809	1.000000	-0.012281

Tie — Infinite Decks

Event	Pays	Combinations	Probability	Return
Win	8	460,601	0.095426	0.763405
Loss	-1	4,366,208	0.904574	-0.904574
Total	4,826,809	1.000000	-0.141170

Player/Banker Pair — Infinite Decks

Event	Pays	Combinations	Probability	Return
Win	11	1	0.076923	0.846154
Loss	-1	12	0.923077	-0.923077
Total	13	1.000000	-0.076923

Reduced Commission Banker Rules

Sign formerly seen at the D casino.

A less than 5% commission on winning banker Bets has been known to happen. As the image above shows, the D casino used to have a 4% commission Banker bet, but no longer do. For a short time in 1989 the Sahara offered 0% commission. The Betfair Internet casino offers a 2.75% commision in their 'Zero Lounge.' The following table shows the expected value on the Banker bet according to the number of decks and commission.

Odd Baccarat Pieces

Reduced Commission on Banker Bet

Commission	1 Deck	6 Decks	8 Decks
5.00%	-1.012%	-1.056%	-1.058%
4.00%	-0.552%	-0.597%	-0.599%
3.00%	-0.092%	-0.139%	-0.141%
2.75%	0.022%	-0.024%	-0.026%
0.00%	1.286%	1.237%	1.235%

Red 8 Rule

Some casinos in London have a 'Red 8' rule, where winning Banker bets pay no commission on certain wins. For example, the Hippodromo casino has a 'red 8' rule where Banker bets pay even money if all of the following are true:

• Banker has two-card total of three points.
• Player has two-card total of five points or less, thus drawing a card.
• Player's third card is an 8.

Assuming eight decks, this lowers the house edge on the Banker bet from 1.06% to 0.81%. The following return table details the possible outcomes of the Banker bet under the Red 8 rule.

Red 8 Rule — Eight Decks

Event	Pays	Combinations	Probability	Return
Red 8 win	1	243,809,554,247,680	0.048778	0.048778
All other banker wins	0.95	2,048,443,012,190,200	0.409820	0.389329
Tie	0	475,627,426,473,216	0.095156	0.000000
Player Wins	-1	2,230,518,282,592,250	0.446247	-0.446247
Total	4,998,398,275,503,350	1.000000	-0.008140

Liberal Player Rules

The 'bonus casino' at the 5 Dimes Internet sports book pays 1.01 to 1 on the Player bet. Assuming eight decks, the house edge on that bet is 1.04%.

Liberal Tie Rules

The Tie has been known to pay 9 to 1 at some casinos. For example, Binion's Horseshoe in Las Vegas used to, and the online casino pays 9 to 1. Here is the house edge on the tie bet if it pays 9 to 1.

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

The 'bonus casino' at the 5 Dimes Internet sports book pays 9.4 to 1 on the Tie bet. Assuming eight decks, the house edge on that bet is 0.79%.

Liberal Pair Rules

I have an unconfirmed report that the Fallsview Casino in Niagara Falls pays 12 to 1 on either a Player pair or Banker pair. Based on an eight deck game, that would reduce the house edge to 2.89%. In a six-deck game the house edge would be 3.86%.

Egalite Bets

Egalities are side bets on specific ties in baccarat. These bets are often seen England and seldom elsewhere. Information about them has been moved off to my page on Egalite bets.

Riding Trends

Every baccarat table will have plenty of score cards and pencils for the players to keep track of the outcome of every hand. Most players will do so religiously and carefully analyze their card for trends as the winning hand switches back and forth from the banker to the player. This is a big waste of time! The smart player will bet on the banker every time and leave the score cards alone.

Calculator

Now you can enter any distribution of cards and my baccarat calculator will give you the odds for the Player, Banker, and Tie bets, broken down by all possible Player and Banker totals.

Practice Game

Practice your game for free with my baccarat game. It features a full history display and exact odds for the next hand.

Practice Mini-Baccarat for Free

Try my mini-baccarat Java applet. This will allow you to get familiar with the rules and how the game is played.

Odd Baccarat Games

Simulations

Odds Baccarat

Before you go and waste your money with a betting system in the casinos, test it out on my 250,000 simulated eight-deck shoes. They were created using a Mersenne Twister random number generator and a Fisher Yates shuffle.

Odd Baccarat Bar

Each file has 25,000 shoes and is 4.06 MB in size. Each line in each file is an individual shoe, starting with the shoe number, and followed by the outcome of each hand (P/B/T). The files are comma delimited for easy loading into Excel. If you need to see every card, I also have these two files.

Wizard Of Odds Free Baccarat

• 1000 6-deck shoes: 1000 computer generated baccarat shoes using six decks. (1.6 MB)
• 1000 8-deck shoes: 1000 computer generated baccarat shoes using eight decks. (1.2 MB)

Written by:Michael Shackleford