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Baccarat - the number of combinations of each possible hand

Last Update: Jun 07, 2010

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 Baccarat
Banker total	Player total
Banker 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 Baccarat
Banker total	Player total
Banker 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	Probability
Total Cards - Six Decks
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	Probability
Total Cards - 8 Decks
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.

Player Cards	Banker Cards	Player Total	Banker Total	Combinations	Probability	Winner
Possible Outcomes in 8-Deck Baccarat
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.

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