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Baccarat - The Number of Combinations of Each Possible Hand

Introduction

The following table shows the number of combinations of each possible combination of the banker's and player's total score. This table assumes an 8-deck shoe.

Combinations in BaccaratExpand

Banker
total		 Player total
0 1 2 3 4 5 6 7 8 9 Total
0 28979901420544 24639193538560 24214066700288 23748210556928 24642962657280 25228523110400 57372381863936 58350707818496 88188576587776 88370543349760 443735067603968
1 24291119898624 20499217668352 20114458411008 19634953029632 20535822327808 21127406358528 48967402696704 49945260759040 60551359631360 60628881760256 346295882541312
2 24211866148864 20469715722240 20006606104576 19563259723776 20465430315008 21055697346560 48870027180032 49787546013696 60325437726720 60401880698880 345157466980352
3 26957461020672 23203622991872 22766694555648 22250510129408 20521555795968 21104461946880 48999874119680 49957422518272 64689446563840 63310197227520 363761246869760
4 46033607532544 40593010774016 38585331400704 38147840425984 36294133463040 34322188435456 51520246087680 49835600281600 64455840866304 67209934278656 466997733545984
5 46171612971008 40755264745472 42895386193920 45111417946112 41782247059456 39684046743808 56928963244032 55257433686016 67329651249152 67443153604608 503359177443584
6 48639204401152 40608057946112 42768049588224 44979215931392 45802040381440 46435736207360 96170001308416 94910008393728 72372438790144 72499777142784 605184530090752
7 54083703291904 46026293614592 45544433684480 45155412082688 45975559438336 46610049437696 100883873370112 101717538899968 77910445719552 78043684069376 641950993608704
8 85076577230848 57774350200832 57197896224768 56875337277440 57616152412160 58332350017536 77985646493696 79056148815872 54879416675072 55165968408576 639959843756800
9 85351454494720 57948328357888 57372732461056 57051058384896 57790127128576 58535141937152 78221719912448 79299874000896 55279842324480 55146054060032 641996333062144
Total 469796508410880 372517055559936 371465655324672 372517215488256 371426030979072 372435601541376 665920136276736 668117541187584 665982456134400 668220074600448 4998398275503360

The next table shows the probability of each final combination based on an eight-deck game.

Combinations in BaccaratExpand

Banker
total		 Player total
0 1 2 3 4 5 6 7 8 9 Total
0 0.005798 0.004929 0.004844 0.004751 0.004930 0.005047 0.011478 0.011674 0.017643 0.017680 0.088775
1 0.004860 0.004101 0.004024 0.003928 0.004108 0.004227 0.009797 0.009992 0.012114 0.01213 0.069281
2 0.004844 0.004095 0.004003 0.003914 0.004094 0.004212 0.009777 0.009961 0.012069 0.012084 0.069054
3 0.005393 0.004642 0.004555 0.004452 0.004106 0.004222 0.009803 0.009995 0.012942 0.012666 0.072776
4 0.009210 0.008121 0.007720 0.007632 0.007261 0.006867 0.010307 0.009970 0.012895 0.013446 0.093429
5 0.009237 0.008154 0.008582 0.009025 0.008359 0.007939 0.011389 0.011055 0.013470 0.013493 0.100704
6 0.009731 0.008124 0.008556 0.008999 0.009163 0.009290 0.019240 0.018988 0.014479 0.014505 0.121076
7 0.010820 0.009208 0.009112 0.009034 0.009198 0.009325 0.020183 0.020350 0.015587 0.015614 0.128431
8 0.017021 0.011559 0.011443 0.011379 0.011527 0.01167 0.015602 0.015816 0.010979 0.011037 0.128033
9 0.017076 0.011593 0.011478 0.011414 0.011562 0.011711 0.015649 0.015865 0.011060 0.011033 0.128440
Total 0.093989 0.074527 0.074317 0.074527 0.074309 0.074511 0.133227 0.133666 0.133239 0.133687 1

The next table shows the probabilities of each total after two cards.

2-Card Combinations in Baccarat

Total 6-deck 8-deck
0 0.147168 0.147359
1 0.094979 0.094903
2 0.094484 0.094532
3 0.094979 0.094903
4 0.094484 0.094532
5 0.094979 0.094903
6 0.094484 0.094532
7 0.094979 0.094903
8 0.094484 0.094532
9 0.094979 0.094903
Total 1 1

The next table shows the number of player points by number of player cards for six decks.

Total Player Points by Number of Cards - Six Decks

Total
Points		 2 Cards 3 Cards Total
0 0.027914 0.065929 0.093843
1 0.018003 0.056574 0.074577
2 0.017909 0.056386 0.074295
3 0.018003 0.056574 0.074577
4 0.01791 0.056375 0.074285
5 0.018003 0.056552 0.074555
6 0.094484 0.038679 0.133163
7 0.094979 0.03877 0.133749
8 0.094484 0.038695 0.13318
9 0.094979 0.038798 0.133777
Total 0.496668 0.503332 1

The next table shows the number of banker points by number of banker cards for six decks.

Total Banker Points by Number of Cards - Six Decks

Total
Points		 2 Cards 3 Cards Total
0 0.027914 0.060717 0.088632
1 0.018003 0.051319 0.069322
2 0.017909 0.051109 0.069018
3 0.022552 0.050269 0.072821
4 0.049534 0.043863 0.093398
5 0.058871 0.041906 0.100778
6 0.085496 0.035526 0.121022
7 0.094979 0.033531 0.12851
8 0.094484 0.033494 0.127978
9 0.094979 0.033543 0.128522
Total 0.564722 0.435278 1

The next table shows the distribution of the total number of cards between banker and player in a six-deck game.

Total Cards - Six Decks

Total
Cards		 Probability
4 0.378925
5 0.30354
6 0.317535
Total cards 1

In a six-deck the average number of player cards is 2.503332474, banker cards is 2.435277628, and total cards is 4.938610102.

The next table shows the number of player points by number of player cards for Eight decks.

Total Player Points by Number of Cards - 8 Decks

Total
Points		 2 Cards 3 Cards Total
0 0.027939 0.066051 0.093989
1 0.017984 0.056544 0.074527
2 0.017913 0.056404 0.074317
3 0.017984 0.056544 0.074527
4 0.017914 0.056395 0.074309
5 0.017984 0.056527 0.074511
6 0.094532 0.038695 0.133227
7 0.094903 0.038764 0.133666
8 0.094532 0.038707 0.133239
9 0.094903 0.038784 0.133687
Total 0.496586 0.503414 1

The next table shows the number of banker points by number of banker cards for Eight decks.

Total Banker Points by Number of Cards - 8 Decks

Total
Points		 2 Cards 3 Cards Total
0 0.027939 0.060837 0.088775
1 0.017984 0.051298 0.069281
2 0.017913 0.05114 0.069054
3 0.022527 0.050249 0.072776
4 0.049551 0.043879 0.093429
5 0.058816 0.041888 0.100704
6 0.085529 0.035547 0.121076
7 0.094903 0.033529 0.128431
8 0.094532 0.033501 0.128033
9 0.094903 0.033538 0.12844
Total 0.564595 0.435405 1

The next table shows the distribution of the total number of cards between banker and player in an Eight-deck game.

Total Cards - 8 Decks

Total
Cards		 Probability
4 0.378868
5 0.303444
6 0.317687
Total 1

In an 8-deck the average number of player cards is 2.50341377, banker cards is 2.43540508, and total cards is 4.93881885.

The next table shows all possible final outcomes in baccarat, by the number of player and banker cards, and the player and banker total.

Possible Outcomes in 8-Deck Baccarat

Player
Cards		 Banker
Cards		 Player
Total		 Banker
Total		 Combinations Probability Winner
2 2 0 8 69,686,799,163,392 0.013942 Banker
2 2 0 9 69,961,111,584,768 0.013997 Banker
2 2 1 8 44,856,843,337,728 0.008974 Banker
2 2 1 9 45,033,013,641,216 0.009009 Banker
2 2 2 8 44,681,366,618,112 0.008939 Banker
2 2 2 9 44,856,843,337,728 0.008974 Banker
2 2 3 8 44,856,843,337,728 0.008974 Banker
2 2 3 9 45,033,013,641,216 0.009009 Banker
2 2 4 8 44,683,447,369,728 0.008940 Banker
2 2 4 9 44,856,843,337,728 0.008974 Banker
2 2 5 8 44,856,843,337,728 0.008974 Banker
2 2 5 9 45,033,013,641,216 0.009009 Banker
2 2 6 6 44,487,098,110,464 0.008900 Tie
2 2 6 7 44,856,843,337,728 0.008974 Banker
2 2 6 8 44,683,447,369,728 0.008940 Banker
2 2 6 9 44,856,843,337,728 0.008974 Banker
2 2 7 6 44,856,843,337,728 0.008974 Player
2 2 7 7 44,838,810,157,056 0.008971 Tie
2 2 7 8 44,856,843,337,728 0.008974 Banker
2 2 7 9 45,033,013,641,216 0.009009 Banker
2 2 8 0 69,686,799,163,392 0.013942 Player
2 2 8 1 44,856,843,337,728 0.008974 Player
2 2 8 2 44,681,366,618,112 0.008939 Player
2 2 8 3 44,856,843,337,728 0.008974 Player
2 2 8 4 44,683,447,369,728 0.008940 Player
2 2 8 5 44,856,843,337,728 0.008974 Player
2 2 8 6 44,683,447,369,728 0.008940 Player
2 2 8 7 44,856,843,337,728 0.008974 Player
2 2 8 8 44,487,098,110,464 0.008900 Tie
2 2 8 9 44,858,924,089,344 0.008975 Banker
2 2 9 0 69,961,111,584,768 0.013997 Player
2 2 9 1 45,033,013,641,216 0.009009 Player
2 2 9 2 44,856,843,337,728 0.008974 Player
2 2 9 3 45,033,013,641,216 0.009009 Player
2 2 9 4 44,856,843,337,728 0.008974 Player
2 2 9 5 45,033,013,641,216 0.009009 Player
2 2 9 6 44,856,843,337,728 0.008974 Player
2 2 9 7 45,033,013,641,216 0.009009 Player
2 2 9 8 44,858,924,089,344 0.008975 Player
2 2 9 9 44,838,810,157,056 0.008971 Tie
2 3 6 0 38,525,195,292,672 0.007708 Player
2 3 6 1 32,964,477,505,536 0.006595 Player
2 3 6 2 32,924,126,748,672 0.006587 Player
2 3 6 3 33,006,509,193,216 0.006603 Player
2 3 6 4 32,923,813,625,856 0.006587 Player
2 3 6 5 33,005,365,284,864 0.006603 Player
2 3 6 6 22,532,336,357,376 0.004508 Tie
2 3 6 7 22,593,907,077,120 0.004520 Banker
2 3 6 8 22,554,580,703,232 0.004512 Banker
2 3 6 9 22,593,912,127,488 0.004520 Banker
2 3 7 0 38,676,532,936,704 0.007738 Player
2 3 7 1 33,094,381,387,776 0.006621 Player
2 3 7 2 33,011,870,158,848 0.006604 Player
2 3 7 3 33,135,282,634,752 0.006629 Player
2 3 7 4 33,053,237,723,136 0.006613 Player
2 3 7 5 33,135,122,706,432 0.006629 Player
2 3 7 6 22,683,042,705,408 0.004538 Player
2 3 7 7 22,661,358,108,672 0.004534 Tie
2 3 7 8 22,642,153,242,624 0.004530 Banker
2 3 7 9 22,682,938,331,136 0.004538 Banker
3 2 0 3 3,456,714,276,864 0.000692 Banker
3 2 0 4 28,249,492,660,224 0.005652 Banker
3 2 0 5 28,361,385,246,720 0.005674 Banker
3 2 0 6 31,665,746,755,584 0.006335 Banker
3 2 0 7 38,676,532,936,704 0.007738 Banker
3 2 1 3 3,455,256,403,968 0.000691 Banker
3 2 1 4 26,104,503,730,176 0.005223 Banker
3 2 1 5 26,207,637,295,104 0.005243 Banker
3 2 1 6 26,105,087,889,408 0.005223 Banker
3 2 1 7 33,094,381,387,776 0.006621 Banker
3 2 2 3 3,456,714,276,864 0.000692 Banker
3 2 2 4 24,074,105,966,592 0.004816 Banker
3 2 2 5 29,582,414,401,536 0.005918 Banker
3 2 2 6 29,467,414,155,264 0.005895 Banker
3 2 2 7 33,011,870,158,848 0.006604 Banker
3 2 3 3 3,470,505,148,416 0.000694 Tie
3 2 3 4 24,155,341,135,872 0.004833 Banker
3 2 3 5 33,120,035,573,760 0.006626 Banker
3 2 3 6 33,006,509,193,216 0.006603 Banker
3 2 3 7 33,135,282,634,752 0.006629 Banker
3 2 4 4 20,627,229,806,592 0.004127 Tie
3 2 4 5 27,638,627,082,240 0.005529 Banker
3 2 4 6 32,923,813,625,856 0.006587 Banker
3 2 4 7 33,053,237,723,136 0.006613 Banker
3 2 5 4 17,254,903,812,096 0.003452 Player
3 2 5 5 24,206,575,435,776 0.004843 Tie
3 2 5 6 33,005,365,284,864 0.006603 Banker
3 2 5 7 33,135,122,706,432 0.006629 Banker
3 2 6 4 3,456,875,888,640 0.000692 Player
3 2 6 5 10,341,852,352,512 0.002069 Player
3 2 6 6 17,217,820,643,328 0.003445 Tie
3 2 6 7 22,683,042,705,408 0.004538 Banker
3 2 7 5 6,911,968,997,376 0.001383 Player
3 2 7 6 13,800,450,416,640 0.002761 Player
3 2 7 7 22,661,358,108,672 0.004534 Tie
3 2 8 3 5,398,882,111,488 0.001080 Player
3 2 8 4 5,377,630,162,944 0.001076 Player
3 2 8 5 8,854,301,810,688 0.001771 Player
3 2 8 6 15,694,721,402,880 0.003140 Player
3 2 8 7 22,642,153,242,624 0.004530 Player
3 2 9 3 3,470,505,148,416 0.000694 Player
3 2 9 4 8,834,344,439,808 0.001767 Player
3 2 9 5 8,869,388,943,360 0.001774 Player
3 2 9 6 15,734,623,518,720 0.003148 Player
3 2 9 7 22,682,938,331,136 0.004538 Player
3 3 0 0 28,979,901,420,544 0.005798 Tie
3 3 0 1 24,291,119,898,624 0.004860 Banker
3 3 0 2 24,211,866,148,864 0.004844 Banker
3 3 0 3 23,500,746,743,808 0.004702 Banker
3 3 0 4 17,784,114,872,320 0.003558 Banker
3 3 0 5 17,810,227,724,288 0.003563 Banker
3 3 0 6 16,973,457,645,568 0.003396 Banker
3 3 0 7 15,407,170,355,200 0.003082 Banker
3 3 0 8 15,389,778,067,456 0.003079 Banker
3 3 0 9 15,390,342,909,952 0.003079 Banker
3 3 1 0 24,639,193,538,560 0.004929 Player
3 3 1 1 20,499,217,668,352 0.004101 Tie
3 3 1 2 20,469,715,722,240 0.004095 Banker
3 3 1 3 19,748,366,587,904 0.003951 Banker
3 3 1 4 14,488,507,043,840 0.002899 Banker
3 3 1 5 14,547,627,450,368 0.002910 Banker
3 3 1 6 14,502,970,056,704 0.002902 Banker
3 3 1 7 12,931,912,226,816 0.002587 Banker
3 3 1 8 12,917,506,863,104 0.002584 Banker
3 3 1 9 12,915,314,716,672 0.002584 Banker
3 3 2 0 24,214,066,700,288 0.004844 Player
3 3 2 1 20,114,458,411,008 0.004024 Player
3 3 2 2 20,006,606,104,576 0.004003 Tie
3 3 2 3 19,309,980,278,784 0.003863 Banker
3 3 2 4 14,511,225,434,112 0.002903 Banker
3 3 2 5 13,312,971,792,384 0.002663 Banker
3 3 2 6 13,300,635,432,960 0.002661 Banker
3 3 2 7 12,532,563,525,632 0.002507 Banker
3 3 2 8 12,516,529,606,656 0.002504 Banker
3 3 2 9 12,515,889,123,328 0.002504 Banker
3 3 3 0 23,748,210,556,928 0.004751 Player
3 3 3 1 19,634,953,029,632 0.003928 Player
3 3 3 2 19,563,259,723,776 0.003914 Player
3 3 3 3 18,780,004,980,992 0.003757 Tie
3 3 3 4 13,992,499,290,112 0.002799 Banker
3 3 3 5 11,991,382,372,352 0.002399 Banker
3 3 3 6 11,972,706,738,176 0.002395 Banker
3 3 3 7 12,020,129,447,936 0.002405 Banker
3 3 3 8 12,018,493,939,712 0.002404 Banker
3 3 3 9 12,018,044,743,680 0.002404 Banker
3 3 4 0 24,642,962,657,280 0.004930 Player
3 3 4 1 20,535,822,327,808 0.004108 Player
3 3 4 2 20,465,430,315,008 0.004094 Player
3 3 4 3 20,521,555,795,968 0.004106 Player
3 3 4 4 15,666,903,656,448 0.003134 Tie
3 3 4 5 14,143,619,977,216 0.002830 Banker
3 3 4 6 12,878,226,755,584 0.002576 Banker
3 3 4 7 12,922,321,715,200 0.002585 Banker
3 3 4 8 12,932,705,042,432 0.002587 Banker
3 3 4 9 12,933,283,790,848 0.002587 Banker
3 3 5 0 25,228,523,110,400 0.005047 Player
3 3 5 1 21,127,406,358,528 0.004227 Player
3 3 5 2 21,055,697,346,560 0.004212 Player
3 3 5 3 21,104,461,946,880 0.004222 Player
3 3 5 4 17,067,284,623,360 0.003415 Player
3 3 5 5 15,477,471,308,032 0.003096 Tie
3 3 5 6 13,430,370,922,496 0.002687 Banker
3 3 5 7 13,474,926,731,264 0.002696 Banker
3 3 5 8 13,475,506,679,808 0.002696 Banker
3 3 5 9 13,502,128,295,936 0.002701 Banker
3 3 6 0 18,847,186,571,264 0.003771 Player
3 3 6 1 16,002,925,191,168 0.003202 Player
3 3 6 2 15,945,900,431,360 0.003190 Player
3 3 6 3 15,993,364,926,464 0.003200 Player
3 3 6 4 15,139,556,573,184 0.003029 Player
3 3 6 5 13,581,745,606,656 0.002717 Player
3 3 6 6 11,932,746,197,248 0.002387 Tie
3 3 6 7 10,750,080,249,856 0.002151 Banker
3 3 6 8 10,747,618,420,736 0.002150 Banker
3 3 6 9 10,770,964,447,232 0.002155 Banker
3 3 7 0 19,674,174,881,792 0.003936 Player
3 3 7 1 16,850,879,371,264 0.003371 Player
3 3 7 2 16,775,675,854,848 0.003356 Player
3 3 7 3 16,822,139,883,520 0.003366 Player
3 3 7 4 16,782,362,558,464 0.003358 Player
3 3 7 5 15,210,341,982,208 0.003043 Player
3 3 7 6 13,569,671,933,952 0.002715 Player
3 3 7 7 11,556,012,525,568 0.002312 Tie
3 3 7 8 11,557,152,235,520 0.002312 Banker
3 3 7 9 11,583,922,028,544 0.002318 Banker
3 3 8 0 18,501,777,424,384 0.003702 Player
3 3 8 1 15,694,516,293,632 0.003140 Player
3 3 8 2 15,644,071,108,608 0.003130 Player
3 3 8 3 14,433,721,114,624 0.002888 Player
3 3 8 4 14,394,763,333,632 0.002880 Player
3 3 8 5 13,618,506,100,736 0.002725 Player
3 3 8 6 11,994,270,017,536 0.002400 Player
3 3 8 7 10,411,449,139,200 0.002083 Player
3 3 8 8 10,392,318,564,608 0.002079 Tie
3 3 8 9 10,420,918,235,136 0.002085 Banker
3 3 9 0 18,409,431,764,992 0.003683 Player
3 3 9 1 15,595,868,119,040 0.003120 Player
3 3 9 2 15,545,037,361,152 0.003110 Player
3 3 9 3 14,806,678,437,888 0.002962 Player
3 3 9 4 13,518,746,501,120 0.002705 Player
3 3 9 5 13,540,751,020,032 0.002709 Player
3 3 9 6 11,908,310,286,336 0.002382 Player
3 3 9 7 10,327,732,097,024 0.002066 Player
3 3 9 8 10,307,044,319,232 0.002062 Player
3 3 9 9 10,307,243,902,976 0.002062 Tie

In case you're wondering, the most common outcome is a tie between a 9-0 Player win and 9-0 Banker win, both with two cards in each hand. The least common outcome is a 3-card Player total of 1 against a 2-card Banker total of 3.

Go back to baccarat
Go to baccarat appendix 2
Go to baccarat appendix 3
Go to baccarat appendix 4
Go to baccarat appendix 5
Go to baccarat appendix 6
Go to baccarat appendix 7

Written by: Michael Shackleford