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1至12副牌的百家樂賠率

介紹

常常有人問我百家樂中不同牌組數的賠率。為了滿足我的所有需求，本頁將顯示從1到12的牌組數對應的組合數、獲勝機率和預期回報。

組合

下表顯示了三種可能結果的組合數，以及牌堆數量。本頁的組合數相當龐大。可惜的是，Excel 只顯示 15 位有效數字，所以請不要寫信告訴我 15 位之後的結果我錯了。請怪罪微軟。

組合

甲板玩家銀行家領帶全部的
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

下表按牌組數量顯示了每種結果的機率。

機率

甲板玩家銀行家領帶
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

下表顯示了不同牌組數量下每種結果的預期回報。請注意，隨著牌組數量的增加，押閒牌和平局的預期回報會增加，押莊牌的預期回報則會減少。

預期收益

甲板玩家銀行家領帶
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

對子投注

下一組表格用於投注閒家對子和莊家對子。預期值表是基於通常的11比1賠率。下表按牌組數量顯示了勝負組合的數量。

對子投注組合

甲板贏損失全部的
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

下表顯示了按牌組數量計算的獲勝和失敗機率。

對子投注機率

甲板機率獲勝機率損失
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

下表顯示了不同牌組數量的預期收益。如你所見，隨著牌組數量的增加，預期收益也會增加。這顯而易見，因為隨著牌組數量的增加，移除牌組的影響會變得不那麼顯著。

對子投注預期回報

甲板預期回報
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

撰寫者: Michael Shackleford