第二節 百家樂收益率的研究
賭戲分析的根本的是研究賭戲的賭博策略和相應的收益率。從表面看來百家樂似乎沒有和二十一點類似的策略，但它和二十一點一樣，用了多副牌，一局牌要玩很多輪，可以預料，百家樂也應該存在著一個浮動的收益率。
從前一節對百家樂基本資料的計算可以看出，這時計算出的是所有的牌都還在牌盒里，一張牌也還沒有發出時的收益率。
在荷官剛洗完牌，游戲尚未開始進行之前的初始狀態，在一副牌中，每種牌平均都有4張；隨著游戲的進行，這種狀態被打破，會出現各種各樣的偏離初始狀態的情形。和研究二十一點採用的方法一樣，我們首先研究比較特殊的情形，即平均到一副牌時，單獨一種牌數量的變化對賭客收益率的影響，以便認識這種牌在百家樂中的作用，從而得到對所有牌作用的認識。
假設某種牌的數量不是4張，而是比其他的牌多出了X張，爲4＋0.5X張，那麼其余的12種牌必須少掉0.5X張才能保持數量上的平衡，爲了研究方便，我們認爲這12種牌的機會都一樣，他們都以相同的概率出現。這樣，在一副牌中多出X張的牌出現的概率爲：
(4＋0.5X )／52＝1/13×(1＋X／8)
其余的牌出現的概率爲：
(4－0.5X／12)／52＝1/13×(1－X／96)
在8副牌的情況下，X的可能取值爲-8爲X爲56。
和前一節的過程類似，我們可以得到對應于每一個X的取值下的收益率，在此我們省略推算過程，直接給出每種牌從少8張到多出25張的情況下，百家樂的收益率。
一 莊
賭客的收益率和一副牌中每種牌多少的關系
| X＝
|
少牌或多牌的種類
|
| 10
|
A
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
| -8
|
-1.003
|
-0.905
|
-0.866
|
-0.921
|
-0.743
|
-1.263
|
-1.528
|
-1.405
|
-1.230
|
-1.159
|
| -7
|
-1.008
|
-0.925
|
-0.894
|
-0.928
|
-0.773
|
-1.250
|
-1.462
|
-1.354
|
-1.211
|
-1.146
|
| -6
|
-1.013
|
-0.945
|
-0.920
|
-0.939
|
-0.806
|
-1.234
|
-1.398
|
-1.306
|
-1.192
|
-1.133
|
| -5
|
-1.020
|
-0.966
|
-0.946
|
-0.953
|
-0.842
|
-1.213
|
-1.336
|
-1.261
|
-1.171
|
-1.120
|
| -4
|
-1.027
|
-0.985
|
-0.971
|
-0.970
|
-0.881
|
-1.189
|
-1.277
|
-1.218
|
-1.151
|
-1.108
|
| -3
|
-1.035
|
-1.005
|
-0.995
|
-0.989
|
-0.923
|
-1.162
|
-1.220
|
-1.177
|
-1.129
|
-1.096
|
| -2
|
-1.044
|
-1.025
|
-1.018
|
-1.012
|
-0.968
|
-1.132
|
-1.166
|
-1.138
|
-1.108
|
-1.085
|
| -1
|
-1.054
|
-1.045
|
-1.041
|
-1.037
|
-1.015
|
-1.099
|
-1.114
|
-1.100
|
-1.086
|
-1.074
|
| 0
|
-1.064
|
-1.064
|
-1.064
|
-1.064
|
-1.064
|
-1.064
|
-1.064
|
-1.064
|
-1.064
|
-1.064
|
| 1
|
-1.075
|
-1.083
|
-1.087
|
-1.094
|
-1.115
|
-1.027
|
-1.016
|
-1.029
|
-1.042
|
-1.054
|
| 2
|
-1.087
|
-1.102
|
-1.109
|
-1.125
|
-1.169
|
-0.987
|
-0.970
|
-0.994
|
-1.019
|
-1.045
|
| 3
|
-1.099
|
-1.121
|
-1.132
|
-1.159
|
-1.224
|
-0.946
|
-0.927
|
-0.960
|
-0.997
|
-1.036
|
| 4
|
-1.113
|
-1.140
|
-1.155
|
-1.195
|
-1.281
|
-0.903
|
-0.885
|
-0.927
|
-0.975
|
-1.028
|
| 5
|
-1.126
|
-1.158
|
-1.179
|
-1.232
|
-1.339
|
-0.859
|
-0.845
|
-0.894
|
-0.953
|
-1.020
|
| 6
|
-1.141
|
-1.176
|
-1.203
|
-1.271
|
-1.398
|
-0.814
|
-0.807
|
-0.862
|
-0.930
|
-1.013
|
| 7
|
-1.156
|
-1.194
|
-1.228
|
-1.312
|
-1.458
|
-0.767
|
-0.771
|
-0.829
|
-0.909
|
-1.007
|
| 8
|
-1.172
|
-1.212
|
-1.254
|
-1.353
|
-1.519
|
-0.720
|
-0.736
|
-0.796
|
-0.887
|
-1.001
|
| 9
|
-1.188
|
-1.230
|
-1.281
|
-1.396
|
-1.580
|
-0.672
|
-0.703
|
-0.763
|
-0.866
|
-0.996
|
| 10
|
-1.204
|
-1.247
|
-1.309
|
-1.440
|
-1.642
|
-0.624
|
-0.672
|
-0.729
|
-0.845
|
-0.992
|
| 11
|
-1.221
|
-1.264
|
-1.339
|
-1.484
|
-1.704
|
-0.575
|
-0.642
|
-0.695
|
-0.824
|
-0.988
|
| 12
|
-1.239
|
-1.281
|
-1.370
|
-1.530
|
-1.766
|
-0.526
|
-0.614
|
-0.661
|
-0.804
|
-0.985
|
| 13
|
-1.257
|
-1.298
|
-1.403
|
-1.576
|
-1.828
|
-0.477
|
-0.587
|
-0.626
|
-0.784
|
-0.982
|
| 14
|
-1.275
|
-1.314
|
-1.437
|
-1.622
|
-1.890
|
-0.428
|
-0.562
|
-0.590
|
-0.765
|
-0.980
|
| 15
|
-1.294
|
-1.330
|
-1.473
|
-1.668
|
-1.951
|
-0.379
|
-0.538
|
-0.553
|
-0.747
|
-0.979
|
| 16
|
-1.313
|
-1.346
|
-1.512
|
-1.715
|
-2.011
|
-0.331
|
-0.515
|
-0.515
|
-0.729
|
-0.979
|
| 17
|
-1.332
|
-1.362
|
-1.552
|
-1.761
|
-2.071
|
-0.284
|
-0.493
|
-0.476
|
-0.712
|
-0.979
|
| 18
|
-1.351
|
-1.377
|
-1.595
|
-1.808
|
-2..130
|
-0.237
|
-0.473
|
-0.436
|
-0.696
|
-0.979
|
| 19
|
-1.371
|
-1.393
|
-1.641
|
-1.854
|
-2.187
|
-0.190
|
-0.453
|
-0.394
|
-0.680
|
-0.981
|
| 20
|
-1.390
|
-1.408
|
-1.689
|
-1.900
|
-2.243
|
-0.145
|
-0.435
|
-0.352
|
-0.665
|
-0.983
|
| 21
|
-1.410
|
-1.422
|
-1.739
|
-1.945
|
-2.298
|
-0.101
|
-0.418
|
-0.308
|
-0.651
|
-0.986
|
| 22
|
-1.429
|
-1.437
|
-1.793
|
-1.989
|
-2.351
|
-0.058
|
-0.402
|
-0.262
|
-0.638
|
-0.989
|
| 23
|
-1.448
|
-1.451
|
-1.850
|
-2.033
|
-2.403
|
-0.016
|
-0.387
|
-0.215
|
-0.626
|
-0.993
|
| 24
|
-1.467
|
-1.465
|
-1.909
|
-2.076
|
-2.452
|
0.024
|
-0.374
|
-0.166
|
-0.614
|
-0.997
|
| 25
|
-1.486
|
-1.479
|
-1.972
|
-2.117
|
-2.500
|
0.063
|
-0.361
|
-0.116
|
-0.604
|
-1.002
|
從表可以看出，X=0時押莊的收益率和前一節計算出的有細微的差別，這是由于這里的計算是根據少牌或多牌的張數對1/13作修正來代替牌實際出現的概率造成的，但結果的精度還是相當高的。
由表可以得出結論，剩牌中“10”、“A”、“2”、“3”、“4”多，押莊的收益率減小，其中以“4”的影響最大，“10“最弱。
剩牌中“5”、“6”、“7”、“8”、“9”多，押莊的收益率增加，其中以“５”的影響最大，“9“最弱。
但押莊的收益率隨X值的變化并不明顯，只有在極爲極端的情況下，才有收益率大于0的情況出現。
爲便于認識每種牌對押莊收益率的改善程度，現在列出相對于X＝0時押莊收益率的變化值。
表7－2－4 賭客的收益率和一副牌中每種牌多少的關系
| X＝
|
少牌或多牌的種類
|
| 10
|
A
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
| -8
|
0.061
|
0.159
|
0.198
|
0.143
|
0.321
|
-0.199
|
-0.464
|
-0.341
|
-0.166
|
-0.095
|
| -7
|
0.056
|
0.139
|
0.170
|
0.136
|
0.291
|
-0.187
|
-0.398
|
-0.290
|
-0.147
|
-0.082
|
| -6
|
0.051
|
0.119
|
0.144
|
0.125
|
0.258
|
-0.170
|
-0.334
|
-0.242
|
-0.128
|
-0.069
|
| -5
|
0.044
|
0.098
|
0.118
|
0.111
|
0.222
|
-0.149
|
-0.272
|
-0.197
|
-0.107
|
-0.056
|
| -4
|
0.037
|
0.079
|
0.093
|
0.094
|
0.183
|
-0.125
|
-0.213
|
-0.154
|
-0.087
|
-0.044
|
| -3
|
0.029
|
0.059
|
0.069
|
0.075
|
0.141
|
-0.098
|
-0.156
|
-0.113
|
-0.065
|
-0.032
|
| -2
|
0.020
|
0.039
|
0.046
|
0.052
|
0.096
|
-0.068
|
-0.102
|
-0.074
|
-0.044
|
-0.021
|
| -1
|
0.010
|
0.019
|
0.023
|
0.027
|
0.049
|
-0.035
|
-0.050
|
-0.036
|
-0.022
|
-0.010
|
| 0
|
0.000
|
0.000
|
0.000
|
0.000
|
0.000
|
0.000
|
0.000
|
0.000
|
0.000
|
0.000
|
| 1
|
-0.011
|
-0.019
|
-0.023
|
-0.030
|
-0.051
|
0.037
|
0.048
|
0.035
|
0.022
|
0.010
|
| 2
|
-0.023
|
-0.038
|
-0.045
|
-0.061
|
-0.105
|
0.077
|
0.094
|
0.070
|
0.045
|
0.019
|
| 3
|
-0.035
|
-0.057
|
-0.068
|
-0.095
|
-0.160
|
0.118
|
0.137
|
0.104
|
0.067
|
0.028
|
| 4
|
-0.049
|
-0.076
|
-0.091
|
-0.131
|
-0.217
|
0.161
|
0.179
|
0.137
|
0.089
|
0.036
|
| 5
|
-0.062
|
-0.094
|
-0.115
|
-0.168
|
-0.275
|
0.205
|
0.219
|
0.170
|
0.111
|
0.044
|
| 6
|
-0.077
|
-0.112
|
-0.139
|
-0.207
|
-0.334
|
0.250
|
0.257
|
0.202
|
0.134
|
0.051
|
| 7
|
-0.092
|
-0.130
|
-0.164
|
-0.248
|
-0.394
|
0.297
|
0.293
|
0.235
|
0.155
|
0.057
|
| 8
|
-0.108
|
-0.148
|
-0.190
|
-0.289
|
-0.455
|
0.344
|
0.328
|
0.268
|
0.177
|
0.063
|
| 9
|
-0.124
|
-0.166
|
-0.217
|
-0.332
|
-0.516
|
0.392
|
0.361
|
0.301
|
0.198
|
0.068
|
| 10
|
-0.140
|
-0.183
|
-0.245
|
-0.376
|
-0.578
|
0.440
|
0.392
|
0.335
|
0.219
|
0.072
|
| 11
|
-0.157
|
-0.200
|
-0.275
|
-0.420
|
-0.640
|
0.489
|
0.422
|
0.369
|
0.240
|
0.076
|
| 12
|
-0.175
|
-0.217
|
-0.306
|
-0.466
|
-0.702
|
0.538
|
0.450
|
0.403
|
0.260
|
0.079
|
| 13
|
-0.193
|
-0.234
|
-0.339
|
-0.512
|
-0.764
|
0.587
|
0.477
|
0.438
|
0.280
|
0.082
|
| 14
|
-0.211
|
-0.250
|
-0.373
|
-0.558
|
-0.826
|
0.636
|
0.502
|
0.474
|
0.299
|
0.084
|
| 15
|
-0.230
|
-0.266
|
-0.409
|
-0.604
|
-0.887
|
0.685
|
0.526
|
0.511
|
0.317
|
0.085
|
| 16
|
-0.249
|
-0.282
|
-0.448
|
-0.651
|
-0.947
|
0.733
|
0.549
|
0.549
|
0.335
|
0.085
|
| 17
|
-0.268
|
-0.298
|
-0.448
|
-0.697
|
-1.007
|
0.780
|
0.571
|
0.588
|
0.352
|
0.085
|
| 18
|
-0.287
|
-0.313
|
-0.531
|
-0.744
|
-1.066
|
0.827
|
0.591
|
0.628
|
0.368
|
0.085
|
| 19
|
-0.307
|
-0.329
|
-0.577
|
-0.790
|
-1.123
|
0.874
|
0.611
|
0.670
|
0.384
|
0.083
|
| 20
|
-0.326
|
-0.344
|
-0.625
|
-0.836
|
-1.179
|
0.919
|
0.629
|
0.712
|
0.399
|
0.081
|
| 21
|
-0.346
|
-0.358
|
-0.675
|
-0.881
|
-1.234
|
0.963
|
0.646
|
0.756
|
0.413
|
0.078
|
| 22
|
-0.365
|
-0.373
|
-0.729
|
-0.925
|
-1.287
|
1.006
|
0.662
|
0.802
|
0.426
|
0.075
|
| 23
|
-0.384
|
-0.387
|
-0.786
|
-0.969
|
-1.339
|
1.048
|
0.677
|
0.849
|
0.438
|
0.071
|
| 24
|
-0.403
|
-0.401
|
-0.845
|
-1.012
|
-1.388
|
1.088
|
0.690
|
0.898
|
0.450
|
0.067
|
| 25
|
-0.422
|
-0.415
|
-0.908
|
-1.053
|
-1.436
|
1.127
|
0.703
|
0.948
|
0.460
|
0.062
|
此表是總結百家樂算牌系統的依據。
二 閑
研究百家樂的收益率是同時得到“莊”、“閑”、“和”的結果，爲了讀起來更方便，我們把“莊”、“閑”、“和”的結果分別列出來，下面是押閑的收益率和一副牌中每種牌多少的關系。
押閑的收益率和一副牌中每種牌多少的關系
| X＝
|
少牌或多牌的種類
|
| 10
|
A
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
| -8
|
-1.285
|
-1.389
|
-1.432
|
-1.373
|
-1.558
|
-1.024
|
-0.759
|
-0.884
|
-1.044
|
-1.122
|
| -7
|
-1.280
|
-1.369
|
-1.404
|
-1.366
|
-1.528
|
-1.038
|
-0.827
|
-0.936
|
-1.066
|
-1.137
|
| -6
|
-1.275
|
-1.348
|
-1.377
|
-1.356
|
-1.494
|
-1.055
|
-0.892
|
-0.985
|
-1.089
|
-1.152
|
| -5
|
-1.269
|
-1.328
|
-1.350
|
-1.342
|
-1.457
|
-1.076
|
-0.954
|
-1.031
|
-1.111
|
-1.166
|
| -4
|
-1.263
|
-1.308
|
-1.325
|
-1.325
|
-1.417
|
-1.101
|
-1.014
|
1.074
|
-1.134
|
-1.180
|
| -3
|
-1.255
|
-1.288
|
-1.300
|
-1.305
|
-1.374
|
-1.128
|
-1.071
|
-1.115
|
-1.158
|
-1.193
|
| -2
|
-1.247
|
-1.268
|
-1.276
|
-1.282
|
-1.328
|
-1.159
|
-1.126
|
-1.155
|
-1.181
|
-1.205
|
| -1
|
-1.238
|
-1.248
|
-1.252
|
-1.256
|
-1.279
|
-1.192
|
-1.178
|
-1.192
|
-1.205
|
-1.217
|
| 0
|
-1.228
|
-1.228
|
-1.228
|
-1.228
|
-1.228
|
-1.228
|
-1.228
|
-1.228
|
-1.228
|
-1.228
|
| 1
|
-1.218
|
-1.209
|
-1.205
|
-1.197
|
-1.175
|
-1.266
|
-1.276
|
-1.263
|
-1.252
|
-1.239
|
| 2
|
-1.206
|
-1.189
|
-1.181
|
-1.164
|
-1.120
|
-1.306
|
-1.321
|
-1.297
|
-1.275
|
-1.249
|
| 3
|
-1.194
|
-1.170
|
-1.157
|
-1.129
|
-1.063
|
-1.348
|
-1.364
|
-1.330
|
-1.298
|
-1.258
|
| 4
|
-1.181
|
-1.151
|
-1.133
|
-1.092
|
-1.004
|
-1.392
|
-1.405
|
-1.362
|
-1.321
|
-1.266
|
| 5
|
-1.168
|
-1.132
|
-1.108
|
-1.053
|
-0.944
|
-1.437
|
-1.444
|
-1.394
|
-1.344
|
-1.274
|
| 6
|
-1.153
|
-1.114
|
-1.083
|
-1.012
|
-0.883
|
-1.483
|
-1.482
|
-1.426
|
-1.367
|
-1.281
|
| 7
|
-1.138
|
-1.095
|
-1.057
|
-0.969
|
-0.821
|
-1.530
|
-1.517
|
-1.458
|
-1.389
|
-1.287
|
| 8
|
-1.123
|
-1.077
|
-1.030
|
-0.926
|
-0.758
|
-1.578
|
-1.551
|
-1.489
|
-1.411
|
-1.293
|
| 9
|
-1.107
|
-1.059
|
-1.002
|
-0.880
|
-0.694
|
-1.627
|
-1.583
|
-1.521
|
-1.432
|
-1.298
|
| 10
|
-1.090
|
-1.042
|
-0.972
|
-0.834
|
-0.630
|
-1.676
|
-1.613
|
-1.553
|
-1.453
|
-1.302
|
| 11
|
-1.072
|
-1.024
|
-0.941
|
-0.787
|
-0.566
|
-1.726
|
-1.641
|
-1.586
|
-1.473
|
-1.306
|
| 12
|
-1.054
|
-1.007
|
-0.909
|
-0.739
|
-0.501
|
-1.776
|
-1.669
|
-1.619
|
-1.493
|
-1.308
|
| 13
|
-1.035
|
-0.990
|
-0.875
|
-0.690
|
-0.437
|
-1.825
|
-1.694
|
-1.653
|
-1.512
|
-1.310
|
| 14
|
-1.016
|
-0.973
|
-0.839
|
-0.640
|
-0.373
|
-1.875
|
-1.719
|
-1.688
|
-1.531
|
-1.311
|
| 15
|
-0.997
|
-0.957
|
-0.801
|
-0.590
|
-0.309
|
-1.925
|
-1.742
|
-1.724
|
-1.548
|
-1.311
|
| 16
|
-0.976
|
-0.940
|
-0.761
|
-0.540
|
-0.246
|
-1.974
|
-1.763
|
-1.761
|
-1.565
|
-1.311
|
| 17
|
-0.956
|
-0.924
|
-0.718
|
-0.489
|
-0.184
|
-2.022
|
-1.784
|
-1.799
|
-1.581
|
-1.310
|
| 18
|
-0.935
|
-0.908
|
-0.673
|
-0.439
|
-0.123
|
-2.070
|
-1.803
|
-1.838
|
-1.596
|
-1.308
|
| 19
|
-0.913
|
-0.893
|
-0.625
|
-0.388
|
-0.063
|
-2.117
|
-1.821
|
-1.878
|
-1.611
|
-1.305
|
| 20
|
-0.891
|
-0.877
|
-0.575
|
-0.338
|
-0.004
|
-2.163
|
-1.838
|
-1.920
|
-1.624
|
-1.301
|
| 21
|
-0.869
|
-0.862
|
-0.521
|
-0.288
|
0.054
|
-0.207
|
-1.854
|
-1.963
|
-1.636
|
-1.297
|
| 22
|
-0.847
|
-0.847
|
-0.465
|
-0.238
|
0.110
|
-2.251
|
-1.869
|
-2.008
|
-1.648
|
-1.292
|
| 23
|
-0.824
|
-0.832
|
-0.405
|
-0.189
|
0.164
|
-2.293
|
-1.883
|
-2.054
|
-1.658
|
-1.286
|
| 24
|
-0.801
|
-0.818
|
-0.343
|
-0.140
|
0.217
|
-2.334
|
-1.896
|
-2.101
|
-1.667
|
-1.279
|
| 25
|
-0.777
|
-0.804
|
-0.276
|
-0.092
|
0.268
|
-2.374
|
-1.907
|
-2.151
|
-1.675
|
-1.272
|
由表可以得出結論，剩牌中“10”、“A”、“2”、“3”、“4”多，押閑的收益率增加，其中以“4”的影響最大，“10“最弱。剩牌中“5”、“6”、“7”、“8”、“9”多，押閑的收益率減少，其中以“５”的影響最大，“9“最弱。但押閑的收益率隨X值的變化也不明顯，只有在極爲極端的情況下，才有收益率大于0的情況出現。爲便于認識每種牌對押閑收益率的改善程度，同樣也列出相對于X＝0時押閑的收益率的變化值。
押閑的收益率和一副牌中每種牌多少的關系
|
X＝ |
少牌或多牌的種類 |
| 10 |
A |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
| -8 |
-0.056 |
-0.161 |
-0.204 |
-0.145 |
-0.330 |
0.204 |
0.469 |
0.345 |
0.184 |
0.107 |
| -7 |
-0.052 |
-0.141 |
-0.176 |
-0.138 |
-0.300 |
0.190 |
0.401 |
0.292 |
0.162 |
0.091 |
| -6 |
-0.047 |
-0.120 |
-0.148 |
-0.127 |
-0.266 |
0.173 |
0.336 |
0.244 |
0.140 |
0.076 |
| -5 |
-0.041 |
-0.100 |
-0.122 |
-0.114 |
-0.229 |
0.152 |
0.274 |
0.198 |
0.117 |
0.062 |
| -4 |
-0.035 |
-0.080 |
-0.097 |
-0.097 |
-0.189 |
0.127 |
0.214 |
0.154 |
0.094 |
0.049 |
| -3 |
-0.027 |
-0.060 |
-0.072 |
-0.077 |
-0.145 |
0.100 |
0.157 |
0.113 |
0.070 |
0.035 |
| -2 |
-0.019 |
-0.040 |
-0.048 |
-0.054 |
-0.099 |
0.069 |
0.102 |
0.074 |
0.047 |
0.023 |
| -1 |
-0.010 |
-0.020 |
-0.024 |
-0.028 |
-0.051 |
0.036 |
0.050 |
0.036 |
0.024 |
0.010 |
| 0 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
| 1 |
0.011 |
0.020 |
0.024 |
0.031 |
0.053 |
-0.038 |
-0.048 |
-0.035 |
-0.024 |
-0.011 |
| 2 |
0.022 |
0.039 |
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