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百家乐——每种可能牌型的组合数

簡介

下表顯示了莊家和玩家總點數每種可能組合的組合數量。此表基於使用八副牌組的牌靴。

百家樂中的組合擴展

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

下表基於八副牌遊戲，展示了每種最終組合的機率。

百家樂中的組合擴展

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

下表顯示兩張牌後每種總和的概率。

百家樂中的兩張牌組合

Total 6副牌 8副牌
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

下表顯示六副牌中玩家持牌數與對應點數的關係。

玩家總點數按卡牌數量計 - 六副牌

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

下表顯示了六副牌中根據莊家牌數所對應的莊家點數。

根據牌數的莊家點數總和 - 六副牌

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

下一張表格顯示了在六副牌遊戲中，莊家與玩家之間總牌數的分佈情況。

總牌數 - 六副牌

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

在六副牌的遊戲中，玩家牌的平均數量為2.503332474，莊家牌的平均數量為2.435277628，總牌數為4.938610102。

下表顯示了使用八副牌時，玩家點數與玩家牌數的對應關係。

玩家總點數按牌數計 - 8副牌

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

下表顯示了使用八副牌時，根據莊家牌數所對應的莊家點數。

根據牌數的莊家總點數 - 8副牌

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

下一張表格顯示了在八副牌的遊戲中，莊家和閒家之間總牌數的分佈情況。

總卡牌數 - 8副牌組

Total
Cards		 Probability
4 0.378868
5 0.303444
6 0.317687
Total 1

在八副牌的遊戲中，玩家牌的平均數量為2.50341377，莊家牌的平均數量為2.43540508，總牌數為4.93881885。

下表按玩家和莊家的牌數及點數總和，列出百家樂所有可能的最終結果。

8副牌百家樂的可能結果

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

如果你好奇的話，最常見的結果是「閒家9-0勝」與「莊家9-0勝」並列，兩者皆為兩張牌成手。而最罕見的結果則是閒家以三張牌總點1，對上莊家兩張牌總點3的局面。

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

撰寫者: Michael Shackleford