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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

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