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Blackjack Appendix 9

Introduction

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

  • 4 decks
  • Dealer stands 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: -0.003471
  • Player may not double after split: -0.004893

 

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

Dealer Player Stand Hit Double Probability
A 2,A -0.665842 -0.060413 -0.617034 0.00059556
A 3,A -0.665502 -0.096307 -0.617555 0.00059556
A 3,2 -0.665146 -0.282222 -1.330292 0.00063526
A 4,A -0.665062 -0.136287 -0.623385 0.00059556
A 4,2 -0.664774 -0.311677 -1.302312 0.00063526
A 4,3 -0.664426 -0.318578 -1.128919 0.00063526
A 5,A -0.664711 -0.175619 -0.631749 0.00059556
A 5,2 -0.664349 -0.315548 -1.128767 0.00063526
A 5,3 -0.664000 -0.201757 -0.806288 0.00063526
A 5,4 -0.663552 -0.066634 -0.423258 0.00063526
A 6,A -0.479196 -0.184673 -0.535285 0.00059556
A 6,2 -0.660560 -0.204007 -0.808090 0.00063526
A 6,3 -0.660208 -0.069118 -0.425523 0.00063526
A 6,4 -0.659759 0.081295 0.000844 0.00063526
A 6,5 -0.659290 0.149949 0.140210 0.00063526
A 7,A -0.100326 -0.096574 -0.362009 0.00059556
A 7,2 -0.661992 -0.069512 -0.425610 0.00063526
A 7,3 -0.661642 0.082724 -0.001287 0.00063526
A 7,4 -0.661155 0.149578 0.135720 0.00063526
A 7,5 -0.660734 -0.356669 -0.831025 0.00063526
A 7,6 -0.657023 -0.408371 -0.894703 0.00063526
A 8,A 0.280663 -0.008408 -0.189670 0.00059556
A 8,2 -0.663434 0.082708 -0.004763 0.00063526
A 8,3 -0.663046 0.149469 0.131759 0.00063526
A 8,4 -0.662608 -0.356489 -0.833283 0.00063526
A 8,5 -0.662189 -0.405365 -0.892240 0.00063526
A 8,6 -0.658485 -0.443296 -0.934274 0.00063526
A 8,7 -0.659935 -0.474112 -0.969744 0.00063526
A 9,A 0.661689 0.078130 -0.020689 0.00059556
A 9,2 -0.664827 0.149272 0.127853 0.00063526
A 9,3 -0.664489 -0.356252 -0.836196 0.00063526
A 9,4 -0.664052 -0.393442 -0.870173 0.00063526
A 9,5 -0.663634 -0.440065 -0.928599 0.00063526
A 9,6 -0.659937 -0.483772 -0.989034 0.00063526
A 9,7 -0.661373 -0.511872 -1.023744 0.00063526
A 9,8 -0.471769 -0.554120 -1.108241 0.00063526
A 10,A 1.500000 0.145297 0.117483 0.00239914
A 10,2 -0.668593 -0.350025 -0.824863 0.00255908
A 10,3 -0.668257 -0.395954 -0.875406 0.00255908
A 10,4 -0.667825 -0.441125 -0.931624 0.00255908
A 10,5 -0.667411 -0.484420 -0.989897 0.00255908
A 10,6 -0.663732 -0.515118 -1.030236 0.00255908
A 10,7 -0.475431 -0.556924 -1.113848 0.00255908
A 10,8 -0.095796 -0.628072 -1.256144 0.00255908
A 10,9 0.284937 -0.729009 -1.458019 0.00255908
2 2,A -0.290409 0.044916 -0.064142 0.00086589
2 3,A -0.290367 0.021065 -0.065064 0.00092361
2 3,2 -0.292406 -0.128787 -0.584812 0.00086589
2 4,A -0.289959 -0.002824 -0.070592 0.00092361
2 4,2 -0.291936 -0.142738 -0.562316 0.00086589
2 4,3 -0.291893 -0.111628 -0.432649 0.00092361
2 5,A -0.286371 -0.023729 -0.074339 0.00092361
2 5,2 -0.288347 -0.106927 -0.427470 0.00086589
2 5,3 -0.288304 -0.019972 -0.194821 0.00092361
2 5,4 -0.287831 0.079581 0.074693 0.00092361
2 6,A -0.147903 0.001295 -0.002371 0.00092361
2 6,2 -0.289067 -0.019475 -0.194365 0.00086589
2 6,3 -0.289021 0.079013 0.073894 0.00092361
2 6,4 -0.288502 0.191062 0.376916 0.00092361
2 6,5 -0.284934 0.249411 0.493862 0.00092361
2 7,A 0.125145 0.063484 0.121608 0.00092361
2 7,2 -0.289814 0.076911 0.074766 0.00086589
2 7,3 -0.289723 0.190187 0.375187 0.00092361
2 7,4 -0.289251 0.246627 0.487895 0.00092361
2 7,5 -0.285685 -0.253225 -0.506450 0.00092361
2 7,6 -0.286410 -0.313654 -0.627309 0.00092361
2 8,A 0.389794 0.122929 0.240518 0.00092361
2 8,2 -0.290474 0.189422 0.374064 0.00086589
2 8,3 -0.290430 0.245388 0.485436 0.00092361
2 8,4 -0.289961 -0.254906 -0.509811 0.00092361
2 8,5 -0.286395 -0.313467 -0.626935 0.00092361
2 8,6 -0.287115 -0.362395 -0.724791 0.00092361
2 8,7 -0.287924 -0.410466 -0.820932 0.00092361
2 9,A 0.643593 0.184222 0.363501 0.00092361
2 9,2 -0.291487 0.244208 0.483144 0.00086589
2 9,3 -0.291444 -0.256493 -0.512986 0.00092361
2 9,4 -0.290977 -0.304134 -0.608268 0.00092361
2 9,5 -0.287408 -0.361422 -0.722844 0.00092361
2 9,6 -0.288190 -0.419909 -0.839819 0.00092361
2 9,7 -0.288941 -0.467478 -0.934956 0.00092361
2 9,8 -0.149103 -0.534848 -1.069696 0.00092361
2 10,A 1.500000 0.240410 0.474326 0.00369444
2 10,2 -0.297344 -0.251066 -0.502132 0.00346354
2 10,3 -0.297301 -0.306815 -0.613631 0.00369444
2 10,4 -0.296832 -0.363641 -0.727281 0.00369444
2 10,5 -0.293322 -0.421269 -0.842538 0.00369444
2 10,6 -0.294045 -0.469652 -0.939305 0.00369444
2 10,7 -0.154260 -0.536693 -1.073386 0.00369444
2 10,8 0.120910 -0.624892 -1.249785 0.00369444
2 10,9 0.386095 -0.734011 -1.468022 0.00369444
3 2,A -0.248982 0.073436 0.002390 0.00092361
3 3,A -0.248581 0.049381 -0.001551 0.00086589
3 3,2 -0.250442 -0.095461 -0.500883 0.00086589
3 4,A -0.245022 0.027614 -0.004340 0.00092361
3 4,2 -0.246883 -0.106586 -0.473380 0.00092361
3 4,3 -0.246415 -0.077047 -0.349816 0.00086589
3 5,A -0.244346 0.006157 -0.010051 0.00092361
3 5,2 -0.246208 -0.072673 -0.347120 0.00092361
3 5,3 -0.245741 0.011365 -0.122335 0.00086589
3 5,4 -0.242154 0.111051 0.139607 0.00092361
3 6,A -0.111445 0.030612 0.059478 0.00092361
3 6,2 -0.246910 0.012000 -0.121865 0.00092361
3 6,3 -0.246397 0.110287 0.137534 0.00086589
3 6,4 -0.242857 0.216800 0.431672 0.00092361
3 6,5 -0.242179 0.273134 0.544523 0.00092361
3 7,A 0.152566 0.090978 0.180203 0.00092361
3 7,2 -0.247610 0.108301 0.138466 0.00092361
3 7,3 -0.247145 0.215722 0.429189 0.00086589
3 7,4 -0.243606 0.270943 0.540133 0.00092361
3 7,5 -0.242931 -0.230978 -0.461956 0.00092361
3 7,6 -0.243634 -0.294470 -0.588941 0.00092361
3 8,A 0.407804 0.154763 0.307584 0.00092361
3 8,2 -0.248619 0.214463 0.426813 0.00092361
3 8,3 -0.248156 0.268664 0.535256 0.00086589
3 8,4 -0.244620 -0.232693 -0.465386 0.00092361
3 8,5 -0.243941 -0.294694 -0.589387 0.00092361
3 8,6 -0.244707 -0.348415 -0.696830 0.00092361
3 8,7 -0.245456 -0.400432 -0.800863 0.00092361
3 9,A 0.648917 0.203979 0.405660 0.00092361
3 9,2 -0.254554 0.262771 0.523244 0.00092361
3 9,3 -0.254093 -0.238694 -0.477387 0.00086589
3 9,4 -0.250554 -0.288354 -0.576708 0.00092361
3 9,5 -0.249933 -0.350253 -0.700505 0.00092361
3 9,6 -0.250641 -0.412320 -0.824640 0.00092361
3 9,7 -0.251392 -0.462570 -0.925140 0.00092361
3 9,8 -0.117610 -0.532554 -1.065107 0.00092361
3 10,A 1.500000 0.262854 0.522236 0.00369444
3 10,2 -0.255715 -0.230175 -0.460350 0.00369444
3 10,3 -0.255253 -0.289105 -0.578209 0.00346354
3 10,4 -0.251772 -0.350253 -0.700507 0.00369444
3 10,5 -0.251093 -0.411772 -0.823544 0.00369444
3 10,6 -0.251802 -0.463189 -0.926377 0.00369444
3 10,7 -0.117647 -0.532869 -1.065738 0.00369444
3 10,8 0.147677 -0.623311 -1.246622 0.00369444
3 10,9 0.399611 -0.724314 -1.448628 0.00369444
4 2,A -0.204628 0.104331 0.072829 0.00092361
4 3,A -0.201074 0.082588 0.071144 0.00092361
4 3,2 -0.202456 -0.056398 -0.404912 0.00092361
4 4,A -0.200381 0.059483 0.064430 0.00086589
4 4,2 -0.201763 -0.068519 -0.383358 0.00086589
4 4,3 -0.198210 -0.036671 -0.257639 0.00086589
4 5,A -0.199799 0.039093 0.058790 0.00092361
4 5,2 -0.201184 -0.035076 -0.261375 0.00092361
4 5,3 -0.197602 0.050221 -0.040958 0.00092361
4 5,4 -0.196904 0.143280 0.208968 0.00086589
4 6,A -0.070209 0.063444 0.126889 0.00092361
4 6,2 -0.201826 0.048818 -0.045273 0.00092361
4 6,3 -0.198291 0.142594 0.207579 0.00092361
4 6,4 -0.197595 0.245162 0.490325 0.00086589
4 6,5 -0.197013 0.299204 0.598409 0.00092361
4 7,A 0.182473 0.127404 0.254809 0.00092361
4 7,2 -0.202892 0.138243 0.203383 0.00092361
4 7,3 -0.199359 0.243622 0.487244 0.00092361
4 7,4 -0.198667 0.297036 0.594072 0.00086589
4 7,5 -0.198081 -0.208602 -0.417205 0.00092361
4 7,6 -0.198831 -0.275166 -0.550331 0.00092361
4 8,A 0.421677 0.178307 0.356613 0.00092361
4 8,2 -0.208819 0.235977 0.471954 0.00092361
4 8,3 -0.205287 0.290239 0.580479 0.00092361
4 8,4 -0.204591 -0.213737 -0.427474 0.00086589
4 8,5 -0.204064 -0.278521 -0.557043 0.00092361
4 8,6 -0.204757 -0.335682 -0.671364 0.00092361
4 8,7 -0.205830 -0.391951 -0.783901 0.00092361
4 9,A 0.659211 0.230121 0.460242 0.00092361
4 9,2 -0.210021 0.287113 0.574226 0.00092361
4 9,3 -0.206487 -0.216336 -0.432672 0.00092361
4 9,4 -0.205849 -0.269488 -0.538976 0.00086589
4 9,5 -0.205264 -0.335223 -0.670446 0.00092361
4 9,6 -0.205958 -0.401510 -0.803019 0.00092361
4 9,7 -0.207031 -0.455585 -0.911171 0.00092361
4 9,8 -0.081417 -0.530243 -1.060485 0.00092361
4 10,A 1.500000 0.286579 0.573026 0.00369444
4 10,2 -0.211151 -0.208764 -0.417528 0.00369444
4 10,3 -0.207676 -0.270178 -0.540356 0.00369444
4 10,4 -0.206980 -0.335306 -0.670612 0.00346354
4 10,5 -0.206395 -0.401063 -0.802127 0.00369444
4 10,6 -0.207089 -0.455938 -0.911877 0.00369444
4 10,7 -0.076682 -0.528969 -1.057938 0.00369444
4 10,8 0.173142 -0.612575 -1.225149 0.00369444
4 10,9 0.418446 -0.723295 -1.446590 0.00369444
5 2,A -0.155477 0.139355 0.146873 0.00092361
5 3,A -0.154833 0.117682 0.144734 0.00092361
5 3,2 -0.152592 -0.013403 -0.305183 0.00092361
5 4,A -0.154230 0.095584 0.137644 0.00092361
5 4,2 -0.151991 -0.024723 -0.285513 0.00092361
5 4,3 -0.151321 0.005793 -0.170158 0.00092361
5 5,A -0.153683 0.075227 0.130673 0.00086589
5 5,2 -0.151416 0.007645 -0.172172 0.00086589
5 5,3 -0.150768 0.087177 0.038529 0.00086589
5 5,4 -0.150162 0.178597 0.282882 0.00086589
5 6,A -0.033358 0.102810 0.205619 0.00092361
5 6,2 -0.152481 0.084884 0.033185 0.00092361
5 6,3 -0.151834 0.176105 0.277834 0.00092361
5 6,4 -0.151230 0.275562 0.551125 0.00092361
5 6,5 -0.150677 0.327443 0.654886 0.00086589
5 7,A 0.204943 0.154268 0.308536 0.00092361
5 7,2 -0.158468 0.167137 0.264547 0.00092361
5 7,3 -0.157824 0.268558 0.537116 0.00092361
5 7,4 -0.157217 0.320261 0.640521 0.00092361
5 7,5 -0.156722 -0.189616 -0.379232 0.00086589
5 7,6 -0.157791 -0.260173 -0.520346 0.00092361
5 8,A 0.444620 0.208811 0.417621 0.00092361
5 8,2 -0.159696 0.265838 0.531677 0.00092361
5 8,3 -0.159049 0.317460 0.634919 0.00092361
5 8,4 -0.158501 -0.191732 -0.383465 0.00092361
5 8,5 -0.157947 -0.260486 -0.520972 0.00086589
5 8,6 -0.159018 -0.320513 -0.641026 0.00092361
5 8,7 -0.165007 -0.382453 -0.764906 0.00092361
5 9,A 0.673137 0.259017 0.518035 0.00092361
5 9,2 -0.160734 0.314945 0.629891 0.00092361
5 9,3 -0.160146 -0.194195 -0.388391 0.00092361
5 9,4 -0.159540 -0.251454 -0.502907 0.00092361
5 9,5 -0.158986 -0.319746 -0.639492 0.00086589
5 9,6 -0.160058 -0.389648 -0.779296 0.00092361
5 9,7 -0.166048 -0.450104 -0.900207 0.00092361
5 9,8 -0.044729 -0.515746 -1.031492 0.00092361
5 10,A 1.500000 0.313124 0.626248 0.00369444
5 10,2 -0.161797 -0.186239 -0.372477 0.00369444
5 10,3 -0.161150 -0.252125 -0.504250 0.00369444
5 10,4 -0.160544 -0.319845 -0.639690 0.00369444
5 10,5 -0.159992 -0.388942 -0.777884 0.00346354
5 10,6 -0.161063 -0.448446 -0.896891 0.00369444
5 10,7 -0.044502 -0.515732 -1.031464 0.00369444
5 10,8 0.200313 -0.609397 -1.218794 0.00369444
5 10,9 0.441414 -0.721883 -1.443767 0.00369444
6 2,A -0.144131 0.163183 0.192050 0.00092361
6 3,A -0.143571 0.140795 0.189743 0.00092361
6 3,2 -0.146040 0.003852 -0.292079 0.00092361
6 4,A -0.143053 0.118505 0.184548 0.00092361
6 4,2 -0.145498 -0.006429 -0.266007 0.00092361
6 4,3 -0.144936 0.036440 -0.119054 0.00092361
6 5,A -0.142573 0.102886 0.188512 0.00092361
6 5,2 -0.145039 0.038950 -0.119648 0.00092361
6 5,3 -0.144478 0.126595 0.111435 0.00092361
6 5,4 -0.143958 0.211908 0.346859 0.00092361
6 6,A 0.012075 0.129810 0.259619 0.00086589
6 6,2 -0.150928 0.119576 0.096823 0.00086589
6 6,3 -0.150369 0.204673 0.332377 0.00086589
6 6,4 -0.149845 0.298463 0.596925 0.00086589
6 6,5 -0.149443 0.345182 0.690365 0.00086589
6 7,A 0.278929 0.191679 0.383357 0.00092361
6 7,2 -0.152067 0.199835 0.327660 0.00092361
6 7,3 -0.151503 0.295346 0.590691 0.00092361
6 7,4 -0.151038 0.341636 0.683273 0.00092361
6 7,5 -0.150578 -0.172221 -0.344443 0.00092361
6 7,6 -0.156468 -0.246032 -0.492065 0.00086589
6 8,A 0.492868 0.240289 0.480578 0.00092361
6 8,2 -0.153013 0.292696 0.585391 0.00092361
6 8,3 -0.152509 0.339171 0.678341 0.00092361
6 8,4 -0.151987 -0.173705 -0.347410 0.00092361
6 8,5 -0.151527 -0.243201 -0.486402 0.00092361
6 8,6 -0.157417 -0.305955 -0.611909 0.00086589
6 8,7 -0.158564 -0.364819 -0.729638 0.00092361
6 9,A 0.701886 0.286099 0.572198 0.00092361
6 9,2 -0.154002 0.336855 0.673709 0.00092361
6 9,3 -0.153440 -0.175959 -0.351918 0.00092361
6 9,4 -0.152919 -0.233779 -0.467558 0.00092361
6 9,5 -0.152459 -0.302569 -0.605138 0.00092361
6 9,6 -0.158358 -0.374064 -0.748128 0.00086589
6 9,7 -0.159416 -0.422744 -0.845487 0.00092361
6 9,8 0.006500 -0.502628 -1.005257 0.00092361
6 10,A 1.500000 0.335539 0.671078 0.00369444
6 10,2 -0.155208 -0.167860 -0.335719 0.00369444
6 10,3 -0.154647 -0.234119 -0.468239 0.00369444
6 10,4 -0.154126 -0.302576 -0.605152 0.00369444
6 10,5 -0.153670 -0.371043 -0.742085 0.00369444
6 10,6 -0.159481 -0.422642 -0.845285 0.00346354
6 10,7 0.006566 -0.502570 -1.005140 0.00369444
6 10,8 0.279797 -0.602339 -1.204678 0.00369444
6 10,9 0.493179 -0.718793 -1.437585 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 Stands on Soft 17 — Player Can Split

Dealer Player Stand Hit Double Split
(DAS=No)		 Split
(DAS=Yes)		 Probability
A A,A -0.666194 -0.023042 -0.618147 0.136338 0.136338 0.00026056
A 2,2 -0.665487 -0.254852 -1.330973 -0.436065 -0.436065 0.00029778
A 3,3 -0.664871 -0.311622 -1.302485 -0.483294 -0.483294 0.00029778
A 4,4 -0.663979 -0.199637 -0.806452 -0.564980 -0.564967 0.00029778
A 5,5 -0.663125 0.083604 0.003395 -0.680970 -0.680825 0.00029778
A 6,6 -0.655570 -0.359475 -0.831783 -0.655819 -0.655646 0.00029778
A 7,7 -0.658474 -0.453109 -0.955310 -0.642188 -0.642188 0.00029778
A 8,8 -0.661381 -0.511736 -1.023472 -0.360130 -0.360130 0.00029778
A 9,9 -0.089453 -0.625904 -1.251808 -0.120073 -0.120073 0.00029778
A 10,10 0.654136 -0.859623 -1.719247 -0.325389 -0.325389 0.00507366
2 A,A -0.288391 0.084868 -0.058748 0.493574 0.493574 0.00043294
2 2,2 -0.292402 -0.114641 -0.584803 -0.150876 -0.073966 0.00037882
2 3,3 -0.292366 -0.142977 -0.563092 -0.208846 -0.131262 0.00043294
2 4,4 -0.291419 -0.018929 -0.198235 -0.262294 -0.184246 0.00043294
2 5,5 -0.284214 0.192598 0.380351 -0.317681 -0.271851 0.00043294
2 6,6 -0.285656 -0.253283 -0.506565 -0.268028 -0.188340 0.00043294
2 7,7 -0.287158 -0.372824 -0.745647 -0.191883 -0.112768 0.00043294
2 8,8 -0.288633 -0.467167 -0.934334 0.010806 0.086230 0.00043294
2 9,9 0.125487 -0.623687 -1.247374 0.158954 0.199424 0.00043294
2 10,10 0.636830 -0.853184 -1.706367 0.095590 0.095590 0.00727344
3 A,A -0.247084 0.107339 0.008117 0.540940 0.540940 0.00043294
3 2,2 -0.250905 -0.081843 -0.501811 -0.093509 -0.001174 0.00043294
3 3,3 -0.249976 -0.108859 -0.479328 -0.137574 -0.045956 0.00037882
3 4,4 -0.242856 0.013335 -0.122047 -0.180556 -0.085894 0.00043294
3 5,5 -0.241474 0.217915 0.433895 -0.241122 -0.184195 0.00043294
3 6,6 -0.242888 -0.231077 -0.462154 -0.189528 -0.087869 0.00043294
3 7,7 -0.244441 -0.358404 -0.716809 -0.119313 -0.019984 0.00043294
3 8,8 -0.246472 -0.460584 -0.921167 0.074372 0.159037 0.00043294
3 9,9 0.142932 -0.624231 -1.248461 0.203200 0.249354 0.00043294
3 10,10 0.646830 -0.852909 -1.705817 0.179776 0.179776 0.00727344
4 A,A -0.203147 0.130517 0.077564 0.591106 0.591106 0.00043294
4 2,2 -0.206009 -0.045498 -0.412018 -0.031764 0.075810 0.00043294
4 3,3 -0.198901 -0.066701 -0.378308 -0.050056 0.059764 0.00043294
4 4,4 -0.197491 0.052735 -0.040638 -0.096531 0.015735 0.00037882
4 5,5 -0.196320 0.245694 0.491389 -0.156041 -0.084203 0.00043294
4 6,6 -0.197701 -0.208262 -0.416523 -0.105007 0.018378 0.00043294
4 7,7 -0.199903 -0.343285 -0.686570 -0.039324 0.084187 0.00043294
4 8,8 -0.211757 -0.457247 -0.914494 0.130449 0.224636 0.00043294
4 9,9 0.173230 -0.612558 -1.225116 0.268250 0.319178 0.00043294
4 10,10 0.657136 -0.852691 -1.705382 0.268547 0.268547 0.00727344
5 A,A -0.157688 0.162849 0.148228 0.643588 0.643588 0.00043294
5 2,2 -0.153234 -0.001167 -0.306468 0.059689 0.186583 0.00043294
5 3,3 -0.151949 -0.024753 -0.285067 0.036452 0.168629 0.00043294
5 4,4 -0.150714 0.090177 0.040095 -0.009388 0.126949 0.00043294
5 5,5 -0.149611 0.277401 0.554803 -0.062490 0.028234 0.00037882
5 6,6 -0.151806 -0.186140 -0.372279 -0.018144 0.128988 0.00043294
5 7,7 -0.163779 -0.333129 -0.666258 0.024132 0.167171 0.00043294
5 8,8 -0.166236 -0.450158 -0.900316 0.202504 0.306923 0.00043294
5 9,9 0.200299 -0.609372 -1.218745 0.339065 0.395015 0.00043294
5 10,10 0.671161 -0.852354 -1.704708 0.366530 0.366530 0.00727344
6 A,A -0.141645 0.189020 0.196249 0.689889 0.689889 0.00043294
6 2,2 -0.146596 0.016371 -0.293191 0.101755 0.247368 0.00043294
6 3,3 -0.145457 -0.006444 -0.265922 0.075920 0.224567 0.00043294
6 4,4 -0.144416 0.129206 0.111986 0.023919 0.176484 0.00043294
6 5,5 -0.143496 0.305341 0.610682 -0.031656 0.070816 0.00043294
6 6,6 -0.155330 -0.175559 -0.351118 -0.001380 0.160640 0.00037882
6 7,7 -0.157606 -0.316272 -0.632543 0.092196 0.249423 0.00043294
6 8,8 -0.159434 -0.422727 -0.845454 0.290792 0.403949 0.00043294
6 9,9 0.279430 -0.602385 -1.204771 0.400449 0.460358 0.00043294
6 10,10 0.702265 -0.851570 -1.703141 0.439670 0.439670 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

 

 

Blackjack Appendix 9 — 4 Decks, Dealer Stands on Soft 17 — 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