Baccarat: The number of combinatinos of each possible hand

Last update: September 6, 2006

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 - 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.

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