Probability Baccarat

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Baccarat is a betting game best played using a strategy. There’re many Baccarat strategies to choose from. But if you were to ask me for my top three, I would have to choose Baccarat strategies that are logical, easy & have the best possible chances of reducing the casinos house edge. Crazy 4 Poker Introduction. Crazy 4 Poker is a poker variation invented by Roger Snow and is marketed by Shufflemaster. It has been around since about 2004 and one of the more successful poker-based casino games. Criss Cross Poker Introduction Photo taken at 2014 Global Gaming Expo. Criss Cross Poker is poker variation by In Bet Gaming.In 2014, I started to hear reports of the game in several casinos in the eastern United States. Live dealer blackjack is nowadays one of the most popular online casino games. It has the same feel and look of the real blackjack found in land-based casinos, but it’s played through a live stream with a professional dealer sitting in front of the camera.

Introduction


Photo taken at 2014 Global Gaming Expo.

Criss Cross Poker is poker variation by In Bet Gaming. In 2014, I started to hear reports of the game in several casinos in the eastern United States. The thrust of the game is the player bets on two hands, in part using five community cards placed in a cross pattern.

Rules

  1. Play begins with the player making Ante Across and Ante Down bets of equal size. The player may also make an optional Five Card Bonus Bet.
  2. The dealer gives the player two hole cards and places five community cards face down in a cross pattern.
  3. The player may make an Across Bet, based on the Across Hand, equal to 1x to 3x the Ante Across bet, or fold.
  4. The dealer will reveal the two outside cards on the horizontal row of the cross.
  5. The player may make a Down bet, based on the Down Hand, equal to 1x to 3x the Ante Down bet, or fold.
  6. The dealer will reveal the two outside cards on the vertical column of the cross.
  7. The player may make a Middle Bet of 1x to 3x either Ante bet or fold.
  8. The dealer will reveal the middle card.
  9. The player uses his two hole cards and the three cards on the horizontal row of the cross to form the Across Hand.
  10. Likewise, the player uses his two hole cards and the three cards on the vertical column of the cross to form the Down Hand.
  11. The Middle Bet pays according to the higher ranking of the Across and Down hands.
  12. All bets pay according to the pay table below. Wins are expressed on a 'to one' basis.

    Criss Cross Poker Pay Table

    HandAnte
    Bets Pay
    Across, Down &
    Middle Bets Pay
    Royal flush1500
    Straight flush1100
    Four of a kind140
    Full house112
    Flush18
    Straight15
    Three of a kind13
    Two pair12
    High pair (J-A)11
    Low pair (6-10)00
    All other-1-1
  13. The Five Card Bonus bet pays according to the five community cards in the cross pattern and following pay table.

    Five Card Bonus Bet Pay Table

    HandPays
    Royal flush250
    Straight flush100
    Four of a kind40
    Full house15
    Flush10
    Straight6
    Three of a kind4
    Two pair3
    Sixes or better1
    Loser-1

Analysis

Baccarat Probability Chart

Gta 5 casino slot machine odds. Between two sets of possible hands, four combinations of raise bets, and folding at various stages of the game, there are 975 possible outcomes. I indicate the probability of all of them in my Criss Cross Poker Full Expected Value Table. However, I don't want to scare you off with that gigantic table here. Instead, I summarize the outcome of each hand below.

The following table shows the possible outcome of both the Ante Across and Across bets. The 'pays' column shows the combined win between the two bets. The lower right cell shows that the player can expect to lose 0.600581 units between the two bets.

Ante Across and Across Bets

HandAcross BetPaysCombinationsProbabilityReturn
Royal flush3150112,9720.0000000.000231
Royal flush1501116,7480.0000010.000694
Straight flush330125,9440.0000000.000093
Straight flush11011,141,5360.0000140.001368
Four of a kind312112,167,7360.0001440.017468
Four of a kind1417,601,5920.0000900.003698
Full house33748,800,6640.0005790.021423
Full house11368,414,3280.0008120.010552
Flush3252,101,4640.0000250.000623
Flush19163,550,9760.0019400.017464
Straight3162,451,7080.0000290.000465
Straight16288,484,3080.0034230.020537
Three of a kind310538,234,2240.0063860.063859
Three of a kind141,170,645,1680.0138890.055557
Two pair37811,632,0960.0096300.067408
Two pair133,010,230,4320.0357150.107146
High pair341,170,593,2800.0138890.055555
High pair129,458,722,6200.1122240.224448
Medium pair301,385,409,6000.0164370.000000
Medium pair1011,610,887,1600.1377590.000000
Loser3-41,240,719,9120.014721-0.058883
Loser1-244,512,795,9320.528127-1.056254
Fold one unit0-14,576,521,6000.054299-0.054299
Fold two units0-24,203,010,8000.049867-0.099734
Fold four units0-400.0000000.000000
Total84,284,272,8001.000000-0.600581

The following table shows the possible outcome of both the Ante Down and Down bets. The 'pays' column shows the combined win between the two bets. The lower right cell shows that the player can expect to lose 0.597666 units on the two bets combined.

Ante Down and Down Bets

HandDown BetPaysCombinationsProbabilityReturn
Royal flush3150145,7920.0000010.000815
Royal flush150183,9280.0000010.000499
Straight flush330137,1160.0000000.000133
Straight flush11011,118,7000.0000130.001341
Four of a kind312112,233,8080.0001450.017563
Four of a kind1417,485,0480.0000890.003641
Full house33748,603,8880.0005770.021337
Full house11368,156,8560.0008090.010513
Flush3256,951,4560.0000820.002062
Flush19157,641,2880.0018700.016833
Straight3164,780,9800.0000570.000908
Straight16285,297,5160.0033850.020310
Three of a kind310540,593,1360.0064140.064139
Three of a kind141,160,967,8160.0137740.055098
Two pair37814,576,1760.0096650.067652
Two pair132,988,662,1840.0354590.106378
High pair341,336,724,4960.0158600.063439
High pair129,281,584,6560.1101220.220245
Medium pair301,417,841,5680.0168220.000000
Medium pair1011,550,329,6160.1370400.000000
Loser3-41,450,469,0880.017209-0.068837
Loser1-244,370,555,2880.526439-1.052879
Fold one unit0-15,013,003,4560.059477-0.059477
Fold two units0-23,766,528,9440.044688-0.089377
Fold four units0-400.0000000.000000
Total84,284,272,8001.000000-0.597666

The following table shows the possible outcome of Middle bet. The lower right cell shows that the player can expect to win 1.111675 units on this bet.

Middle Bet

HandMiddle BetPaysCombinationsProbabilityReturn
Royal flush31500259,4400.0000030.004617
Royal flush150000.0000000.000000
Straight flush33002,323,1040.0000280.008269
Straight flush110000.0000000.000000
Four of a kind312039,488,1840.0004690.056221
Four of a kind14000.0000000.000000
Full house336231,865,8480.0027510.099036
Full house11200.0000000.000000
Flush324325,921,0560.0038670.092806
Flush1800.0000000.000000
Straight315375,011,4120.0044490.066740
Straight15199,154,2680.0023630.011814
Three of a kind392,751,292,1880.0326430.293787
Three of a kind13399,927,5640.0047450.014235
Two pair365,494,091,3280.0651850.391111
Two pair121,329,787,3680.0157770.031555
High pair3311,485,872,2400.1362750.408826
High pair114,170,202,7160.0494780.049478
Medium pair3012,309,204,6360.1460440.000000
Medium pair104,837,617,2280.0573960.000000
Loser3-31,789,381,7400.021230-0.063691
Loser1-129,763,340,0800.353130-0.353130
Fold008,779,532,4000.1041660.000000
Total84,284,272,8001.0000001.111675

The next table summarizes the expected win of all three groups of bets.

Summary

Bet(s)Expected Win
Ante Across & Across-0.600581
Ante Down & Down-0.597666
Middle1.111675
Total-0.086571

The lower right cell shows an expected loss of 0.086571 units. Since the player must wager two units at least, I divided that by two to get a house edge of 4.33%.

Considering the raise bets, the average total bet by the end of the hand is 5.86529 units. Thus the element of risk, which is the ratio of expected loss to the total amount wagered, is 8.6571/5.8629 = 1.48%, which isn't bad for a poker variant game.

The standard deviation, relative to the combined initial wager, is 4.802.

Strategy

With all decision points, the only viable options are fold, minimum 1x raise, or maximum 3x raise. Never raise anything between 1x and 3x.

Across Strategy

  • If the hole cards are unsuited, unpaired, and the largest one is five or less, then fold.
  • If the hole cards are suited J/Q or any pair, then make the large raise.
  • With all other hole cards, make the small raise.

Down Strategy

This is a very similar decision to raising on the Across hand. However, the player has the benefit of seeing two extra cards. This can help in borderline situations. I have not implicitly identified these deviations from the Across strategy. However, I can say that knowing them, whatever they are, will increase the expected return by 0.29% the amount of the Ante Down bet, or lower the house edge by 0.15%.

Middle Strategy

With four cards seen, there are thousands of possible situations in terms of a Middle bet strategy. However, a perfect decision can be made by averaging the win over all possible 46 cards left in the deck under both a small and large Middle bet. I can say that if the player made either 3x bet, then he shouldn't fold. I can also say that 4.5% of the time the player will make both a small Across and Down bet and then fold the Middle bet.

Five Card Bonus Bet

The following table presents the possible outcomes of the Five Card Bonus Bet. The lower right cell shows a house edge of 3.53%, which for a side bet isn't bad.

Five Card Bonus Bet

HandPaysCombinationsProbabilityReturn
Royal flush25040.0000020.000385
Straight flush100360.0000140.001385
Four of a kind406240.0002400.009604
Full house153,7440.0014410.021609
Flush105,1080.0019650.019654
Straight610,2000.0039250.023548
Three of a kind454,9120.0211280.084514
Two pair3123,5520.0475390.142617
Sixes or better1760,3200.2925480.292548
Loser-11,640,4600.631199-0.631199
Total2,598,9601.000000-0.035336

Methodology

The game was analyzed by me in C++ by looping through all combin(52,2)*combin(50,2)*combin(48,2)*46 = 84,284,272,800 possible combinations. The program is about 600 lines of code and takes about six hours to run.

Internal Links

Full expected value table. This page shows the probability and return from all 974 possible outcomes.


Written by:Michael Shackleford

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Introduction

Crazy 4 Poker is a poker variation invented by Roger Snow and is marketed by Shufflemaster. It has been around since about 2004 and one of the more successful poker-based casino games.

Video Tutorial

Video uses our practice Crazy 4 Poker game.

Rules

  1. Play starts with the player making equal bets on the Ante and Super Bonus. The player may also bet on the Queens Up side bet at this time.
  2. Following is the ranking of hands from highest to lowest:
    • Four of a kind.
    • Straight flush
    • Three of a kind
    • Flush
    • Straight
    • Two pair
    • Pair
    • Four singletons
  3. All player and dealer get five cards each.
  4. The player decides to fold or raise by making a Play wager.
  5. If the player folds he forfeits all bets.
  6. The Play bet may be up to three times the Ante bet if the player has at least a pair of aces. Otherwise, the Play bet must be exactly equal to the Ante bet.
  7. Players make their best four-card poker hand, and discard the fifth card.
  8. After all decisions have been made, the dealer will turn over his cards and select the best four out of five.
  9. The player's hand shall be compared to the dealer's hand, the higher hand winning.
  10. For purposes of the Ante bet only, the dealer needs at least a king high to open.
  11. The Ante bet pays as follows:
    • Dealer does not open: Ante pushes.
    • Dealer opens and player wins: Ante wins.
    • Dealer opens and ties: Ante pushes.
    • Dealer opens and wins: Ante loses.
  12. The Play bet pays as follows:
    • Dealer does not open: Play wins.
    • Dealer opens and player wins: Play wins.
    • Dealer opens and ties player: Play pushes.
    • Dealer opens and wins: Play loses.
  13. The Super Bonus bet pays as follows. It is not pertinent whether or not the dealer opens.
    • Player has straight or higher (beating dealer not required): Super Bonus wins according to pay table below.
    • Player has less than straight and wins or pushes: Super Bonus pushes.
    • Player has less than straight and loses: Super Bonus loses.

Super Bonus Pay Table

Player HandPays
Four aces200
Four 2-K30
Straight flush15
Three of a kind2
Flush1.5
Straight1

Strategy

Optimal strategy would be tedious and complicated memorize. However, the player can get extremely close to it with this simple strategy. Follow the first rule to apply.

  1. Make large raise when allowed (with pair of aces or higher).
  2. Make small raise with K-Q-8-4 or higher.
  3. Fold all other.

The increase in house edge with the KQ84 strategy, compared to optimal, is 0.000089%.

Analysis

The next table shows the return of the Ante bet under optimal player strategy.

Ante Bet

EventPaysCombinationsProbabilityReturn
Win11,415,369,375,1480.3550280.355028
Push0473,003,972,8920.1186470.000000
Loss-12,098,272,755,4000.526325-0.526325
Total3,986,646,103,4401.000000-0.171298

The next table shows the return of the Play bet under optimal player strategy. A win of 0 also includes folding, in which case a raise bet was never made.

Raise Bet

WinCombinationsProbabilityReturn
3671,609,661,9480.1684650.505394
11,215,649,215,6840.3049300.304930
0938,265,298,8240.2353520.000000
-11,093,014,959,1960.274169-0.274169
-368,106,967,7880.017084-0.051251
Total3,986,646,103,4401.0000000.484904

The next table shows the return of the Super Bonus bet under optimal player strategy.

Super Bonus Bet

EventPaysCombinationsProbabilityReturn
Four aces20073,629,0720.0000180.003694
Four 2-K30883,548,8640.0002220.006649
Straight flush153,178,321,6080.0007970.011959
Three of a kind289,974,725,9840.0225690.045138
Flush1.5175,813,952,4240.0441010.066151
Straight1156,167,261,7120.0391730.039173
Push01,485,273,310,1400.3725620.000000
Loss-12,075,281,353,6360.520558-0.520558
Total3,986,646,103,4401.000000-0.347795

The next table summarizes the Ante, Play, and Super Bonus bets. The sum shows the player can expect to lose 3.48% for every hand played, compared to the size of his Ante (or Super Bonus) bet. For example, if the player started with $10 on both the Ante and Super Bonus, then he could expect to lose 34.8¢, assuming optimal strategy.

Summary

BetReturn
Ante-0.171298
Raise0.484904
Super Bonus-0.347795
Total-0.034189

The next table shows the net overall win between the Ante, Play, and Super Bonus under optimal player strategy.

Net Win

WinCombinationsProbabilityReturn
20456,580,4320.0000140.002895
20317,048,6400.0000040.000868
34764,060,8080.0001920.006516
33119,340,4800.0000300.000988
26147,5760.0000000.000001
192,708,500,2160.0006790.012908
18467,451,2040.0001170.002111
15239,5440.0000000.000001
112,130,6440.0000010.000006
675,428,689,4240.0189200.113522
5.5140,729,630,9760.0353000.194151
5132,528,726,0360.0332430.166216
4.526,782,817,4360.0067180.030232
4240,544,812,5160.0603380.241351
351,462,003,7800.0129090.038726
2859,165,302,4440.2155110.431022
1.511,157,3840.0000030.000004
1356,744,817,3360.0894850.089485
0842,169,3840.0002110.000000
-2938,364,828,4960.235377-0.470754
-2.58,290,346,6280.002080-0.005199
-31,106,499,736,0320.277552-0.832655
-545,115,566,0240.011317-0.056583
Total3,986,646,103,4401.000000-0.034189

The bottom right cell of the table above shows a house edge of 3.42%. This is the ratio of the expected player loss to the Ante bet. One might argue that since the Super Bonus bet is required I define the house edge as the expected loss to the sum of the required starting bets. However, in the interests of consistency with how the term is defined in other games, I choose to base the house edge on the Ante only. So, for every $100 you bet on the Ante you can expect to lose $3.42 between the Ante, Raise, and Super Bonus combined.

Probability Of Baccarat

The standard deviation is 3.13, based on the Ante bet.

Overall the player has a 18.56% chance of making a big raise, 57.93% for a small raise, and 23.51% for folding, for an average final wager of 3.14 units. Thus, the element of risk of the game (ratio of expected loss to average total bet) is 3.42%/3.14 = 1.09%.

Queens Up

As far as I know, there are four pay tables available for the Queens Up, according to the choice of casino management. Most Las Vegas casinos use pay table 4.

Queens Up Pay Tables

Player HandPay Table 1Pay Table 2Pay Table 3Pay Table 4
Four of a kind50 to 150 to 150 to 150 to 1
Straight flush30 to 140 to 130 to 140 to 1
Three of a kind9 to 18 to 18 to 17 to 1
Flush4 to 14 to 14 to 14 to 1
Straight3 to 13 to 13 to 13 to 1
Two pair2 to 12 to 12 to 12 to 1
Pair of queens or better1 to 11 to 11 to 11 to 1

Queens Up — Pay Table 1

Probability Baccarat
EventPaysCombinationsProbabilityReturn
Four of a kind506240.0002400.012005
Straight flush302,0720.0007970.023917
Three of a kind958,6560.0225690.203121
Flush4114,6160.0441010.176403
Straight3101,8080.0391730.117518
Two pair2123,5520.0475390.095078
Pair of Qs to As1242,9160.0934670.093467
Loser-11,954,7160.752115-0.752115
Total2,598,9601.000000-0.030606

Queens Up — Pay Table 2

EventPaysCombinationsProbabilityReturn
Four of a kind506240.0002400.012005
Straight flush402,0720.0007970.031890
Three of a kind858,6560.0225690.180552
Flush4114,6160.0441010.176403
Straight3101,8080.0391730.117518
Two pair2123,5520.0475390.095078
Pair of Qs to As1242,9160.0934670.093467
Loser-11,954,7160.752115-0.752115
Total2,598,9601.000000-0.045203

Queens Up — Pay Table 3

EventPaysCombinationsProbabilityReturn
Four of a kind506240.0002400.012005
Straight flush302,0720.0007970.023917
Three of a kind858,6560.0225690.180552
Flush4114,6160.0441010.176403
Straight3101,8080.0391730.117518
Two pair2123,5520.0475390.095078
Pair of Qs to As1242,9160.0934670.093467
Loser-11,954,7160.752115-0.752115
Total2,598,9601.000000-0.053175
Probability Baccarat

Queens Up — Pay Table 4

EventPaysCombinationsProbabilityReturn
Four of a kind506240.0002400.012005
Straight flush402,0720.0007970.031890
Three of a kind758,6560.0225690.157983
Flush4114,6160.0441010.176403
Straight3101,8080.0391730.117518
Two pair2123,5520.0475390.095078
Pair of Qs to As1242,9160.0934670.093467
Loser-11,954,7160.752115-0.752115
Total2,598,9601.000000-0.067772

6-Card Bonus

Some casinos add on a side bet known as the 6-Card Bonus. This side bet is found on multiple poker-derivative games, so I created a special page for it. For more information, please see my page on the 6-Card Bonus.

Millionaire Progressive

This is a $5 'red light' progressive side bet that pays $1,000,000 for a royal flush in spades, using the player's five cards. For all the rules and analysis, please see my page on the Millionaire Progressive.

Practice Game

Before you play for real money, practice your Crazy 4 Poker game right here.

Internal Links

There is also a similar game called Four Card Poker.

External Links

Baccarat Probability Calculator

Shufflemaster's official web site for Crazy 4 Poker.


Probability Baccarat

Written by:Michael Shackleford