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Blackjack expected Returns for four decks and dealer hits on soft 17.

Introduction

Following are the expected returns for each initial hands in blackjack under the following rules.

  • Four decks
  • Dealer hits on soft 17
  • Player may resplit to four hands, except aces
  • No drawing to split aces

There are three tables, the first for when the player has two cards of different ranks and no dealer blackjack, the second when the player has two cards of the same rank and no dealer blackjack, and the third for when the dealer has a blackjack. In the second table there are two columns for splitting, one in which the player may double after a split and the other in which the player may not.

The values in this table were created using a recursive combinatorial program that considered the exact hand composition at every step. For splitting values the program only considered how many hands the player had already split, but not the other cards drawn in those hands.

Expected return of game:

  • Player may double after split: -0.005568
  • Player may not double after split: -0.007018

 

Blackjack Appendix 9 — 4 Decks, Dealer Hits on Soft 17 — Player Can't Split

Dealer Player Stand Hit Double Probability
A 2,A -0.596474 -0.101172 -0.581196 0.00059556
A 3,A -0.595965 -0.135655 -0.580368 0.00059556
A 3,2 -0.595457 -0.318446 -1.190914 0.00063526
A 4,A -0.595416 -0.173615 -0.585335 0.00059556
A 4,2 -0.594949 -0.346188 -1.177880 0.00063526
A 4,3 -0.594461 -0.355533 -1.083119 0.00063526
A 5,A -0.595655 -0.212274 -0.594241 0.00059556
A 5,2 -0.595170 -0.353184 -1.084618 0.00063526
A 5,3 -0.594650 -0.267129 -0.828680 0.00063526
A 5,4 -0.594091 -0.124178 -0.448239 0.00063526
A 6,A -0.513575 -0.222214 -0.546802 0.00059556
A 6,2 -0.594186 -0.267109 -0.830184 0.00063526
A 6,3 -0.593668 -0.124752 -0.450110 0.00063526
A 6,4 -0.593115 0.035513 -0.020386 0.00063526
A 6,5 -0.593241 0.111710 0.134492 0.00063526
A 7,A -0.225191 -0.161674 -0.417624 0.00059556
A 7,2 -0.592793 -0.128143 -0.452358 0.00063526
A 7,3 -0.592278 0.034387 -0.023485 0.00063526
A 7,4 -0.591680 0.110619 0.131911 0.00063526
A 7,5 -0.591886 -0.389279 -0.837167 0.00063526
A 7,6 -0.590945 -0.437537 -0.911017 0.00063526
A 8,A 0.190271 -0.065708 -0.228736 0.00059556
A 8,2 -0.594420 0.034369 -0.027004 0.00063526
A 8,3 -0.593863 0.109865 0.126656 0.00063526
A 8,4 -0.593314 -0.388890 -0.839859 0.00063526
A 8,5 -0.593521 -0.435648 -0.909178 0.00063526
A 8,6 -0.592581 -0.470803 -0.961917 0.00063526
A 8,7 -0.591245 -0.501146 -1.010099 0.00063526
A 9,A 0.607410 0.029283 -0.042944 0.00059556
A 9,2 -0.595990 0.109244 0.121847 0.00063526
A 9,3 -0.595483 -0.389079 -0.843659 0.00063526
A 9,4 -0.594935 -0.423791 -0.887279 0.00063526
A 9,5 -0.595142 -0.468465 -0.956977 0.00063526
A 9,6 -0.594189 -0.509542 -1.027457 0.00063526
A 9,7 -0.592862 -0.536913 -1.073826 0.00063526
A 9,8 -0.509705 -0.575986 -1.151972 0.00063526
A 10,A 1.500000 0.105024 0.111257 0.00239914
A 10,2 -0.599391 -0.383001 -0.832454 0.00255908
A 10,3 -0.598886 -0.426448 -0.893557 0.00255908
A 10,4 -0.598343 -0.468814 -0.958468 0.00255908
A 10,5 -0.598541 -0.510904 -1.029171 0.00255908
A 10,6 -0.597634 -0.538953 -1.077906 0.00255908
A 10,7 -0.512810 -0.578724 -1.157449 0.00255908
A 10,8 -0.222188 -0.640412 -1.280824 0.00255908
A 10,9 0.193983 -0.734426 -1.468852 0.00255908
2 2,A -0.284357 0.045095 -0.060966 0.00086589
2 3,A -0.284272 0.021735 -0.061684 0.00092361
2 3,2 -0.285926 -0.126132 -0.571853 0.00086589
2 4,A -0.284141 -0.001763 -0.067212 0.00092361
2 4,2 -0.285749 -0.139859 -0.551276 0.00086589
2 4,3 -0.285663 -0.111691 -0.428461 0.00092361
2 5,A -0.280296 -0.022271 -0.070719 0.00092361
2 5,2 -0.281886 -0.107068 -0.423230 0.00086589
2 5,3 -0.281800 -0.022559 -0.196717 0.00092361
2 5,4 -0.281617 0.077524 0.072626 0.00092361
2 6,A -0.150884 0.000433 -0.003095 0.00092361
2 6,2 -0.282627 -0.022065 -0.196309 0.00086589
2 6,3 -0.282538 0.076868 0.071671 0.00092361
2 6,4 -0.282309 0.189665 0.375075 0.00092361
2 6,5 -0.278471 0.248657 0.493353 0.00092361
2 7,A 0.114542 0.060729 0.117057 0.00092361
2 7,2 -0.283393 0.074734 0.072461 0.00086589
2 7,3 -0.283259 0.188722 0.373259 0.00092361
2 7,4 -0.283077 0.246002 0.487602 0.00092361
2 7,5 -0.279241 -0.253535 -0.507070 0.00092361
2 7,6 -0.279988 -0.314431 -0.628862 0.00092361
2 8,A 0.382107 0.120840 0.237347 0.00092361
2 8,2 -0.284070 0.187995 0.372141 0.00086589
2 8,3 -0.283983 0.244672 0.485011 0.00092361
2 8,4 -0.283803 -0.255220 -0.510439 0.00092361
2 8,5 -0.279969 -0.314244 -0.628489 0.00092361
2 8,6 -0.280710 -0.363686 -0.727371 0.00092361
2 8,7 -0.281537 -0.412240 -0.824480 0.00092361
2 9,A 0.638969 0.182844 0.361686 0.00092361
2 9,2 -0.285101 0.243453 0.482637 0.00086589
2 9,3 -0.285014 -0.256863 -0.513727 0.00092361
2 9,4 -0.284835 -0.304881 -0.609761 0.00092361
2 9,5 -0.280999 -0.362697 -0.725395 0.00092361
2 9,6 -0.281801 -0.421677 -0.843354 0.00092361
2 9,7 -0.282572 -0.469673 -0.939345 0.00092361
2 9,8 -0.152453 -0.536788 -1.073577 0.00092361
2 10,A 1.500000 0.239713 0.473824 0.00369444
2 10,2 -0.290878 -0.251443 -0.502887 0.00346354
2 10,3 -0.290792 -0.307652 -0.615303 0.00369444
2 10,4 -0.290614 -0.364794 -0.729587 0.00369444
2 10,5 -0.286832 -0.423019 -0.846038 0.00369444
2 10,6 -0.287576 -0.471853 -0.943706 0.00369444
2 10,7 -0.157582 -0.538641 -1.077282 0.00369444
2 10,8 0.109414 -0.625997 -1.251995 0.00369444
2 10,9 0.377810 -0.734499 -1.468997 0.00369444
3 2,A -0.243738 0.073581 0.005080 0.00092361
3 3,A -0.243614 0.049907 0.001226 0.00086589
3 3,2 -0.245133 -0.093277 -0.490266 0.00086589
3 4,A -0.239733 0.028628 -0.001187 0.00092361
3 4,2 -0.241230 -0.103957 -0.463292 0.00092361
3 4,3 -0.241056 -0.077103 -0.346213 0.00086589
3 5,A -0.239109 0.007407 -0.006943 0.00092361
3 5,2 -0.240610 -0.072797 -0.343447 0.00092361
3 5,3 -0.240434 0.009252 -0.123873 0.00086589
3 5,4 -0.236505 0.109220 0.137766 0.00092361
3 6,A -0.114018 0.029863 0.058843 0.00092361
3 6,2 -0.241330 0.009754 -0.123541 0.00092361
3 6,3 -0.241108 0.108573 0.135757 0.00086589
3 6,4 -0.237227 0.215503 0.430006 0.00092361
3 6,5 -0.236606 0.272474 0.544064 0.00092361
3 7,A 0.143418 0.088592 0.176300 0.00092361
3 7,2 -0.242047 0.106454 0.136505 0.00092361
3 7,3 -0.241872 0.214502 0.427622 0.00086589
3 7,4 -0.237993 0.270365 0.539846 0.00092361
3 7,5 -0.237375 -0.231244 -0.462489 0.00092361
3 7,6 -0.238096 -0.295172 -0.590345 0.00092361
3 8,A 0.401226 0.152991 0.304809 0.00092361
3 8,2 -0.243071 0.213200 0.425153 0.00092361
3 8,3 -0.242897 0.268070 0.534888 0.00086589
3 8,4 -0.239022 -0.232977 -0.465953 0.00092361
3 8,5 -0.238399 -0.295395 -0.590791 0.00092361
3 8,6 -0.239183 -0.349513 -0.699027 0.00092361
3 8,7 -0.239949 -0.401918 -0.803836 0.00092361
3 9,A 0.644935 0.202765 0.404101 0.00092361
3 9,2 -0.249021 0.262106 0.522783 0.00092361
3 9,3 -0.248848 -0.238994 -0.477988 0.00086589
3 9,4 -0.244970 -0.289064 -0.578127 0.00092361
3 9,5 -0.244406 -0.351338 -0.702676 0.00092361
3 9,6 -0.245131 -0.413800 -0.827601 0.00092361
3 9,7 -0.245899 -0.464469 -0.928938 0.00092361
3 9,8 -0.120503 -0.534231 -1.068463 0.00092361
3 10,A 1.500000 0.262244 0.521784 0.00369444
3 10,2 -0.250114 -0.230500 -0.461000 0.00369444
3 10,3 -0.249943 -0.289818 -0.579635 0.00346354
3 10,4 -0.246120 -0.351286 -0.702572 0.00369444
3 10,5 -0.245496 -0.413237 -0.826474 0.00369444
3 10,6 -0.246223 -0.465092 -0.930185 0.00369444
3 10,7 -0.120516 -0.534554 -1.069107 0.00369444
3 10,8 0.137755 -0.624273 -1.248545 0.00369444
3 10,9 0.392518 -0.724746 -1.449491 0.00369444
4 2,A -0.199743 0.105013 0.075410 0.00092361
4 3,A -0.195867 0.083700 0.074056 0.00092361
4 3,2 -0.197210 -0.053618 -0.394420 0.00092361
4 4,A -0.195168 0.060928 0.067459 0.00086589
4 4,2 -0.196511 -0.065491 -0.373989 0.00086589
4 4,3 -0.192612 -0.036176 -0.253886 0.00086589
4 5,A -0.194637 0.040776 0.061847 0.00092361
4 5,2 -0.195983 -0.034700 -0.257969 0.00092361
4 5,3 -0.192060 0.048507 -0.042578 0.00092361
4 5,4 -0.191356 0.141932 0.207128 0.00086589
4 6,A -0.072750 0.063127 0.126253 0.00092361
4 6,2 -0.196642 0.047189 -0.046843 0.00092361
4 6,3 -0.192766 0.141224 0.205691 0.00092361
4 6,4 -0.192065 0.244349 0.488698 0.00086589
4 6,5 -0.191538 0.298997 0.597993 0.00092361
4 7,A 0.173450 0.125459 0.250918 0.00092361
4 7,2 -0.197724 0.136946 0.201533 0.00092361
4 7,3 -0.193851 0.242809 0.485617 0.00092361
4 7,4 -0.193153 0.296913 0.593827 0.00086589
4 7,5 -0.192622 -0.208860 -0.417720 0.00092361
4 7,6 -0.193390 -0.275811 -0.551622 0.00092361
4 8,A 0.415146 0.176956 0.353913 0.00092361
4 8,2 -0.203664 0.235210 0.470420 0.00092361
4 8,3 -0.199794 0.290066 0.580131 0.00092361
4 8,4 -0.199092 -0.214012 -0.428025 0.00086589
4 8,5 -0.198620 -0.279167 -0.558334 0.00092361
4 8,6 -0.199330 -0.336760 -0.673519 0.00092361
4 8,7 -0.200418 -0.393452 -0.786904 0.00092361
4 9,A 0.655298 0.229357 0.458714 0.00092361
4 9,2 -0.204880 0.286916 0.573832 0.00092361
4 9,3 -0.201009 -0.216647 -0.433293 0.00092361
4 9,4 -0.200364 -0.270141 -0.540282 0.00086589
4 9,5 -0.199835 -0.336288 -0.672576 0.00092361
4 9,6 -0.200545 -0.403005 -0.806010 0.00092361
4 9,7 -0.201635 -0.457452 -0.914905 0.00092361
4 9,8 -0.084264 -0.531893 -1.063787 0.00092361
4 10,A 1.500000 0.286307 0.572613 0.00369444
4 10,2 -0.205947 -0.209063 -0.418127 0.00369444
4 10,3 -0.202129 -0.270878 -0.541756 0.00369444
4 10,4 -0.201428 -0.336319 -0.672639 0.00346354
4 10,5 -0.200897 -0.402544 -0.805088 0.00369444
4 10,6 -0.201609 -0.457810 -0.915620 0.00369444
4 10,7 -0.079505 -0.530627 -1.061254 0.00369444
4 10,8 0.163389 -0.613514 -1.227029 0.00369444
4 10,9 0.411428 -0.723708 -1.447417 0.00369444
5 2,A -0.153288 0.139648 0.148011 0.00092361
5 3,A -0.152642 0.118117 0.145904 0.00092361
5 3,2 -0.150084 -0.012095 -0.300168 0.00092361
5 4,A -0.152036 0.096157 0.138832 0.00092361
5 4,2 -0.149480 -0.023294 -0.281007 0.00092361
5 4,3 -0.148807 0.005990 -0.168453 0.00092361
5 5,A -0.151511 0.075888 0.131876 0.00086589
5 5,2 -0.148930 0.007799 -0.170525 0.00086589
5 5,3 -0.148280 0.086370 0.037764 0.00086589
5 5,4 -0.147671 0.177950 0.282001 0.00086589
5 6,A -0.034467 0.102635 0.205270 0.00092361
5 6,2 -0.150003 0.084068 0.032397 0.00092361
5 6,3 -0.149354 0.175448 0.276933 0.00092361
5 6,4 -0.148748 0.275170 0.550340 0.00092361
5 6,5 -0.148219 0.327316 0.654632 0.00086589
5 7,A 0.201050 0.153409 0.306817 0.00092361
5 7,2 -0.155998 0.166475 0.263608 0.00092361
5 7,3 -0.155351 0.268166 0.536332 0.00092361
5 7,4 -0.154742 0.320172 0.640345 0.00092361
5 7,5 -0.154272 -0.189740 -0.379479 0.00086589
5 7,6 -0.155349 -0.260473 -0.520946 0.00092361
5 8,A 0.441785 0.208204 0.416408 0.00092361
5 8,2 -0.157233 0.265445 0.530890 0.00092361
5 8,3 -0.156583 0.317348 0.634696 0.00092361
5 8,4 -0.156032 -0.191864 -0.383728 0.00092361
5 8,5 -0.155504 -0.260786 -0.521573 0.00086589
5 8,6 -0.156582 -0.321029 -0.642057 0.00092361
5 8,7 -0.162578 -0.383177 -0.766354 0.00092361
5 9,A 0.671436 0.258672 0.517344 0.00092361
5 9,2 -0.158278 0.314817 0.629635 0.00092361
5 9,3 -0.157687 -0.194343 -0.388686 0.00092361
5 9,4 -0.157078 -0.251758 -0.503515 0.00092361
5 9,5 -0.156550 -0.320256 -0.640512 0.00086589
5 9,6 -0.157629 -0.390370 -0.780739 0.00092361
5 9,7 -0.163626 -0.450989 -0.901978 0.00092361
5 9,8 -0.046053 -0.516508 -1.033015 0.00092361
5 10,A 1.500000 0.313008 0.626016 0.00369444
5 10,2 -0.159310 -0.186390 -0.372780 0.00369444
5 10,3 -0.158660 -0.252450 -0.504900 0.00369444
5 10,4 -0.158051 -0.320331 -0.640663 0.00369444
5 10,5 -0.157523 -0.389656 -0.779313 0.00346354
5 10,6 -0.158603 -0.449333 -0.898667 0.00369444
5 10,7 -0.045816 -0.516496 -1.032993 0.00369444
5 10,8 0.195823 -0.609827 -1.219655 0.00369444
5 10,9 0.438165 -0.722070 -1.444140 0.00369444
6 2,A -0.112532 0.167174 0.208324 0.00092361
6 3,A -0.111935 0.146848 0.206477 0.00092361
6 3,2 -0.112255 0.021325 -0.224509 0.00092361
6 4,A -0.111383 0.126567 0.201535 0.00092361
6 4,2 -0.111676 0.012471 -0.205841 0.00092361
6 4,3 -0.111075 0.038569 -0.097207 0.00092361
6 5,A -0.111221 0.112080 0.205032 0.00092361
6 5,2 -0.111557 0.040496 -0.098602 0.00092361
6 5,3 -0.110956 0.115305 0.100214 0.00092361
6 5,4 -0.110400 0.202961 0.334470 0.00092361
6 6,A -0.004442 0.126798 0.253595 0.00086589
6 6,2 -0.117554 0.108168 0.085294 0.00086589
6 6,3 -0.116955 0.195592 0.319694 0.00086589
6 6,4 -0.116397 0.293011 0.586022 0.00086589
6 6,5 -0.116337 0.343758 0.687516 0.00086589
6 7,A 0.222101 0.178870 0.357740 0.00092361
6 7,2 -0.118793 0.190690 0.314474 0.00092361
6 7,3 -0.118192 0.289896 0.579793 0.00092361
6 7,4 -0.117691 0.340736 0.681471 0.00092361
6 7,5 -0.117572 -0.173630 -0.347259 0.00092361
6 7,6 -0.123570 -0.250102 -0.500204 0.00086589
6 8,A 0.451752 0.231294 0.462588 0.00092361
6 8,2 -0.119828 0.287230 0.574460 0.00092361
6 8,3 -0.119285 0.337956 0.675913 0.00092361
6 8,4 -0.118726 -0.175220 -0.350439 0.00092361
6 8,5 -0.118607 -0.247268 -0.494537 0.00092361
6 8,6 -0.124606 -0.312919 -0.625837 0.00086589
6 8,7 -0.125846 -0.374576 -0.749152 0.00092361
6 9,A 0.677213 0.280966 0.561933 0.00092361
6 9,2 -0.120900 0.335424 0.670847 0.00092361
6 9,3 -0.120299 -0.177686 -0.355373 0.00092361
6 9,4 -0.119741 -0.237889 -0.475778 0.00092361
6 9,5 -0.119622 -0.309449 -0.618897 0.00092361
6 9,6 -0.125622 -0.383780 -0.767560 0.00086589
6 9,7 -0.126781 -0.434864 -0.869727 0.00092361
6 9,8 -0.011876 -0.513210 -1.026419 0.00092361
6 10,A 1.500000 0.334133 0.668266 0.00369444
6 10,2 -0.121695 -0.169633 -0.339266 0.00369444
6 10,3 -0.121095 -0.238517 -0.477033 0.00369444
6 10,4 -0.120537 -0.309137 -0.618274 0.00369444
6 10,5 -0.120418 -0.380669 -0.761338 0.00369444
6 10,6 -0.126338 -0.434795 -0.869590 0.00346354
6 10,7 -0.011673 -0.513194 -1.026389 0.00369444
6 10,8 0.218639 -0.608353 -1.216705 0.00369444
6 10,9 0.449198 -0.721432 -1.442864 0.00369444
7 2,A -0.472081 0.119029 -0.177283 0.00092361
7 3,A -0.471601 0.075026 -0.181438 0.00092361
7 3,2 -0.473744 -0.119489 -0.947488 0.00092361
7 4,A -0.471066 0.036105 -0.173744 0.00092361
7 4,2 -0.473230 -0.154678 -0.888294 0.00092361
7 4,3 -0.472747 -0.069274 -0.578029 0.00092361
7 5,A -0.473288 -0.009400 -0.184635 0.00092361
7 5,2 -0.475452 -0.068561 -0.583473 0.00092361
7 5,3 -0.474970 0.084676 -0.174035 0.00092361
7 5,4 -0.474451 0.178579 0.125159 0.00092361
7 6,A -0.102349 0.055160 -0.006438 0.00092361
7 6,2 -0.476595 0.084790 -0.177547 0.00092361
7 6,3 -0.476108 0.178192 0.121837 0.00092361
7 6,4 -0.475650 0.263931 0.412852 0.00092361
7 6,5 -0.477867 0.293989 0.473101 0.00092361
7 7,A 0.402983 0.172399 0.225480 0.00086589
7 7,2 -0.477635 0.175421 0.118269 0.00086589
7 7,3 -0.477207 0.262042 0.405886 0.00086589
7 7,4 -0.476692 0.293167 0.469940 0.00086589
7 7,5 -0.478909 -0.223059 -0.524096 0.00086589
7 7,6 -0.480052 -0.283238 -0.616348 0.00086589
7 8,A 0.615612 0.221109 0.320950 0.00092361
7 8,2 -0.478446 0.259428 0.398326 0.00092361
7 8,3 -0.477960 0.292163 0.465353 0.00092361
7 8,4 -0.477445 -0.220527 -0.520166 0.00092361
7 8,5 -0.479663 -0.282775 -0.615300 0.00092361
7 8,6 -0.480814 -0.327645 -0.683560 0.00092361
7 8,7 -0.481778 -0.358883 -0.729046 0.00086589
7 9,A 0.773288 0.253501 0.382647 0.00092361
7 9,2 -0.479752 0.291519 0.461327 0.00092361
7 9,3 -0.479268 -0.220984 -0.523830 0.00092361
7 9,4 -0.478752 -0.270153 -0.590256 0.00092361
7 9,5 -0.480975 -0.327523 -0.681924 0.00092361
7 9,6 -0.482037 -0.368223 -0.747781 0.00092361
7 9,7 -0.483097 -0.405216 -0.810432 0.00086589
7 9,8 -0.111160 -0.475319 -0.950637 0.00092361
7 10,A 1.500000 0.290734 0.464189 0.00369444
7 10,2 -0.474716 -0.212542 -0.503932 0.00369444
7 10,3 -0.474233 -0.269367 -0.586268 0.00369444
7 10,4 -0.473721 -0.326238 -0.677342 0.00369444
7 10,5 -0.475899 -0.368503 -0.745856 0.00369444
7 10,6 -0.477058 -0.405505 -0.811011 0.00369444
7 10,7 -0.109970 -0.475803 -0.951607 0.00346354
7 10,8 0.396838 -0.585413 -1.170826 0.00369444
7 10,9 0.614575 -0.711392 -1.422783 0.00369444
8 2,A -0.509765 0.050688 -0.313928 0.00092361
8 3,A -0.509230 0.018138 -0.300276 0.00092361
8 3,2 -0.511100 -0.186664 -1.022199 0.00092361
8 4,A -0.511398 -0.029088 -0.314247 0.00092361
8 4,2 -0.513268 -0.221006 -1.006429 0.00092361
8 4,3 -0.512753 -0.214655 -0.847489 0.00092361
8 5,A -0.511319 -0.070735 -0.318548 0.00092361
8 5,2 -0.513188 -0.212223 -0.848329 0.00092361
8 5,3 -0.512670 -0.059228 -0.446067 0.00092361
8 5,4 -0.514892 0.100702 -0.019341 0.00092361
8 6,A -0.382888 -0.071144 -0.249396 0.00092361
8 6,2 -0.514137 -0.058875 -0.448845 0.00092361
8 6,3 -0.513679 0.102668 -0.016898 0.00092361
8 6,4 -0.515843 0.200444 0.294105 0.00092361
8 6,5 -0.515758 0.230237 0.354804 0.00092361
8 7,A 0.109163 0.041697 -0.027229 0.00092361
8 7,2 -0.515041 0.100447 -0.019183 0.00092361
8 7,3 -0.514524 0.202193 0.295685 0.00092361
8 7,4 -0.516689 0.228163 0.348281 0.00092361
8 7,5 -0.516605 -0.283101 -0.638037 0.00092361
8 7,6 -0.517566 -0.340317 -0.726862 0.00092361
8 8,A 0.596957 0.153259 0.193419 0.00086589
8 8,2 -0.516352 0.199835 0.288079 0.00086589
8 8,3 -0.515836 0.227823 0.345738 0.00086589
8 8,4 -0.518002 -0.282865 -0.640233 0.00086589
8 8,5 -0.517922 -0.338532 -0.723484 0.00086589
8 8,6 -0.518794 -0.371252 -0.768217 0.00086589
8 8,7 -0.519657 -0.407971 -0.826337 0.00086589
8 9,A 0.790078 0.191569 0.272846 0.00092361
8 9,2 -0.511321 0.226418 0.345105 0.00092361
8 9,3 -0.510806 -0.282234 -0.636741 0.00092361
8 9,4 -0.512975 -0.327234 -0.698857 0.00092361
8 9,5 -0.512851 -0.371382 -0.765021 0.00092361
8 9,6 -0.513819 -0.417733 -0.844886 0.00092361
8 9,7 -0.514682 -0.451179 -0.902358 0.00092361
8 9,8 -0.390252 -0.498754 -0.997508 0.00086589
8 10,A 1.500000 0.227782 0.346321 0.00369444
8 10,2 -0.512196 -0.272223 -0.618185 0.00369444
8 10,3 -0.511684 -0.324828 -0.694547 0.00369444
8 10,4 -0.513809 -0.368434 -0.759805 0.00369444
8 10,5 -0.513733 -0.416983 -0.842012 0.00369444
8 10,6 -0.514701 -0.450434 -0.900867 0.00369444
8 10,7 -0.385343 -0.498411 -0.996822 0.00369444
8 10,8 0.103131 -0.584919 -1.169839 0.00346354
8 10,9 0.589698 -0.709749 -1.419499 0.00369444
9 2,A -0.537051 -0.032240 -0.436487 0.00092361
9 3,A -0.539227 -0.071521 -0.441293 0.00092361
9 3,2 -0.540768 -0.265667 -1.081536 0.00092361
9 4,A -0.539104 -0.112536 -0.448202 0.00092361
9 4,2 -0.540645 -0.295486 -1.062540 0.00092361
9 4,3 -0.542817 -0.290016 -0.954433 0.00092361
9 5,A -0.538949 -0.152841 -0.455455 0.00092361
9 5,2 -0.540486 -0.285462 -0.949837 0.00092361
9 5,3 -0.542716 -0.212011 -0.714383 0.00092361
9 5,4 -0.542587 -0.052189 -0.295603 0.00092361
9 6,A -0.419274 -0.146009 -0.387684 0.00092361
9 6,2 -0.541476 -0.209728 -0.711897 0.00092361
9 6,3 -0.543647 -0.051952 -0.297836 0.00092361
9 6,4 -0.543520 0.116835 0.149459 0.00092361
9 6,5 -0.543360 0.156958 0.231125 0.00092361
9 7,A -0.182351 -0.097328 -0.282050 0.00092361
9 7,2 -0.542697 -0.052173 -0.294949 0.00092361
9 7,3 -0.544869 0.116772 0.146517 0.00092361
9 7,4 -0.544742 0.156165 0.226910 0.00092361
9 7,5 -0.544586 -0.351710 -0.758593 0.00092361
9 7,6 -0.545439 -0.394601 -0.822584 0.00092361
9 8,A 0.287634 0.007206 -0.070222 0.00092361
9 8,2 -0.537670 0.117102 0.150972 0.00092361
9 8,3 -0.539843 0.153842 0.224717 0.00092361
9 8,4 -0.539720 -0.353168 -0.760375 0.00092361
9 8,5 -0.539520 -0.393419 -0.817534 0.00092361
9 8,6 -0.540469 -0.432288 -0.880220 0.00092361
9 8,7 -0.541703 -0.464749 -0.935196 0.00092361
9 9,A 0.760068 0.111528 0.135641 0.00086589
9 9,2 -0.538549 0.154592 0.224297 0.00086589
9 9,3 -0.540726 -0.352393 -0.761887 0.00086589
9 9,4 -0.540559 -0.381891 -0.796071 0.00086589
9 9,5 -0.540406 -0.429775 -0.873961 0.00086589
9 9,6 -0.541355 -0.473445 -0.952638 0.00086589
9 9,7 -0.542589 -0.502797 -1.005594 0.00086589
9 9,8 -0.420344 -0.548382 -1.096764 0.00086589
9 10,A 1.500000 0.155893 0.224949 0.00369444
9 10,2 -0.539371 -0.341050 -0.739516 0.00369444
9 10,3 -0.541503 -0.380451 -0.794521 0.00369444
9 10,4 -0.541384 -0.426843 -0.869023 0.00369444
9 10,5 -0.541232 -0.472492 -0.949345 0.00369444
9 10,6 -0.542181 -0.502126 -1.004252 0.00369444
9 10,7 -0.421644 -0.547224 -1.094448 0.00369444
9 10,8 -0.186228 -0.610958 -1.221915 0.00369444
9 10,9 0.282025 -0.711335 -1.422671 0.00346354
10 2,A -0.540060 -0.101005 -0.505785 0.00342412
10 3,A -0.539920 -0.135860 -0.506008 0.00342412
10 3,2 -0.542021 -0.312335 -1.084042 0.00340610
10 4,A -0.539817 -0.172939 -0.509750 0.00342412
10 4,2 -0.541913 -0.339672 -1.064902 0.00340610
10 4,3 -0.541832 -0.322763 -0.951751 0.00340610
10 5,A -0.539637 -0.211321 -0.519456 0.00342412
10 5,2 -0.541797 -0.319734 -0.951709 0.00340610
10 5,3 -0.541651 -0.249688 -0.744354 0.00340610
10 5,4 -0.541541 -0.149368 -0.459610 0.00340610
10 6,A -0.419454 -0.194959 -0.452642 0.00342412
10 6,2 -0.543112 -0.249702 -0.746678 0.00340610
10 6,3 -0.542967 -0.150299 -0.461865 0.00340610
10 6,4 -0.542858 0.027559 -0.004039 0.00340610
10 6,5 -0.542681 0.118138 0.177799 0.00340610
10 7,A -0.180201 -0.142446 -0.340658 0.00342412
10 7,2 -0.537660 -0.153134 -0.459236 0.00340610
10 7,3 -0.537516 0.026491 -0.003201 0.00340610
10 7,4 -0.537411 0.117111 0.177694 0.00340610
10 7,5 -0.537186 -0.381902 -0.795549 0.00340610
10 7,6 -0.538516 -0.431265 -0.877595 0.00340610
10 8,A 0.063422 -0.087360 -0.232658 0.00342412
10 8,2 -0.538613 0.026652 -0.005191 0.00340610
10 8,3 -0.538473 0.117046 0.175515 0.00340610
10 8,4 -0.538321 -0.382253 -0.799070 0.00340610
10 8,5 -0.538148 -0.430314 -0.877566 0.00340610
10 8,6 -0.539478 -0.465986 -0.937945 0.00340610
10 8,7 -0.534086 -0.497373 -0.994746 0.00340610
10 9,A 0.554561 0.021363 -0.018877 0.00342412
10 9,2 -0.539505 0.115827 0.170985 0.00340610
10 9,3 -0.539317 -0.382961 -0.802777 0.00340610
10 9,4 -0.539216 -0.418939 -0.855830 0.00340610
10 9,5 -0.539043 -0.463099 -0.931619 0.00340610
10 9,6 -0.540373 -0.505801 -1.012641 0.00340610
10 9,7 -0.534982 -0.533145 -1.066291 0.00340610
10 9,8 -0.412670 -0.578154 -1.156309 0.00340610
10 10,A 1.500000 0.115859 0.169912 0.01348247
10 10,2 -0.540283 -0.372900 -0.783223 0.01341151
10 10,3 -0.540147 -0.417688 -0.854216 0.01341151
10 10,4 -0.540046 -0.461443 -0.928883 0.01341151
10 10,5 -0.539873 -0.503624 -1.007248 0.01341151
10 10,6 -0.541203 -0.532053 -1.064107 0.01341151
10 10,7 -0.417830 -0.577834 -1.155668 0.01341151
10 10,8 -0.172653 -0.641750 -1.283500 0.01341151
10 10,9 0.072642 -0.724834 -1.449668 0.01341151

 

Blackjack Appendix 9 — 4 Decks, Dealer Hits on Soft 17 — Player Can Split

Dealer Player Stand Hit Double Split
(DAS=No)		 Split
(DAS=Yes)		 Probability
A A,A -0.596977 -0.065388 -0.582574 0.129936 0.129936 0.00026056
A 2,2 -0.595967 -0.292870 -1.191935 -0.523610 -0.520445 0.00029778
A 3,3 -0.594987 -0.346177 -1.177947 -0.567817 -0.563775 0.00029778
A 4,4 -0.593902 -0.264116 -0.827063 -0.639149 -0.634071 0.00029778
A 5,5 -0.594290 0.035263 -0.020254 -0.758331 -0.752908 0.00029778
A 6,6 -0.592290 -0.388892 -0.836346 -0.727365 -0.721978 0.00029778
A 7,7 -0.589617 -0.481972 -0.984407 -0.728085 -0.723218 0.00029778
A 8,8 -0.592875 -0.536768 -1.073536 -0.512423 -0.509113 0.00029778
A 9,9 -0.216334 -0.638282 -1.276564 -0.236875 -0.235107 0.00029778
A 10,10 0.599679 -0.860952 -1.721905 -0.462283 -0.462283 0.00507366
2 A,A -0.282726 0.083156 -0.055862 0.493092 0.493092 0.00043294
2 2,2 -0.285960 -0.112376 -0.571921 -0.151336 -0.074506 0.00037882
2 3,3 -0.285842 -0.139940 -0.551452 -0.204755 -0.127194 0.00043294
2 4,4 -0.285483 -0.021159 -0.199796 -0.256983 -0.178934 0.00043294
2 5,5 -0.277730 0.191055 0.378266 -0.311423 -0.265558 0.00043294
2 6,6 -0.279215 -0.253537 -0.507075 -0.261251 -0.181597 0.00043294
2 7,7 -0.280755 -0.374122 -0.748245 -0.193574 -0.113433 0.00043294
2 8,8 -0.282263 -0.469361 -0.938721 0.003451 0.078851 0.00043294
2 9,9 0.114027 -0.624788 -1.249576 0.155731 0.196240 0.00043294
2 10,10 0.631826 -0.853306 -1.706611 0.094617 0.094617 0.00727344
3 A,A -0.242201 0.105864 0.010606 0.540506 0.540506 0.00043294
3 2,2 -0.245303 -0.079865 -0.490606 -0.093934 -0.001666 0.00043294
3 3,3 -0.244961 -0.106527 -0.470379 -0.134480 -0.042884 0.00037882
3 4,4 -0.237152 0.011189 -0.123541 -0.175308 -0.080635 0.00043294
3 5,5 -0.235882 0.216556 0.432104 -0.235624 -0.178683 0.00043294
3 6,6 -0.237332 -0.231295 -0.462590 -0.183642 -0.082047 0.00043294
3 7,7 -0.238920 -0.359510 -0.719019 -0.120676 -0.020455 0.00043294
3 8,8 -0.240979 -0.462481 -0.924963 0.068145 0.152774 0.00043294
3 9,9 0.133040 -0.625189 -1.250378 0.200515 0.246702 0.00043294
3 10,10 0.642519 -0.853018 -1.706035 0.178826 0.178826 0.00727344
4 A,A -0.198317 0.130565 0.080030 0.590708 0.590708 0.00043294
4 2,2 -0.201106 -0.043137 -0.402213 -0.028869 0.078443 0.00043294
4 3,3 -0.193311 -0.063477 -0.368335 -0.044836 0.064732 0.00043294
4 4,4 -0.191887 0.051068 -0.042119 -0.089919 0.022424 0.00037882
4 5,5 -0.190827 0.244820 0.489639 -0.149285 -0.077328 0.00043294
4 6,6 -0.192244 -0.208472 -0.416945 -0.098864 0.024172 0.00043294
4 7,7 -0.194479 -0.344370 -0.688740 -0.039508 0.084642 0.00043294
4 8,8 -0.206360 -0.459113 -0.918225 0.125332 0.219354 0.00043294
4 9,9 0.163507 -0.613494 -1.226987 0.266631 0.317521 0.00043294
4 10,10 0.652914 -0.852794 -1.705588 0.268647 0.268647 0.00727344
5 A,A -0.155794 0.162955 0.149183 0.643381 0.643381 0.00043294
5 2,2 -0.150729 0.000025 -0.301459 0.061760 0.188657 0.00043294
5 3,3 -0.149439 -0.023324 -0.280561 0.038776 0.170963 0.00043294
5 4,4 -0.148198 0.089391 0.039393 -0.006405 0.129939 0.00043294
5 5,5 -0.147145 0.276981 0.553963 -0.059539 0.031214 0.00037882
5 6,6 -0.149356 -0.186242 -0.372484 -0.015431 0.131522 0.00043294
5 7,7 -0.161344 -0.333648 -0.667296 0.023989 0.167275 0.00043294
5 8,8 -0.163814 -0.451043 -0.902086 0.200119 0.304450 0.00043294
5 9,9 0.195823 -0.609801 -1.219603 0.338258 0.394184 0.00043294
5 10,10 0.669203 -0.852399 -1.704799 0.366853 0.366853 0.00727344
6 A,A -0.112187 0.190740 0.211466 0.687177 0.687177 0.00043294
6 2,2 -0.112850 0.032260 -0.225700 0.129375 0.275171 0.00043294
6 3,3 -0.111632 0.012463 -0.205750 0.106855 0.255791 0.00043294
6 4,4 -0.110518 0.118208 0.101627 0.063726 0.216500 0.00043294
6 5,5 -0.110282 0.299512 0.599024 0.008196 0.111071 0.00043294
6 6,6 -0.122332 -0.176676 -0.353352 0.034309 0.193783 0.00037882
6 7,7 -0.124808 -0.323281 -0.646562 0.089068 0.249902 0.00043294
6 8,8 -0.126803 -0.434843 -0.869686 0.257753 0.371721 0.00043294
6 9,9 0.218471 -0.608375 -1.216749 0.389160 0.448796 0.00043294
6 10,10 0.675762 -0.852223 -1.704446 0.443399 0.443399 0.00727344
7 A,A -0.469898 0.163757 -0.172621 0.481997 0.481997 0.00043294
7 2,2 -0.474248 -0.089107 -0.948496 -0.048593 0.009444 0.00043294
7 3,3 -0.473262 -0.154760 -0.888339 -0.108412 -0.050171 0.00043294
7 4,4 -0.472234 0.088868 -0.168701 -0.229879 -0.169963 0.00043294
7 5,5 -0.476727 0.262157 0.410917 -0.333366 -0.299294 0.00043294
7 6,6 -0.479008 -0.224556 -0.527096 -0.307410 -0.246964 0.00043294
7 7,7 -0.481104 -0.337152 -0.704102 -0.104738 -0.048004 0.00037882
7 8,8 -0.482548 -0.405174 -0.810348 0.263126 0.317582 0.00043294
7 9,9 0.399617 -0.585162 -1.170324 0.334607 0.363647 0.00043294
7 10,10 0.771374 -0.849732 -1.699465 0.257538 0.257538 0.00727344
8 A,A -0.507875 0.094344 -0.310003 0.364258 0.364258 0.00043294
8 2,2 -0.511613 -0.155451 -1.023226 -0.215345 -0.176815 0.00043294
8 3,3 -0.510585 -0.220202 -1.001065 -0.269360 -0.229910 0.00043294
8 4,4 -0.514918 -0.058680 -0.450579 -0.365868 -0.325509 0.00043294
8 5,5 -0.514807 0.200331 0.296188 -0.485644 -0.461764 0.00043294
8 6,6 -0.516710 -0.283253 -0.638127 -0.452532 -0.411820 0.00043294
8 7,7 -0.518333 -0.380634 -0.788316 -0.428743 -0.390152 0.00043294
8 8,8 -0.520981 -0.450837 -0.901675 -0.067474 -0.032440 0.00037882
8 9,9 0.095884 -0.585162 -1.170324 0.206929 0.227585 0.00043294
8 10,10 0.789718 -0.849395 -1.698790 0.006145 0.006145 0.00727344
9 A,A -0.535511 -0.000541 -0.448071 0.242792 0.242792 0.00043294
9 2,2 -0.538592 -0.236442 -1.077184 -0.403033 -0.382969 0.00043294
9 3,3 -0.542943 -0.296613 -1.067142 -0.448178 -0.428832 0.00043294
9 4,4 -0.542748 -0.209639 -0.714277 -0.526941 -0.506881 0.00043294
9 5,5 -0.542426 0.116751 0.151645 -0.654780 -0.640367 0.00043294
9 6,6 -0.544301 -0.350806 -0.756276 -0.617811 -0.597494 0.00043294
9 7,7 -0.546673 -0.441447 -0.902100 -0.589299 -0.571146 0.00043294
9 8,8 -0.536732 -0.503887 -1.007774 -0.410528 -0.391112 0.00043294
9 9,9 -0.186289 -0.611291 -1.222581 -0.095054 -0.083302 0.00037882
9 10,10 0.754917 -0.848731 -1.697462 -0.279489 -0.279489 0.00727344
10 A,A -0.537981 -0.064423 -0.502927 0.183155 0.183155 0.00161350
10 2,2 -0.542162 -0.286004 -1.084324 -0.507807 -0.497438 0.00159661
10 3,3 -0.541877 -0.339159 -1.064826 -0.553395 -0.542421 0.00159661
10 4,4 -0.541723 -0.247273 -0.744401 -0.627410 -0.616031 0.00159661
10 5,5 -0.541359 0.028133 -0.001792 -0.740571 -0.729207 0.00159661
10 6,6 -0.543907 -0.382407 -0.798973 -0.714449 -0.703285 0.00159661
10 7,7 -0.533125 -0.477970 -0.961397 -0.660761 -0.649067 0.00159661
10 8,8 -0.535048 -0.533098 -1.066195 -0.483352 -0.472463 0.00159661
10 9,9 -0.167428 -0.642116 -1.284232 -0.315720 -0.306151 0.00159661
10 10,10 0.561473 -0.846180 -1.692361 -0.418713 -0.418713 0.02598479

 

 

Appendix 9 — 4 Decks — Dealer Has Blackjack

Player Stand Probability
BJ 0 0.00212881
Non BJ -1 0.04543715
 

For similar tables under other rules visit my blackjack appendix 9.

For blackjack in general, visit my blackjack section.

Written by: Michael Shackleford