Exact Winning Side and Total in Baccarat

Introduction

Some baccarat games offer side bets on the exact winning side (or tie) and the winning (or tieing total) total. The image below shows an example of such bets as seen on an electronic table game at the 2024 Global Gaming Expo.

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Analysis

The following table shows the number of combinations of all 28 possible combinations of winning side and total (including ties).

Total	Player Win	Banker Win	Tie
9	613,074,020,540,416	586,850,279,002,112	55,146,054,060,032
8	555,823,197,134,848	529,914,458,673,152	54,879,416,675,072
7	408,043,979,470,848	384,279,324,919,808	101,717,538,899,968
6	312,658,895,192,064	269,232,304,455,680	96,170,001,308,416
5	122,838,277,197,824	216,715,928,915,968	39,684,046,743,808
4	86,165,771,096,064	163,359,790,133,248	36,294,133,463,040
3	62,946,423,310,336	72,927,778,568,192	22,250,510,129,408
2	44,328,525,111,296	44,681,581,871,104	20,006,606,104,576
1	24,639,193,538,560	24,291,119,898,624	20,499,217,668,352
0	0	0	28,979,901,420,544
Total	2,230,518,282,592,260	2,292,252,566,437,890	475,627,426,473,216

The next table shows the probability of the 28 possible outcomes listed above.

Total	Player Win	Banker Win	Tie
9	0.122654	0.117408	0.011033
8	0.111200	0.106017	0.010979
7	0.081635	0.076880	0.020350
6	0.062552	0.053864	0.019240
5	0.024576	0.043357	0.007939
4	0.017239	0.032682	0.007261
3	0.012593	0.014590	0.004452
2	0.008869	0.008939	0.004003
1	0.004929	0.004860	0.004101
0	0.000000	0.000000	0.005798
Total	0.446247	0.458597	0.095156

The next table shows the fair odds on a "to one" basis for all 28 bets above. In other words, these are what a bet should pay to have zero house advantage.

Total	Player Win	Banker Win	Tie
9	7.15	7.52	89.64
8	7.99	8.43	90.08
7	11.25	12.01	48.14
6	14.99	17.57	50.97
5	39.69	22.06	124.95
4	57.01	29.60	136.72
3	78.41	67.54	223.64
2	111.76	110.87	248.84
1	201.86	204.77	242.83
0			171.48

The next table shows the actual pays based on the image shown above.

Total	Player Win	Banker Win	Tie
9	6	7	77
8	7	7.5	77
7	10	10.5	41
6	13	15	44
5	34	19	107
4	49	26	117
3	67	58	190
2	95	95	212
1	172	174	207
0			146

The next table shows the house advantage based on the pay table above.

Total	Player Win	Banker Win	Tie
9	14.14%	6.07%	13.94%
8	11.04%	9.89%	14.36%
7	10.20%	11.59%	14.53%
6	12.43%	13.82%	13.42%
5	13.99%	13.29%	14.25%
4	13.81%	11.76%	14.32%
3	14.37%	13.92%	14.98%
2	14.86%	14.18%	14.74%
1	14.72%	14.95%	14.70%
0			14.77%

For any other pays, I will let you do that math for yourself. I will present two different formulas to get the house advantage.

Key:
F = Fair odds (on "to one" basis)
A = Actual odds (on "to one" basis)
P = Probability of winning

Method 1 — 1 - (A+1)×P
Method 2 — (F-A)/(F+1)

Note that method 2 doesn't require knowing P explicitly.

Let's look at the case of a winning Player total of 9, for example. In this case the variables equal:

Key:
F = 7.153009
A = 6
P = 0.122654

Method 1 — 1 - (6+1)×0.122654 = 14.14%
Method 2 — (7.153009-6)/(7.153009+1) = 14.14%

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