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Exact Winning Side and Total in Baccarat
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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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