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# Blackjack Appendix 9 — Eight decks, dealer hits on soft 17, no hole card.

## Introduction

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

• 8 decks
• Dealer hits on soft 17
• No dealer hole card. Player loses everything, including doubles and splits, if dealer gets a blackjack.
• Player may resplit to four hands, except aces
• No drawing to split aces

There are two tables, the first for when the player has two cards of different ranks, the second when the player has two cards of the same rank. 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: -0.007562.
• Player may not double after split: -0.0089865

### Blackjack Appendix 9 - 8 Decks, Dealer Hits on Soft 17

Dealer Player Stand Hit Double Probability
A 2,A -0.722307 -0.378980 -1.024082 0.00061298
A 3,A -0.722128 -0.403033 -1.023795 0.00061298
A 3,2 -0.721955 -0.528987 -1.443909 0.00063275
A 4,A -0.721938 -0.428099 -1.025496 0.00061298
A 4,2 -0.721771 -0.546899 -1.434987 0.00063275
A 4,3 -0.721595 -0.553054 -1.369253 0.00063275
A 5,A -0.722008 -0.453214 -1.028535 0.00061298
A 5,2 -0.721838 -0.552263 -1.369770 0.00063275
A 5,3 -0.721657 -0.492985 -1.193543 0.00063275
A 5,4 -0.721465 -0.395636 -0.932679 0.00063275
A 6,A -0.665036 -0.462428 -0.998987 0.00061298
A 6,2 -0.721522 -0.492995 -1.194090 0.00063275
A 6,3 -0.721341 -0.395857 -0.933368 0.00063275
A 6,4 -0.721150 -0.286345 -0.639456 0.00063275
A 6,5 -0.721201 -0.235977 -0.537869 0.00063275
A 7,A -0.465508 -0.420237 -0.908247 0.00061298
A 7,2 -0.721047 -0.396987 -0.934124 0.00063275
A 7,3 -0.720867 -0.286720 -0.640513 0.00063275
A 7,4 -0.720668 -0.236353 -0.538760 0.00063275
A 7,5 -0.720733 -0.576756 N/V 0.00063275
A 7,6 -0.720423 -0.608597 N/V 0.00063275
A 8,A -0.179419 -0.354184 -0.777187 0.00061298
A 8,2 -0.721601 -0.286725 -0.641713 0.00063275
A 8,3 -0.721414 -0.236609 -0.540563 0.00063275
A 8,4 -0.721224 -0.576638 N/V 0.00063275
A 8,5 -0.721289 -0.607925 N/V 0.00063275
A 8,6 -0.720980 -0.634179 N/V 0.00063275
A 8,7 -0.720514 -0.657747 N/V 0.00063275
A 9,A 0.107241 -0.288474 -0.647190 0.00061298
A 9,2 -0.722146 -0.236819 -0.542207 0.00063275
A 9,3 -0.721967 -0.576694 N/V 0.00063275
A 9,4 -0.721777 -0.603866 N/V 0.00063275
A 9,5 -0.721842 -0.633363 N/V 0.00063275
A 9,6 -0.721530 -0.660628 N/V 0.00063275
A 9,7 -0.721067 -0.682238 N/V 0.00063275
A 9,8 -0.663720 N/V N/V 0.00063275
A 10,A 1.500000 -0.235585 -0.540727 0.00246051
A 10,2 -0.722342 -0.573114 N/V 0.00253989
A 10,3 -0.722163 -0.603400 N/V 0.00253989
A 10,4 -0.721973 -0.632203 N/V 0.00253989
A 10,5 -0.722037 -0.659901 N/V 0.00253989
A 10,6 -0.721732 -0.681829 N/V 0.00253989
A 10,7 -0.663606 N/V N/V 0.00253989
A 10,8 -0.462603 N/V N/V 0.00253989
A 10,9 -0.175266 N/V N/V 0.00253989
2 2,A -0.285456 0.045865 -0.064702 0.00088828
2 3,A -0.285385 0.022282 -0.065038 0.00091693
2 3,2 -0.286200 -0.125977 -0.572401 0.00088828
2 4,A -0.285310 -0.000502 -0.067778 0.00091693
2 4,2 -0.286114 -0.138923 -0.552109 0.00088828
2 4,3 -0.286043 -0.110560 -0.430104 0.00091693
2 5,A -0.283432 -0.021034 -0.069590 0.00091693
2 5,2 -0.284232 -0.108335 -0.427609 0.00088828
2 5,3 -0.284161 -0.023497 -0.201636 0.00091693
2 5,4 -0.284073 0.074907 0.065735 0.00091693
2 6,A -0.153662 -0.000625 -0.005768 0.00091693
2 6,2 -0.284604 -0.023277 -0.201488 0.00088828
2 6,3 -0.284532 0.074560 0.065231 0.00091693
2 6,4 -0.284433 0.185304 0.365961 0.00091693
2 6,5 -0.282557 0.243091 0.481665 0.00091693
2 7,A 0.112397 0.060298 0.115866 0.00091693
2 7,2 -0.284984 0.073498 0.065587 0.00088828
2 7,3 -0.284901 0.184826 0.365041 0.00091693
2 7,4 -0.284813 0.241774 0.478818 0.00091693
2 7,5 -0.282938 -0.253645 N/V 0.00091693
2 7,6 -0.283311 -0.311517 N/V 0.00091693
2 8,A 0.380082 0.121257 0.237818 0.00091693
2 8,2 -0.285334 0.184433 0.364431 0.00088828
2 8,3 -0.285262 0.241095 0.477494 0.00091693
2 8,4 -0.285175 -0.254479 N/V 0.00091693
2 8,5 -0.283301 -0.311423 N/V 0.00091693
2 8,6 -0.283672 -0.363607 N/V 0.00091693
2 8,7 -0.284067 -0.415350 N/V 0.00091693
2 9,A 0.636989 0.181885 0.359249 0.00091693
2 9,2 -0.285842 0.240483 0.476300 0.00088828
2 9,3 -0.285771 -0.255298 N/V 0.00091693
2 9,4 -0.285684 -0.306761 N/V 0.00091693
2 9,5 -0.283809 -0.363110 N/V 0.00091693
2 9,6 -0.284196 -0.420028 N/V 0.00091693
2 9,7 -0.284577 -0.471493 N/V 0.00091693
2 9,8 -0.154450 N/V N/V 0.00091693
2 10,A 1.500000 0.238638 0.471942 0.00366774
2 10,2 -0.288712 -0.252610 N/V 0.00355312
2 10,3 -0.288641 -0.308136 N/V 0.00366774
2 10,4 -0.288555 -0.364138 N/V 0.00366774
2 10,5 -0.286693 -0.420689 N/V 0.00366774
2 10,6 -0.287065 -0.472569 N/V 0.00366774
2 10,7 -0.156996 N/V N/V 0.00366774
2 10,8 0.109835 N/V N/V 0.00366774
2 10,9 0.377972 N/V N/V 0.00366774
3 2,A -0.245149 0.073858 0.000384 0.00091693
3 3,A -0.245077 0.050574 -0.001503 0.00088828
3 3,2 -0.245816 -0.093181 -0.491632 0.00088828
3 4,A -0.243173 0.029292 -0.002766 0.00091693
3 4,2 -0.243908 -0.104369 -0.468510 0.00091693
3 4,3 -0.243823 -0.076947 -0.351216 0.00088828
3 5,A -0.242875 0.008799 -0.005661 0.00091693
3 5,2 -0.243610 -0.074848 -0.349880 0.00091693
3 5,3 -0.243525 0.007457 -0.130965 0.00088828
3 5,4 -0.241611 0.104241 0.128239 0.00091693
3 6,A -0.117176 0.028886 0.056347 0.00091693
3 6,2 -0.243978 0.007675 -0.130851 0.00091693
3 6,3 -0.243882 0.103892 0.127204 0.00088828
3 6,4 -0.241979 0.210091 0.418702 0.00091693
3 6,5 -0.241681 0.266014 0.530603 0.00091693
3 7,A 0.140675 0.088051 0.174678 0.00091693
3 7,2 -0.244337 0.102836 0.127551 0.00091693
3 7,3 -0.244253 0.209576 0.417492 0.00088828
3 7,4 -0.242350 0.264965 0.528507 0.00091693
3 7,5 -0.242053 -0.232647 N/V 0.00091693
3 7,6 -0.242421 -0.293573 N/V 0.00091693
3 8,A 0.399077 0.150105 0.298662 0.00091693
3 8,2 -0.244844 0.208922 0.416261 0.00091693
3 8,3 -0.244760 0.263825 0.526052 0.00088828
3 8,4 -0.242858 -0.233497 N/V 0.00091693
3 8,5 -0.242560 -0.293672 N/V 0.00091693
3 8,6 -0.242943 -0.349725 N/V 0.00091693
3 8,7 -0.243314 -0.404927 N/V 0.00091693
3 9,A 0.645407 0.203764 0.405843 0.00091693
3 9,2 -0.247778 0.260884 0.520083 0.00091693
3 9,3 -0.247695 -0.236468 N/V 0.00088828
3 9,4 -0.245792 -0.290506 N/V 0.00091693
3 9,5 -0.245509 -0.350606 N/V 0.00091693
3 9,6 -0.245878 -0.410802 N/V 0.00091693
3 9,7 -0.246249 -0.465145 N/V 0.00091693
3 9,8 -0.120372 N/V N/V 0.00091693
3 10,A 1.500000 0.260935 0.519541 0.00366774
3 10,2 -0.248340 -0.232264 N/V 0.00366774
3 10,3 -0.248256 -0.290886 N/V 0.00355312
3 10,4 -0.246367 -0.350586 N/V 0.00366774
3 10,5 -0.246069 -0.410526 N/V 0.00366774
3 10,6 -0.246438 -0.465458 N/V 0.00366774
3 10,7 -0.120413 N/V N/V 0.00366774
3 10,8 0.137888 N/V N/V 0.00366774
3 10,9 0.394786 N/V N/V 0.00366774
4 2,A -0.202783 0.104016 0.068239 0.00091693
4 3,A -0.200882 0.082320 0.067545 0.00091693
4 3,2 -0.201546 -0.056246 -0.403093 0.00091693
4 4,A -0.200550 0.060679 0.064202 0.00088828
4 4,2 -0.201215 -0.067786 -0.383499 0.00088828
4 4,3 -0.199309 -0.039525 -0.266631 0.00088828
4 5,A -0.200288 0.041090 0.061395 0.00091693
4 5,2 -0.200953 -0.038781 -0.268623 0.00091693
4 5,3 -0.199041 0.042717 -0.055481 0.00091693
4 5,4 -0.198708 0.134693 0.193429 0.00088828
4 6,A -0.078144 0.060917 0.121835 0.00091693
4 6,2 -0.201298 0.042057 -0.057600 0.00091693
4 6,3 -0.199397 0.134353 0.192741 0.00091693
4 6,4 -0.199065 0.236916 0.473832 0.00088828
4 6,5 -0.198803 0.290820 0.581640 0.00091693
4 7,A 0.169828 0.120881 0.241762 0.00091693
4 7,2 -0.201818 0.132236 0.190685 0.00091693
4 7,3 -0.199918 0.236172 0.472344 0.00091693
4 7,4 -0.199586 0.289790 0.579580 0.00088828
4 7,5 -0.199324 -0.211388 N/V 0.00091693
4 7,6 -0.199695 -0.275388 N/V 0.00091693
4 8,A 0.415771 0.175525 0.351051 0.00091693
4 8,2 -0.204749 0.232415 0.464830 0.00091693
4 8,3 -0.202850 0.286419 0.572839 0.00091693
4 8,4 -0.202518 -0.213931 N/V 0.00088828
4 8,5 -0.202270 -0.277039 N/V 0.00091693
4 8,6 -0.202627 -0.336396 N/V 0.00091693
4 8,7 -0.203148 -0.395297 N/V 0.00091693
4 9,A 0.656117 0.229480 0.458960 0.00091693
4 9,2 -0.205361 0.284846 0.569693 0.00091693
4 9,3 -0.203461 -0.215228 N/V 0.00091693
4 9,4 -0.203143 -0.272555 N/V 0.00088828
4 9,5 -0.202881 -0.336164 N/V 0.00091693
4 9,6 -0.203238 -0.400046 N/V 0.00091693
4 9,7 -0.203760 -0.457833 N/V 0.00091693
4 9,8 -0.083846 N/V N/V 0.00091693
4 10,A 1.500000 0.284544 0.569088 0.00366774
4 10,2 -0.205889 -0.211461 N/V 0.00366774
4 10,3 -0.204003 -0.272906 N/V 0.00366774
4 10,4 -0.203670 -0.336170 N/V 0.00355312
4 10,5 -0.203408 -0.399814 N/V 0.00366774
4 10,6 -0.203765 -0.458008 N/V 0.00366774
4 10,7 -0.081488 N/V N/V 0.00366774
4 10,8 0.164845 N/V N/V 0.00366774
4 10,9 0.413885 N/V N/V 0.00366774
5 2,A -0.159015 0.136612 0.137558 0.00091693
5 3,A -0.158697 0.115190 0.136507 0.00091693
5 3,2 -0.157469 -0.017467 -0.314937 0.00091693
5 4,A -0.158403 0.094321 0.132980 0.00091693
5 4,2 -0.157175 -0.028476 -0.296263 0.00091693
5 4,3 -0.156850 -0.000668 -0.185034 0.00091693
5 5,A -0.158146 0.074998 0.129483 0.00088828
5 5,2 -0.156911 0.000229 -0.186052 0.00088828
5 5,3 -0.156593 0.078071 0.019971 0.00088828
5 5,4 -0.156297 0.167540 0.261824 0.00088828
5 6,A -0.040456 0.096732 0.193463 0.00091693
5 6,2 -0.157428 0.076942 0.017342 0.00091693
5 6,3 -0.157110 0.166315 0.259345 0.00091693
5 6,4 -0.156815 0.265409 0.530817 0.00091693
5 6,5 -0.156557 0.317168 0.634336 0.00088828
5 7,A 0.197978 0.149995 0.299990 0.00091693
5 7,2 -0.160373 0.161897 0.252802 0.00091693
5 7,3 -0.160055 0.261967 0.523935 0.00091693
5 7,4 -0.159759 0.313657 0.627314 0.00091693
5 7,5 -0.159515 -0.191597 N/V 0.00088828
5 7,6 -0.160033 -0.259068 N/V 0.00091693
5 8,A 0.438984 0.205232 0.410465 0.00091693
5 8,2 -0.160988 0.260620 0.521239 0.00091693
5 8,3 -0.160670 0.312262 0.624524 0.00091693
5 8,4 -0.160389 -0.192649 N/V 0.00091693
5 8,5 -0.160130 -0.259226 N/V 0.00088828
5 8,6 -0.160649 -0.321492 N/V 0.00091693
5 8,7 -0.163593 -0.384688 N/V 0.00091693
5 9,A 0.669897 0.257245 0.514491 0.00091693
5 9,2 -0.161530 0.310991 0.621982 0.00091693
5 9,3 -0.161225 -0.193886 N/V 0.00091693
5 9,4 -0.160930 -0.254748 N/V 0.00091693
5 9,5 -0.160672 -0.321109 N/V 0.00088828
5 9,6 -0.161190 -0.388279 N/V 0.00091693
5 9,7 -0.164136 -0.450710 N/V 0.00091693
5 9,8 -0.046183 N/V N/V 0.00091693
5 10,A 1.500000 0.310105 0.620210 0.00366774
5 10,2 -0.162025 -0.189934 N/V 0.00366774
5 10,3 -0.161706 -0.255084 N/V 0.00366774
5 10,4 -0.161411 -0.321139 N/V 0.00366774
5 10,5 -0.161153 -0.387915 N/V 0.00355312
5 10,6 -0.161672 -0.449904 N/V 0.00366774
5 10,7 -0.046071 N/V N/V 0.00366774
5 10,8 0.195388 N/V N/V 0.00366774
5 10,9 0.437176 N/V N/V 0.00366774
6 2,A -0.116815 0.166144 0.202168 0.00091693
6 3,A -0.116523 0.145750 0.201233 0.00091693
6 3,2 -0.116691 0.018226 -0.233383 0.00091693
6 4,A -0.116252 0.126128 0.198770 0.00091693
6 4,2 -0.116414 0.008594 -0.215234 0.00091693
6 4,3 -0.116121 0.034461 -0.107934 0.00091693
6 5,A -0.116167 0.110070 0.200517 0.00091693
6 5,2 -0.116339 0.035424 -0.108590 0.00091693
6 5,3 -0.116046 0.109541 0.087946 0.00091693
6 5,4 -0.115774 0.195062 0.319528 0.00091693
6 6,A -0.005284 0.125698 0.251397 0.00088828
6 6,2 -0.119286 0.106038 0.080627 0.00088828
6 6,3 -0.118993 0.191438 0.312261 0.00088828
6 6,4 -0.118720 0.287722 0.575443 0.00088828
6 6,5 -0.118660 0.338026 0.676052 0.00088828
6 7,A 0.222828 0.178233 0.356466 0.00091693
6 7,2 -0.119903 0.189030 0.309711 0.00091693
6 7,3 -0.119610 0.286189 0.572377 0.00091693
6 7,4 -0.119352 0.336525 0.673051 0.00091693
6 7,5 -0.119277 -0.172996 N/V 0.00091693
6 7,6 -0.122223 -0.245002 N/V 0.00088828
6 8,A 0.452450 0.230918 0.461836 0.00091693
6 8,2 -0.120443 0.284853 0.569707 0.00091693
6 8,3 -0.120164 0.335134 0.670268 0.00091693
6 8,4 -0.119892 -0.173804 N/V 0.00091693
6 8,5 -0.119817 -0.243618 N/V 0.00091693
6 8,6 -0.122764 -0.310248 N/V 0.00088828
6 8,7 -0.123382 -0.374905 N/V 0.00091693
6 9,A 0.677749 0.281742 0.563484 0.00091693
6 9,2 -0.120958 0.333887 0.667774 0.00091693
6 9,3 -0.120665 -0.175023 N/V 0.00091693
6 9,4 -0.120393 -0.238963 N/V 0.00091693
6 9,5 -0.120318 -0.308532 N/V 0.00091693
6 9,6 -0.123265 -0.379473 N/V 0.00088828
6 9,7 -0.123863 -0.438881 N/V 0.00091693
6 9,8 -0.009010 N/V N/V 0.00091693
6 10,A 1.500000 0.333231 0.666462 0.00366774
6 10,2 -0.121374 -0.171034 N/V 0.00366774
6 10,3 -0.121081 -0.239285 N/V 0.00366774
6 10,4 -0.120809 -0.308390 N/V 0.00366774
6 10,5 -0.120734 -0.377945 N/V 0.00366774
6 10,6 -0.123661 -0.438855 N/V 0.00355312
6 10,7 -0.008904 N/V N/V 0.00366774
6 10,8 0.221064 N/V N/V 0.00366774
6 10,9 0.451179 N/V N/V 0.00366774
7 2,A -0.473735 0.120737 -0.180581 0.00091693
7 3,A -0.473491 0.077289 -0.182645 0.00091693
7 3,2 -0.474559 -0.119475 -0.949118 0.00091693
7 4,A -0.473232 0.036559 -0.178852 0.00091693
7 4,2 -0.474305 -0.153286 -0.891184 0.00091693
7 4,3 -0.474060 -0.069042 -0.583711 0.00091693
7 5,A -0.474314 -0.007128 -0.184212 0.00091693
7 5,2 -0.475388 -0.068693 -0.586381 0.00091693
7 5,3 -0.475143 0.083432 -0.180897 0.00091693
7 5,4 -0.474889 0.175175 0.114653 0.00091693
7 6,A -0.104568 0.054483 -0.010074 0.00091693
7 6,2 -0.475959 0.083509 -0.182622 0.00091693
7 6,3 -0.475713 0.175017 0.113018 0.00091693
7 6,4 -0.475473 0.260407 0.402606 0.00091693
7 6,5 -0.476555 0.293108 0.468053 0.00091693
7 7,A 0.401292 0.171590 0.222746 0.00088828
7 7,2 -0.476479 0.173688 0.111258 0.00088828
7 7,3 -0.476248 0.259479 0.399175 0.00088828
7 7,4 -0.475994 0.292702 0.466480 0.00088828
7 7,5 -0.477075 -0.217865 N/V 0.00088828
7 7,6 -0.477646 -0.276056 N/V 0.00088828
7 8,A 0.615795 0.220876 0.320372 0.00091693
7 8,2 -0.476910 0.258155 0.395355 0.00091693
7 8,3 -0.476665 0.292162 0.464122 0.00091693
7 8,4 -0.476411 -0.216650 N/V 0.00091693
7 8,5 -0.477493 -0.275854 N/V 0.00091693
7 8,6 -0.478066 -0.324429 N/V 0.00091693
7 8,7 -0.478567 -0.364365 N/V 0.00088828
7 9,A 0.773263 0.255216 0.387570 0.00091693
7 9,2 -0.477554 0.291847 0.462131 0.00091693
7 9,3 -0.477309 -0.216882 N/V 0.00091693
7 9,4 -0.477055 -0.269599 N/V 0.00091693
7 9,5 -0.478138 -0.324368 N/V 0.00091693
7 9,6 -0.478689 -0.369001 N/V 0.00091693
7 9,7 -0.479214 -0.410030 N/V 0.00088828
7 9,8 -0.109007 N/V N/V 0.00091693
7 10,A 1.500000 0.291454 0.463548 0.00366774
7 10,2 -0.475044 -0.212688 N/V 0.00366774
7 10,3 -0.474799 -0.269217 N/V 0.00366774
7 10,4 -0.474546 -0.323738 N/V 0.00366774
7 10,5 -0.475618 -0.369137 N/V 0.00366774
7 10,6 -0.476193 -0.410173 N/V 0.00366774
7 10,7 -0.108354 N/V N/V 0.00355312
7 10,8 0.398196 N/V N/V 0.00366774
7 10,9 0.615281 N/V N/V 0.00366774
8 2,A -0.510138 0.052400 -0.314185 0.00091693
8 3,A -0.509879 0.015663 -0.307428 0.00091693
8 3,2 -0.510811 -0.187408 -1.021621 0.00091693
8 4,A -0.510949 -0.028065 -0.314334 0.00091693
8 4,2 -0.511880 -0.219089 -1.003819 0.00091693
8 4,3 -0.511626 -0.212600 -0.847284 0.00091693
8 5,A -0.510897 -0.068732 -0.316464 0.00091693
8 5,2 -0.511829 -0.211400 -0.847677 0.00091693
8 5,3 -0.511574 -0.059567 -0.449031 0.00091693
8 5,4 -0.512657 0.099532 -0.022871 0.00091693
8 6,A -0.382427 -0.072047 -0.252297 0.00091693
8 6,2 -0.512328 -0.059384 -0.450460 0.00091693
8 6,3 -0.512087 0.100484 -0.021718 0.00091693
8 6,4 -0.513156 0.199206 0.290356 0.00091693
8 6,5 -0.513103 0.230133 0.352774 0.00091693
8 7,A 0.107517 0.040692 -0.028645 0.00091693
8 7,2 -0.512783 0.099394 -0.022901 0.00091693
8 7,3 -0.512528 0.200030 0.291084 0.00091693
8 7,4 -0.513597 0.229092 0.349503 0.00091693
8 7,5 -0.513545 -0.277274 N/V 0.00091693
8 7,6 -0.514046 -0.331884 N/V 0.00091693
8 8,A 0.595375 0.152736 0.194301 0.00088828
8 8,2 -0.513429 0.198861 0.287319 0.00088828
8 8,3 -0.513175 0.228923 0.348233 0.00088828
8 8,4 -0.514244 -0.277171 N/V 0.00088828
8 8,5 -0.514192 -0.330999 N/V 0.00088828
8 8,6 -0.514672 -0.371589 N/V 0.00088828
8 8,7 -0.515118 -0.412415 N/V 0.00088828
8 9,A 0.790948 0.194779 0.279775 0.00091693
8 9,2 -0.510922 0.228210 0.347912 0.00091693
8 9,3 -0.510668 -0.276866 N/V 0.00091693
8 9,4 -0.511737 -0.325404 N/V 0.00091693
8 9,5 -0.511675 -0.371645 N/V 0.00091693
8 9,6 -0.512179 -0.417255 N/V 0.00091693
8 9,7 -0.512624 -0.454840 N/V 0.00091693
8 9,8 -0.386104 N/V N/V 0.00088828
8 10,A 1.500000 0.228882 0.348517 0.00366774
8 10,2 -0.511354 -0.271893 N/V 0.00366774
8 10,3 -0.511101 -0.324221 N/V 0.00366774
8 10,4 -0.512159 -0.370188 N/V 0.00366774
8 10,5 -0.512109 -0.416875 N/V 0.00366774
8 10,6 -0.512612 -0.454468 N/V 0.00366774
8 10,7 -0.383670 N/V N/V 0.00366774
8 10,8 0.104526 N/V N/V 0.00355312
8 10,9 0.591784 N/V N/V 0.00366774
9 2,A -0.540131 -0.035002 -0.446509 0.00091693
9 3,A -0.541202 -0.073354 -0.448880 0.00091693
9 3,2 -0.541971 -0.266148 -1.083943 0.00091693
9 4,A -0.541140 -0.112360 -0.452323 0.00091693
9 4,2 -0.541909 -0.294053 -1.065215 0.00091693
9 4,3 -0.542979 -0.287676 -0.955758 0.00091693
9 5,A -0.541066 -0.150716 -0.455924 0.00091693
9 5,2 -0.541835 -0.285428 -0.953491 0.00091693
9 5,3 -0.542919 -0.211080 -0.716426 0.00091693
9 5,4 -0.542856 -0.052188 -0.298309 0.00091693
9 6,A -0.421226 -0.147898 -0.394384 0.00091693
9 6,2 -0.542313 -0.209957 -0.715210 0.00091693
9 6,3 -0.543383 -0.052069 -0.299392 0.00091693
9 6,4 -0.543320 0.116689 0.146896 0.00091693
9 6,5 -0.543245 0.157633 0.229476 0.00091693
9 7,A -0.182777 -0.099039 -0.286193 0.00091693
9 7,2 -0.542938 -0.052171 -0.298024 0.00091693
9 7,3 -0.544008 0.116649 0.145410 0.00091693
9 7,4 -0.543945 0.157219 0.227339 0.00091693
9 7,5 -0.543872 -0.345808 N/V 0.00091693
9 7,6 -0.544316 -0.390848 N/V 0.00091693
9 8,A 0.287611 0.007557 -0.071618 0.00091693
9 8,2 -0.540433 0.116784 0.147584 0.00091693
9 8,3 -0.541503 0.156077 0.226257 0.00091693
9 8,4 -0.541441 -0.346546 N/V 0.00091693
9 8,5 -0.541357 -0.390263 N/V 0.00091693
9 8,6 -0.541825 -0.431593 N/V 0.00091693
9 8,7 -0.542453 -0.468185 N/V 0.00091693
9 9,A 0.759204 0.114032 0.139970 0.00088828
9 9,2 -0.540867 0.156445 0.226041 0.00088828
9 9,3 -0.541939 -0.346151 N/V 0.00088828
9 9,4 -0.541866 -0.384540 N/V 0.00088828
9 9,5 -0.541793 -0.430361 N/V 0.00088828
9 9,6 -0.542261 -0.472500 N/V 0.00088828
9 9,7 -0.542889 -0.506083 N/V 0.00088828
9 9,8 -0.421755 N/V N/V 0.00088828
9 10,A 1.500000 0.157075 0.226367 0.00366774
9 10,2 -0.541272 -0.340528 N/V 0.00366774
9 10,3 -0.542332 -0.383833 N/V 0.00366774
9 10,4 -0.542271 -0.428907 N/V 0.00366774
9 10,5 -0.542198 -0.472034 N/V 0.00366774
9 10,6 -0.542666 -0.505749 N/V 0.00366774
9 10,7 -0.422418 N/V N/V 0.00366774
9 10,8 -0.184682 N/V N/V 0.00366774
9 10,9 0.284831 N/V N/V 0.00355312
10 2,A -0.574759 -0.170282 -0.621791 0.00339243
10 3,A -0.574696 -0.202427 -0.621897 0.00339243
10 3,2 -0.576778 -0.366148 -1.153556 0.00338355
10 4,A -0.574637 -0.235399 -0.623600 0.00339243
10 4,2 -0.576718 -0.390032 -1.136143 0.00338355
10 4,3 -0.576668 -0.373544 -1.032544 0.00338355
10 5,A -0.574558 -0.268805 -0.628058 0.00339243
10 5,2 -0.576654 -0.372158 -1.032528 0.00338355
10 5,3 -0.576589 -0.307704 -0.842668 0.00338355
10 5,4 -0.576529 -0.216968 -0.582237 0.00338355
10 6,A -0.463168 -0.256283 -0.571461 0.00339243
10 6,2 -0.577277 -0.307701 -0.843776 0.00338355
10 6,3 -0.577213 -0.217393 -0.583325 0.00338355
10 6,4 -0.577153 -0.053111 -0.160846 0.00338355
10 6,5 -0.577075 0.032111 0.009954 0.00338355
10 7,A -0.240852 -0.207463 -0.468104 0.00339243
10 7,2 -0.574772 -0.218687 -0.582087 0.00338355
10 7,3 -0.574708 -0.053600 -0.160458 0.00338355
10 7,4 -0.574649 0.031631 0.009895 0.00338355
10 7,5 -0.574560 -0.429420 N/V 0.00338355
10 7,6 -0.575187 -0.472540 N/V 0.00338355
10 8,A -0.016539 -0.156221 -0.366613 0.00339243
10 8,2 -0.575207 -0.053527 -0.161365 0.00338355
10 8,3 -0.575143 0.031603 0.008891 0.00338355
10 8,4 -0.575073 -0.429583 N/V 0.00338355
10 8,5 -0.574996 -0.472127 N/V 0.00338355
10 8,6 -0.575623 -0.507515 N/V 0.00338355
10 8,7 -0.573132 -0.539616 N/V 0.00338355
10 9,A 0.437866 -0.053467 -0.162810 0.00339243
10 9,2 -0.575611 0.031051 0.006825 0.00338355
10 9,3 -0.575537 -0.429911 N/V 0.00338355
10 9,4 -0.575479 -0.466919 N/V 0.00338355
10 9,5 -0.575402 -0.506207 N/V 0.00338355
10 9,6 -0.576028 -0.543461 N/V 0.00338355
10 9,7 -0.573537 -0.572420 N/V 0.00338355
10 9,8 -0.461445 N/V N/V 0.00338355
10 10,A 1.500000 0.033761 0.011586 0.01346372
10 10,2 -0.575975 -0.425294 N/V 0.01342848
10 10,3 -0.575913 -0.466330 N/V 0.01342848
10 10,4 -0.575855 -0.505451 N/V 0.01342848
10 10,5 -0.575778 -0.542476 N/V 0.01342848
10 10,6 -0.576404 -0.571928 N/V 0.01342848
10 10,7 -0.463807 N/V N/V 0.01342848
10 10,8 -0.239373 N/V N/V 0.01342848
10 10,9 -0.014886 N/V N/V 0.01342848

### Blackjack Appendix 9 -- 8 Decks, Dealer Hits on Soft 17

Dealer Player Stand Hit Double Split
DAS -- No
Split
DAS -- Yes
Probability
A A,A -0.722480 -0.354395 -1.024545 -0.539428 -0.539428 0.00028733
A 2,2 -0.722134 -0.511819 -1.444267 -1.033122 -1.033122 0.00030649
A 3,3 -0.721782 -0.546902 -1.435007 -1.064395 -1.064395 0.00030649
A 4,4 -0.721403 -0.491952 -1.192972 -1.112964 -1.112964 0.00030649
A 5,5 -0.721529 -0.286424 -0.639356 -1.191566 -1.191566 0.00030649
A 6,6 -0.720890 -0.576632 N/V -1.175449 -1.175449 0.00030649
A 7,7 -0.719959 -0.638001 N/V -1.177474 -1.177474 0.00030649
A 8,8 -0.721070 -0.682189 N/V -1.029094 -1.029094 0.00030649
A 9,9 -0.462484 N/V N/V -0.840183 -0.840183 0.00030649
A 10,10 0.112335 N/V N/V -0.742847 -0.742847 0.00505771
2 A,A -0.284645 0.081466 -0.062173 0.481540 0.481540 0.00044414
2 2,2 -0.286259 -0.112626 -0.572519 -0.155196 -0.080006 0.00041638
2 3,3 -0.286129 -0.138949 -0.552136 -0.206369 -0.130638 0.00044414
2 4,4 -0.285956 -0.022792 -0.203115 -0.258929 -0.182945 0.00044414
2 5,5 -0.282186 0.185996 0.367548 -0.319547 -0.275097 0.00044414
2 6,6 -0.282929 -0.253650 N/V -0.270820 -0.194232 0.00044414
2 7,7 -0.283691 -0.368785 N/V -0.200588 -0.123412 0.00044414
2 8,8 -0.284429 -0.471348 N/V -0.002055 0.072611 0.00044414
2 9,9 0.112141 N/V N/V 0.153189 0.194489 0.00044414
2 10,10 0.633452 N/V N/V -0.023125 -0.023125 0.00727817
3 A,A -0.244405 0.103750 0.003080 0.528868 0.528868 0.00044414
3 2,2 -0.245899 -0.080271 -0.491799 -0.098512 -0.008962 0.00044414
3 3,3 -0.245732 -0.105607 -0.471977 -0.137081 -0.047839 0.00041638
3 4,4 -0.241915 0.008401 -0.130794 -0.183062 -0.092303 0.00044414
3 5,5 -0.241313 0.210620 0.419754 -0.245336 -0.191185 0.00044414
3 6,6 -0.242051 -0.232674 N/V -0.194733 -0.098149 0.00044414
3 7,7 -0.242807 -0.354696 N/V -0.129305 -0.032580 0.00044414
3 8,8 -0.243822 -0.464173 N/V 0.062177 0.146169 0.00044414
3 9,9 0.135585 N/V N/V 0.205014 0.251428 0.00044414
3 10,10 0.644190 N/V N/V 0.025664 0.025664 0.00727817
4 A,A -0.202098 0.128661 0.070502 0.578111 0.578111 0.00044414
4 2,2 -0.203452 -0.045028 -0.406904 -0.033961 0.070794 0.00044414
4 3,3 -0.199640 -0.066794 -0.380712 -0.056360 0.049514 0.00044414
4 4,4 -0.198971 0.043972 -0.055286 -0.103434 0.003880 0.00041638
4 5,5 -0.198447 0.237153 0.474306 -0.164048 -0.095272 0.00044414
4 6,6 -0.199159 -0.211201 N/V -0.113919 0.004017 0.00044414
4 7,7 -0.200216 -0.340184 N/V -0.052113 0.067081 0.00044414
4 8,8 -0.206080 -0.458635 N/V 0.124280 0.217652 0.00044414
4 9,9 0.164860 N/V N/V 0.268756 0.319779 0.00044414
4 10,10 0.654940 N/V N/V 0.076800 0.076800 0.00727817
5 A,A -0.160232 0.159763 0.138175 0.628892 0.628892 0.00044414
5 2,2 -0.157786 -0.005663 -0.315573 0.049078 0.170547 0.00044414
5 3,3 -0.157151 -0.028484 -0.296039 0.023708 0.149738 0.00044414
5 4,4 -0.156555 0.079564 0.020767 -0.022598 0.107326 0.00044414
5 5,5 -0.156040 0.266294 0.532588 -0.078941 0.007421 0.00041638
5 6,6 -0.157089 -0.189878 N/V -0.032778 0.107528 0.00044414
5 7,7 -0.162978 -0.327749 N/V 0.018045 0.157179 0.00044414
5 8,8 -0.164209 -0.450726 N/V 0.196814 0.299919 0.00044414
5 9,9 0.195376 N/V N/V 0.337066 0.392814 0.00044414
5 10,10 0.668790 N/V N/V 0.132181 0.132181 0.00727817
6 A,A -0.116641 0.188937 0.203740 0.675893 0.675893 0.00044414
6 2,2 -0.116983 0.029218 -0.233966 0.122449 0.264417 0.00044414
6 3,3 -0.116393 0.008593 -0.215191 0.097860 0.243244 0.00044414
6 4,4 -0.115849 0.110966 0.088623 0.053572 0.202590 0.00044414
6 5,5 -0.115700 0.290927 0.581855 -0.002754 0.098205 0.00044414
6 6,6 -0.121606 -0.174496 N/V 0.033551 0.190853 0.00041638
6 7,7 -0.122841 -0.315381 N/V 0.091084 0.249695 0.00044414
6 8,8 -0.123902 -0.438879 N/V 0.261636 0.374948 0.00044414
6 9,9 0.221002 N/V N/V 0.395347 0.455232 0.00044414
6 10,10 0.677004 N/V N/V 0.182772 0.182772 0.00727817
7 A,A -0.472657 0.164619 -0.178282 0.472412 0.472412 0.00044414
7 2,2 -0.474809 -0.088690 -0.949618 -0.048579 0.008153 0.00044414
7 3,3 -0.474314 -0.153324 -0.891195 -0.108462 -0.051586 0.00044414
7 4,4 -0.473807 0.085502 -0.178278 -0.225917 -0.168204 0.00044414
7 5,5 -0.475985 0.259518 0.401662 -0.329653 -0.296742 0.00044414
7 6,6 -0.477126 -0.218600 N/V -0.313801 -0.255866 0.00044414
7 7,7 -0.478169 -0.329129 N/V -0.105023 -0.048918 0.00041638
7 8,8 -0.478988 -0.410029 N/V 0.264576 0.319500 0.00044414
7 9,9 0.399563 N/V N/V 0.334481 0.364447 0.00044414
7 10,10 0.772322 N/V N/V 0.061681 0.061681 0.00727817
8 A,A -0.509202 0.094712 -0.312243 0.357443 0.357443 0.00044414
8 2,2 -0.511064 -0.157443 -1.022129 -0.215029 -0.176800 0.00044414
8 3,3 -0.510557 -0.218685 -1.001173 -0.269707 -0.231032 0.00044414
8 4,4 -0.512695 -0.059294 -0.451269 -0.365079 -0.325947 0.00044414
8 5,5 -0.512604 0.199141 0.291445 -0.481348 -0.458202 0.00044414
8 6,6 -0.513603 -0.277348 N/V -0.458017 -0.418724 0.00044414
8 7,7 -0.514468 -0.376254 N/V -0.429551 -0.391334 0.00044414
8 8,8 -0.515767 -0.454672 N/V -0.064082 -0.027641 0.00041638
8 9,9 0.100941 N/V N/V 0.209863 0.231110 0.00044414
8 10,10 0.790770 N/V N/V -0.090050 -0.090050 0.00727817
9 A,A -0.539362 -0.000231 -0.452269 0.235248 0.235248 0.00044414
9 2,2 -0.540900 -0.238579 -1.081800 -0.403973 -0.385126 0.00044414
9 3,3 -0.543042 -0.294599 -1.067483 -0.451476 -0.432996 0.00044414
9 4,4 -0.542930 -0.209943 -0.716400 -0.528123 -0.509275 0.00044414
9 5,5 -0.542781 0.116614 0.147967 -0.651132 -0.637534 0.00044414
9 6,6 -0.543711 -0.345349 N/V -0.623318 -0.604348 0.00044414
9 7,7 -0.544944 -0.436155 N/V -0.592570 -0.574705 0.00044414
9 8,8 -0.539962 -0.506614 N/V -0.407864 -0.389375 0.00044414
9 9,9 -0.184712 N/V N/V -0.092854 -0.080698 0.00041638
9 10,10 0.756649 N/V N/V -0.257088 -0.257088 0.00727817
10 A,A -0.572677 -0.135004 -0.616845 0.022953 0.022953 0.00164751
10 2,2 -0.576842 -0.342832 -1.153683 -0.641336 -0.641336 0.00163891
10 3,3 -0.576714 -0.389720 -1.136132 -0.683773 -0.683773 0.00163891
10 4,4 -0.576608 -0.306594 -0.842677 -0.750924 -0.750924 0.00163891
10 5,5 -0.576450 -0.052878 -0.159774 -0.851550 -0.851550 0.00163891
10 6,6 -0.577678 -0.429630 N/V -0.833092 -0.833092 0.00163891
10 7,7 -0.572696 -0.513025 N/V -0.781172 -0.781172 0.00163891
10 8,8 -0.573568 -0.572399 N/V -0.617779 -0.617779 0.00163891
10 9,9 -0.236981 N/V N/V -0.465030 -0.465030 0.00163891
10 10,10 0.437272 N/V N/V -0.422795 -0.422795 0.02643732

N/V = Not a viable play.

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