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Baccarat Odds for 1 to 12 Decks
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.
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 |