Probability Baccarat

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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

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.
The dealer gives the player two hole cards and places five community cards face down in a cross pattern.
The player may make an Across Bet, based on the Across Hand, equal to 1x to 3x the Ante Across bet, or fold.
The dealer will reveal the two outside cards on the horizontal row of the cross.
The player may make a Down bet, based on the Down Hand, equal to 1x to 3x the Ante Down bet, or fold.
The dealer will reveal the two outside cards on the vertical column of the cross.
The player may make a Middle Bet of 1x to 3x either Ante bet or fold.
The dealer will reveal the middle card.
The player uses his two hole cards and the three cards on the horizontal row of the cross to form the Across Hand.
Likewise, the player uses his two hole cards and the three cards on the vertical column of the cross to form the Down Hand.
The Middle Bet pays according to the higher ranking of the Across and Down hands.

All bets pay according to the pay table below. Wins are expressed on a 'to one' basis.

Criss Cross Poker Pay Table

Hand	Ante Bets Pay	Across, Down & Middle Bets Pay
Royal flush	1	500
Straight flush	1	100
Four of a kind	1	40
Full house	1	12
Flush	1	8
Straight	1	5
Three of a kind	1	3
Two pair	1	2
High pair (J-A)	1	1
Low pair (6-10)	0	0
All other	-1	-1

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
Hand Pays
Royal flush 250
Straight flush 100
Four of a kind 40
Full house 15
Flush 10
Straight 6
Three of a kind 4
Two pair 3
Sixes or better 1
Loser -1

Hand	Pays
Royal flush	250
Straight flush	100
Four of a kind	40
Full house	15
Flush	10
Straight	6
Three of a kind	4
Two pair	3
Sixes or better	1
Loser	-1

Analysis

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

Hand	Across Bet	Pays	Combinations	Probability	Return
Royal flush	3	1501	12,972	0.000000	0.000231
Royal flush	1	501	116,748	0.000001	0.000694
Straight flush	3	301	25,944	0.000000	0.000093
Straight flush	1	101	1,141,536	0.000014	0.001368
Four of a kind	3	121	12,167,736	0.000144	0.017468
Four of a kind	1	41	7,601,592	0.000090	0.003698
Full house	3	37	48,800,664	0.000579	0.021423
Full house	1	13	68,414,328	0.000812	0.010552
Flush	3	25	2,101,464	0.000025	0.000623
Flush	1	9	163,550,976	0.001940	0.017464
Straight	3	16	2,451,708	0.000029	0.000465
Straight	1	6	288,484,308	0.003423	0.020537
Three of a kind	3	10	538,234,224	0.006386	0.063859
Three of a kind	1	4	1,170,645,168	0.013889	0.055557
Two pair	3	7	811,632,096	0.009630	0.067408
Two pair	1	3	3,010,230,432	0.035715	0.107146
High pair	3	4	1,170,593,280	0.013889	0.055555
High pair	1	2	9,458,722,620	0.112224	0.224448
Medium pair	3	0	1,385,409,600	0.016437	0.000000
Medium pair	1	0	11,610,887,160	0.137759	0.000000
Loser	3	-4	1,240,719,912	0.014721	-0.058883
Loser	1	-2	44,512,795,932	0.528127	-1.056254
Fold one unit	0	-1	4,576,521,600	0.054299	-0.054299
Fold two units	0	-2	4,203,010,800	0.049867	-0.099734
Fold four units	0	-4	0	0.000000	0.000000
Total	84,284,272,800	1.000000	-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

Hand	Down Bet	Pays	Combinations	Probability	Return
Royal flush	3	1501	45,792	0.000001	0.000815
Royal flush	1	501	83,928	0.000001	0.000499
Straight flush	3	301	37,116	0.000000	0.000133
Straight flush	1	101	1,118,700	0.000013	0.001341
Four of a kind	3	121	12,233,808	0.000145	0.017563
Four of a kind	1	41	7,485,048	0.000089	0.003641
Full house	3	37	48,603,888	0.000577	0.021337
Full house	1	13	68,156,856	0.000809	0.010513
Flush	3	25	6,951,456	0.000082	0.002062
Flush	1	9	157,641,288	0.001870	0.016833
Straight	3	16	4,780,980	0.000057	0.000908
Straight	1	6	285,297,516	0.003385	0.020310
Three of a kind	3	10	540,593,136	0.006414	0.064139
Three of a kind	1	4	1,160,967,816	0.013774	0.055098
Two pair	3	7	814,576,176	0.009665	0.067652
Two pair	1	3	2,988,662,184	0.035459	0.106378
High pair	3	4	1,336,724,496	0.015860	0.063439
High pair	1	2	9,281,584,656	0.110122	0.220245
Medium pair	3	0	1,417,841,568	0.016822	0.000000
Medium pair	1	0	11,550,329,616	0.137040	0.000000
Loser	3	-4	1,450,469,088	0.017209	-0.068837
Loser	1	-2	44,370,555,288	0.526439	-1.052879
Fold one unit	0	-1	5,013,003,456	0.059477	-0.059477
Fold two units	0	-2	3,766,528,944	0.044688	-0.089377
Fold four units	0	-4	0	0.000000	0.000000
Total	84,284,272,800	1.000000	-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

Hand	Middle Bet	Pays	Combinations	Probability	Return
Royal flush	3	1500	259,440	0.000003	0.004617
Royal flush	1	500	0	0.000000	0.000000
Straight flush	3	300	2,323,104	0.000028	0.008269
Straight flush	1	100	0	0.000000	0.000000
Four of a kind	3	120	39,488,184	0.000469	0.056221
Four of a kind	1	40	0	0.000000	0.000000
Full house	3	36	231,865,848	0.002751	0.099036
Full house	1	12	0	0.000000	0.000000
Flush	3	24	325,921,056	0.003867	0.092806
Flush	1	8	0	0.000000	0.000000
Straight	3	15	375,011,412	0.004449	0.066740
Straight	1	5	199,154,268	0.002363	0.011814
Three of a kind	3	9	2,751,292,188	0.032643	0.293787
Three of a kind	1	3	399,927,564	0.004745	0.014235
Two pair	3	6	5,494,091,328	0.065185	0.391111
Two pair	1	2	1,329,787,368	0.015777	0.031555
High pair	3	3	11,485,872,240	0.136275	0.408826
High pair	1	1	4,170,202,716	0.049478	0.049478
Medium pair	3	0	12,309,204,636	0.146044	0.000000
Medium pair	1	0	4,837,617,228	0.057396	0.000000
Loser	3	-3	1,789,381,740	0.021230	-0.063691
Loser	1	-1	29,763,340,080	0.353130	-0.353130
Fold	0	0	8,779,532,400	0.104166	0.000000
Total	84,284,272,800	1.000000	1.111675

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
Middle	1.111675
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

Baccarat Probability Theory

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.