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.g    Blackjack Appendix 9 - 1 deck, dealer stands on soft 17 Last Update: March 27, 2005 Following are the expected returns for each initial hands in blackjack under the following rules. 1 deck Dealer stands on soft 17 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: 0.001839 Player may not double after split: 0.000423 Blackjack Appendix 9 - 1 deck, dealer stands on soft 17 Dealer Player Stand Hit Double Probability A 2,A -0.661732 -0.067808 -0.593281 0.00048758 A 3,A -0.661047 -0.100568 -0.595803 0.00048758 A 3,2 -0.659375 -0.291565 -1.318751 0.00065011 A 4,A -0.658818 -0.153931 -0.619882 0.00048758 A 4,2 -0.658712 -0.334901 -1.295655 0.00065011 A 4,3 -0.657967 -0.345048 -1.126423 0.00065011 A 5,A -0.659151 -0.206061 -0.656070 0.00048758 A 5,2 -0.657529 -0.331115 -1.125237 0.00065011 A 5,3 -0.656761 -0.217133 -0.793964 0.00065011 A 5,4 -0.654533 -0.070215 -0.395771 0.00065011 A 6,A -0.482815 -0.199899 -0.527213 0.00048758 A 6,2 -0.639160 -0.225772 -0.799095 0.00065011 A 6,3 -0.638344 -0.078495 -0.401329 0.00065011 A 6,4 -0.636118 0.081506 0.048615 0.00065011 A 6,5 -0.634013 0.172946 0.240259 0.00065011 A 7,A -0.101015 -0.108386 -0.357778 0.00048758 A 7,2 -0.645559 -0.082785 -0.402057 0.00065011 A 7,3 -0.644759 0.086843 0.040179 0.00065011 A 7,4 -0.641782 0.171306 0.222041 0.00065011 A 7,5 -0.640619 -0.376769 -0.837456 0.00065011 A 7,6 -0.623767 -0.443347 -0.937178 0.00065011 A 8,A 0.289743 -0.017242 -0.192057 0.00048758 A 8,2 -0.651889 0.086448 0.025084 0.00065011 A 8,3 -0.650336 0.170920 0.205672 0.00065011 A 8,4 -0.648266 -0.374915 -0.846660 0.00065011 A 8,5 -0.647201 -0.433030 -0.930925 0.00065011 A 8,6 -0.630436 -0.453389 -0.941763 0.00065011 A 8,7 -0.637158 -0.455656 -0.928015 0.00065011 A 9,A 0.680745 0.067505 -0.040657 0.00048758 A 9,2 -0.657160 0.169644 0.188935 0.00065011 A 9,3 -0.656577 -0.374483 -0.858217 0.00065011 A 9,4 -0.654534 -0.383075 -0.838019 0.00065011 A 9,5 -0.653430 -0.441215 -0.920605 0.00065011 A 9,6 -0.636809 -0.495853 -1.008370 0.00065011 A 9,7 -0.643247 -0.495493 -0.990987 0.00065011 A 9,8 -0.451875 -0.544608 -1.089217 0.00065011 A 10,A 1.500000 0.152526 0.145194 0.00200942 A 10,2 -0.672788 -0.348923 -0.810930 0.00267923 A 10,3 -0.672168 -0.392503 -0.859049 0.00267923 A 10,4 -0.670237 -0.444961 -0.931613 0.00267923 A 10,5 -0.669157 -0.498741 -1.012900 0.00267923 A 10,6 -0.652909 -0.508750 -1.017500 0.00267923 A 10,7 -0.467041 -0.555802 -1.111604 0.00267923 A 10,8 -0.082020 -0.633177 -1.266355 0.00267923 A 10,9 0.307676 -0.742645 -1.485290 0.00267923 2 2,A -0.282713 0.039266 -0.042020 0.00072398 2 3,A -0.284212 0.016914 -0.046786 0.00096531 2 3,2 -0.293290 -0.131387 -0.586579 0.00072398 2 4,A -0.283157 -0.011697 -0.069980 0.00096531 2 4,2 -0.290967 -0.150708 -0.560134 0.00072398 2 4,3 -0.292521 -0.122957 -0.429083 0.00096531 2 5,A -0.266382 -0.031724 -0.081847 0.00096531 2 5,2 -0.274153 -0.099369 -0.401783 0.00072398 2 5,3 -0.275744 -0.016523 -0.167800 0.00096531 2 5,4 -0.273342 0.093338 0.113509 0.00096531 2 6,A -0.131767 0.007098 0.013321 0.00096531 2 6,2 -0.277108 -0.012997 -0.162783 0.00072398 2 6,3 -0.278474 0.092149 0.111874 0.00096531 2 6,4 -0.275236 0.217494 0.432187 0.00096531 2 6,5 -0.258788 0.284031 0.567187 0.00096531 2 7,A 0.135802 0.065248 0.127578 0.00096531 2 7,2 -0.280229 0.083488 0.117402 0.00072398 2 7,3 -0.280805 0.214421 0.425980 0.00096531 2 7,4 -0.278423 0.272341 0.541765 0.00096531 2 7,5 -0.262070 -0.252632 -0.505264 0.00096531 2 7,6 -0.265046 -0.331966 -0.663932 0.00096531 2 8,A 0.401625 0.120029 0.237194 0.00096531 2 8,2 -0.282092 0.213045 0.424480 0.00072398 2 8,3 -0.283577 0.268218 0.533572 0.00096531 2 8,4 -0.281306 -0.259826 -0.519652 0.00096531 2 8,5 -0.264895 -0.331281 -0.662563 0.00096531 2 8,6 -0.267800 -0.362615 -0.725231 0.00096531 2 8,7 -0.272058 -0.390377 -0.780755 0.00096531 2 9,A 0.655985 0.190762 0.379768 0.00096531 2 9,2 -0.286593 0.263585 0.524828 0.00072398 2 9,3 -0.288030 -0.266255 -0.532510 0.00096531 2 9,4 -0.285726 -0.293008 -0.586015 0.00096531 2 9,5 -0.269314 -0.358926 -0.717852 0.00096531 2 9,6 -0.273299 -0.430257 -0.860513 0.00096531 2 9,7 -0.276548 -0.455974 -0.911949 0.00096531 2 9,8 -0.136521 -0.530171 -1.060342 0.00096531 2 10,A 1.500000 0.247095 0.486979 0.00386124 2 10,2 -0.310993 -0.243408 -0.486825 0.00289593 2 10,3 -0.312390 -0.304215 -0.608429 0.00386124 2 10,4 -0.310107 -0.368778 -0.737556 0.00386124 2 10,5 -0.294783 -0.436219 -0.872437 0.00386124 2 10,6 -0.297664 -0.465396 -0.930791 0.00386124 2 10,7 -0.158128 -0.538453 -1.076907 0.00386124 2 10,8 0.118877 -0.632537 -1.265075 0.00386124 2 10,9 0.384834 -0.749661 -1.499321 0.00386124 3 2,A -0.241149 0.070664 0.028414 0.00096531 3 3,A -0.240149 0.044160 0.010855 0.00072398 3 3,2 -0.248452 -0.098250 -0.496903 0.00072398 3 4,A -0.223953 0.023369 0.002628 0.00096531 3 4,2 -0.232274 -0.106138 -0.445516 0.00096531 3 4,3 -0.229954 -0.081490 -0.323683 0.00072398 3 5,A -0.220511 -0.001897 -0.019208 0.00096531 3 5,2 -0.228858 -0.061451 -0.310676 0.00096531 3 5,3 -0.226529 0.019485 -0.083156 0.00072398 3 5,4 -0.209737 0.141082 0.195605 0.00096531 3 6,A -0.093231 0.036948 0.073896 0.00096531 3 6,2 -0.231142 0.023800 -0.078252 0.00096531 3 6,3 -0.227965 0.138944 0.189258 0.00072398 3 6,4 -0.212068 0.250439 0.500878 0.00096531 3 6,5 -0.208559 0.314733 0.629467 0.00096531 3 7,A 0.166817 0.094467 0.188934 0.00096531 3 7,2 -0.233898 0.130825 0.194602 0.00096531 3 7,3 -0.231628 0.246306 0.492141 0.00072398 3 7,4 -0.215710 0.305578 0.611156 0.00096531 3 7,5 -0.212344 -0.221955 -0.443909 0.00096531 3 7,6 -0.214771 -0.304029 -0.608059 0.00096531 3 8,A 0.419872 0.173024 0.346048 0.00096531 3 8,2 -0.238252 0.241477 0.482478 0.00096531 3 8,3 -0.235978 0.295198 0.590396 0.00072398 3 8,4 -0.220209 -0.229755 -0.459509 0.00096531 3 8,5 -0.216700 -0.305572 -0.611145 0.00096531 3 8,6 -0.220210 -0.346969 -0.693938 0.00096531 3 8,7 -0.223854 -0.381276 -0.762552 0.00096531 3 9,A 0.644126 0.196434 0.392234 0.00096531 3 9,2 -0.264051 0.269339 0.538044 0.00096531 3 9,3 -0.261815 -0.255712 -0.511424 0.00072398 3 9,4 -0.245979 -0.280361 -0.560722 0.00096531 3 9,5 -0.243558 -0.356045 -0.712089 0.00096531 3 9,6 -0.245987 -0.432276 -0.864553 0.00096531 3 9,7 -0.249700 -0.459006 -0.918012 0.00096531 3 9,8 -0.120664 -0.535486 -1.070971 0.00096531 3 10,A 1.500000 0.270668 0.536351 0.00386124 3 10,2 -0.267867 -0.219298 -0.438596 0.00386124 3 10,3 -0.265648 -0.283224 -0.566448 0.00289593 3 10,4 -0.250870 -0.355683 -0.711366 0.00386124 3 10,5 -0.247378 -0.429758 -0.859516 0.00386124 3 10,6 -0.249876 -0.461402 -0.922803 0.00386124 3 10,7 -0.118943 -0.536375 -1.072751 0.00386124 3 10,8 0.144414 -0.633653 -1.267307 0.00386124 3 10,9 0.383557 -0.712572 -1.425143 0.00386124 4 2,A -0.186079 0.110213 0.115097 0.00096531 4 3,A -0.169927 0.090755 0.109144 0.00096531 4 3,2 -0.175850 -0.041018 -0.351700 0.00096531 4 4,A -0.166175 0.061445 0.084883 0.00072398 4 4,2 -0.172106 -0.055247 -0.323648 0.00072398 4 4,3 -0.155934 -0.016569 -0.177708 0.00072398 4 5,A -0.163592 0.037976 0.062592 0.00096531 4 5,2 -0.169574 -0.010320 -0.194641 0.00096531 4 5,3 -0.152839 0.086611 0.041092 0.00096531 4 5,4 -0.148879 0.189608 0.297004 0.00072398 4 6,A -0.036667 0.077274 0.154548 0.00096531 4 6,2 -0.171327 0.081062 0.023309 0.00096531 4 6,3 -0.155475 0.186021 0.289638 0.00096531 4 6,4 -0.151621 0.292761 0.585523 0.00072398 4 6,5 -0.149063 0.351969 0.703939 0.00096531 4 7,A 0.203974 0.156342 0.312684 0.00096531 4 7,2 -0.176727 0.167601 0.271815 0.00096531 4 7,3 -0.160912 0.285179 0.570357 0.00096531 4 7,4 -0.157223 0.342543 0.685086 0.00072398 4 7,5 -0.154588 -0.191775 -0.383549 0.00096531 4 7,6 -0.158372 -0.277040 -0.554079 0.00096531 4 8,A 0.415490 0.186537 0.373075 0.00096531 4 8,2 -0.202468 0.252465 0.504929 0.00096531 4 8,3 -0.186681 0.312585 0.625171 0.00096531 4 8,4 -0.182858 -0.214040 -0.428081 0.00072398 4 8,5 -0.181302 -0.291820 -0.583640 0.00096531 4 8,6 -0.184078 -0.337542 -0.675083 0.00096531 4 8,7 -0.189590 -0.380044 -0.760089 0.00096531 4 9,A 0.653883 0.229578 0.459156 0.00096531 4 9,2 -0.207033 0.300049 0.600097 0.00096531 4 9,3 -0.191249 -0.225500 -0.451000 0.00096531 4 9,4 -0.188473 -0.254283 -0.508566 0.00072398 4 9,5 -0.185886 -0.335518 -0.671036 0.00096531 4 9,6 -0.188622 -0.419615 -0.839231 0.00096531 4 9,7 -0.194143 -0.452965 -0.905930 0.00096531 4 9,8 -0.084414 -0.541139 -1.082279 0.00096531 4 10,A 1.500000 0.298209 0.595071 0.00386124 4 10,2 -0.211839 -0.193955 -0.387911 0.00386124 4 10,3 -0.197109 -0.257952 -0.515904 0.00386124 4 10,4 -0.193331 -0.336459 -0.672917 0.00289593 4 10,5 -0.190687 -0.417990 -0.835980 0.00386124 4 10,6 -0.193442 -0.454566 -0.909133 0.00386124 4 10,7 -0.064395 -0.535108 -1.070216 0.00386124 4 10,8 0.164240 -0.597317 -1.194633 0.00386124 4 10,9 0.404114 -0.711696 -1.423391 0.00386124 5 2,A -0.118846 0.158731 0.212295 0.00096531 5 3,A -0.116020 0.136564 0.203578 0.00096531 5 3,2 -0.104257 0.021505 -0.208514 0.00096531 5 4,A -0.113097 0.107857 0.174985 0.00096531 5 4,2 -0.101371 0.008890 -0.185536 0.00096531 5 4,3 -0.097993 0.048986 -0.061355 0.00096531 5 5,A -0.110736 0.082113 0.148256 0.00072398 5 5,2 -0.098373 0.056768 -0.070318 0.00072398 5 5,3 -0.095495 0.141067 0.154983 0.00072398 5 5,4 -0.092477 0.246436 0.414995 0.00072398 5 6,A 0.004662 0.140016 0.280033 0.00096531 5 6,2 -0.103819 0.130630 0.130583 0.00096531 5 6,3 -0.100904 0.234973 0.391603 0.00096531 5 6,4 -0.097978 0.338825 0.677649 0.00096531 5 6,5 -0.095545 0.393653 0.787307 0.00072398 5 7,A 0.222295 0.174547 0.349094 0.00096531 5 7,2 -0.130567 0.195515 0.332121 0.00096531 5 7,3 -0.127734 0.307505 0.615011 0.00096531 5 7,4 -0.124742 0.361679 0.723359 0.00096531 5 7,5 -0.123356 -0.177736 -0.355472 0.00072398 5 7,6 -0.128821 -0.269028 -0.538057 0.00096531 5 8,A 0.460792 0.226529 0.453059 0.00096531 5 8,2 -0.135542 0.295987 0.591974 0.00096531 5 8,3 -0.132654 0.349536 0.699071 0.00096531 5 8,4 -0.130749 -0.186621 -0.373241 0.00096531 5 8,5 -0.128264 -0.270015 -0.540029 0.00072398 5 8,6 -0.133747 -0.317192 -0.634384 0.00096531 5 8,7 -0.160466 -0.373118 -0.746237 0.00096531 5 9,A 0.682074 0.268101 0.536203 0.00096531 5 9,2 -0.138632 0.339824 0.679649 0.00096531 5 9,3 -0.136834 -0.196612 -0.393224 0.00096531 5 9,4 -0.133859 -0.232059 -0.464118 0.00096531 5 9,5 -0.131352 -0.313835 -0.627670 0.00072398 5 9,6 -0.136855 -0.402035 -0.804070 0.00096531 5 9,7 -0.163593 -0.452057 -0.904114 0.00096531 5 9,8 -0.044394 -0.492582 -0.985164 0.00096531 5 10,A 1.500000 0.331496 0.662991 0.00386124 5 10,2 -0.144042 -0.163570 -0.327141 0.00386124 5 10,3 -0.141141 -0.235303 -0.470606 0.00386124 5 10,4 -0.138177 -0.314638 -0.629277 0.00386124 5 10,5 -0.135683 -0.399584 -0.799167 0.00289593 5 10,6 -0.141194 -0.444223 -0.888446 0.00386124 5 10,7 -0.043148 -0.492434 -0.984867 0.00386124 5 10,8 0.202290 -0.590755 -1.181511 0.00386124 5 10,9 0.447849 -0.708740 -1.417480 0.00386124 6 2,A -0.114167 0.168495 0.230210 0.00096531 6 3,A -0.111568 0.147164 0.221860 0.00096531 6 3,2 -0.121915 0.019233 -0.243829 0.00096531 6 4,A -0.109194 0.120257 0.200721 0.00096531 6 4,2 -0.119123 0.014113 -0.215036 0.00096531 6 4,3 -0.116458 0.059192 -0.056865 0.00096531 6 5,A -0.107596 0.115918 0.216665 0.00096531 6 5,2 -0.117900 0.069596 -0.061038 0.00096531 6 5,3 -0.115243 0.163696 0.189929 0.00096531 6 5,4 -0.112881 0.263316 0.443310 0.00096531 6 6,A 0.010435 0.133243 0.266486 0.00072398 6 6,2 -0.144181 0.132139 0.124194 0.00072398 6 6,3 -0.141538 0.231326 0.379215 0.00072398 6 6,4 -0.139190 0.331648 0.663295 0.00072398 6 6,5 -0.138999 0.380696 0.761392 0.00072398 6 7,A 0.262174 0.192430 0.384860 0.00096531 6 7,2 -0.148738 0.209711 0.357361 0.00096531 6 7,3 -0.146112 0.318050 0.636099 0.00096531 6 7,4 -0.144697 0.365741 0.731482 0.00096531 6 7,5 -0.143473 -0.178846 -0.357692 0.00096531 6 7,6 -0.169692 -0.280666 -0.561333 0.00072398 6 8,A 0.482354 0.241313 0.482626 0.00096531 6 8,2 -0.151323 0.307533 0.615067 0.00096531 6 8,3 -0.149705 0.356117 0.712234 0.00096531 6 8,4 -0.147293 -0.184088 -0.368175 0.00096531 6 8,5 -0.146040 -0.267603 -0.535207 0.00096531 6 8,6 -0.172288 -0.323254 -0.646508 0.00072398 6 8,7 -0.176976 -0.362116 -0.724232 0.00096531 6 9,A 0.694187 0.279876 0.559752 0.00096531 6 9,2 -0.156394 0.345816 0.691631 0.00096531 6 9,3 -0.153745 -0.193794 -0.387588 0.00096531 6 9,4 -0.151345 -0.228134 -0.456268 0.00096531 6 9,5 -0.150100 -0.308744 -0.617488 0.00096531 6 9,6 -0.176503 -0.400876 -0.801753 0.00072398 6 9,7 -0.179634 -0.397210 -0.794420 0.00096531 6 9,8 -0.011411 -0.483777 -0.967554 0.00096531 6 10,A 1.500000 0.341117 0.682234 0.00386124 6 10,2 -0.160379 -0.159436 -0.318872 0.00386124 6 10,3 -0.157773 -0.228909 -0.457819 0.00386124 6 10,4 -0.155349 -0.308093 -0.616186 0.00386124 6 10,5 -0.154176 -0.387327 -0.774653 0.00386124 6 10,6 -0.179023 -0.396278 -0.792555 0.00289593 6 10,7 -0.011287 -0.483343 -0.966686 0.00386124 6 10,8 0.268101 -0.586115 -1.172230 0.00386124 6 10,9 0.484093 -0.706989 -1.413978 0.00386124 7 2,A -0.461877 0.107387 -0.157238 0.00096531 7 3,A -0.460210 0.060465 -0.174539 0.00096531 7 3,2 -0.468910 -0.118962 -0.937820 0.00096531 7 4,A -0.457617 0.033764 -0.140937 0.00096531 7 4,2 -0.466638 -0.163473 -0.870342 0.00096531 7 4,3 -0.464878 -0.070036 -0.542577 0.00096531 7 5,A -0.467986 -0.023768 -0.189067 0.00096531 7 5,2 -0.477078 -0.066929 -0.567247 0.00096531 7 5,3 -0.475347 0.093112 -0.132299 0.00096531 7 5,4 -0.473057 0.201986 0.190459 0.00096531 7 6,A -0.089639 0.059646 0.014177 0.00096531 7 6,2 -0.481633 0.091845 -0.148120 0.00096531 7 6,3 -0.479852 0.197720 0.175697 0.00096531 7 6,4 -0.478620 0.285588 0.475371 0.00096531 7 6,5 -0.488938 0.297369 0.500525 0.00096531 7 7,A 0.411952 0.174667 0.240224 0.00072398 7 7,2 -0.485803 0.183614 0.160008 0.00072398 7 7,3 -0.485024 0.277201 0.444697 0.00072398 7 7,4 -0.482795 0.293770 0.487454 0.00072398 7 7,5 -0.493091 -0.258201 -0.585150 0.00072398 7 7,6 -0.497620 -0.330723 -0.714109 0.00072398 7 8,A 0.614504 0.221937 0.325331 0.00096531 7 8,2 -0.487638 0.267639 0.416638 0.00096531 7 8,3 -0.485835 0.291794 0.472587 0.00096531 7 8,4 -0.483611 -0.245545 -0.563593 0.00096531 7 8,5 -0.493922 -0.327475 -0.706532 0.00096531 7 8,6 -0.498604 -0.348533 -0.734594 0.00096531 7 8,7 -0.501385 -0.324114 -0.666031 0.00072398 7 9,A 0.773194 0.242711 0.351279 0.00096531 7 9,2 -0.493342 0.288889 0.455401 0.00096531 7 9,3 -0.491583 -0.247104 -0.578757 0.00096531 7 9,4 -0.489331 -0.274092 -0.600140 0.00096531 7 9,5 -0.499719 -0.348015 -0.727641 0.00096531 7 9,6 -0.502845 -0.363208 -0.744596 0.00096531 7 9,7 -0.507327 -0.374945 -0.749890 0.00072398 7 9,8 -0.122900 -0.448819 -0.897639 0.00096531 7 10,A 1.500000 0.285776 0.467593 0.00386124 7 10,2 -0.472817 -0.212020 -0.495828 0.00386124 7 10,3 -0.471067 -0.270391 -0.582990 0.00386124 7 10,4 -0.468887 -0.342199 -0.706808 0.00386124 7 10,5 -0.478497 -0.364503 -0.736265 0.00386124 7 10,6 -0.483324 -0.376194 -0.752387 0.00386124 7 10,7 -0.121287 -0.451963 -0.903925 0.00289593 7 10,8 0.388746 -0.567259 -1.134517 0.00386124 7 10,9 0.610121 -0.698546 -1.397092 0.00386124 8 2,A -0.507685 0.039129 -0.312387 0.00096531 8 3,A -0.505092 0.035047 -0.254214 0.00096531 8 3,2 -0.512722 -0.180629 -1.025445 0.00096531 8 4,A -0.514496 -0.035471 -0.314134 0.00096531 8 4,2 -0.522155 -0.233910 -1.023132 0.00096531 8 4,3 -0.519886 -0.227966 -0.848583 0.00096531 8 5,A -0.514869 -0.084296 -0.332550 0.00096531 8 5,2 -0.522459 -0.217532 -0.853460 0.00096531 8 5,3 -0.520172 -0.056500 -0.428185 0.00096531 8 5,4 -0.530618 0.108485 0.000739 0.00096531 8 6,A -0.385254 -0.064896 -0.229744 0.00096531 8 6,2 -0.524927 -0.055928 -0.437135 0.00096531 8 6,3 -0.523699 0.117522 0.014199 0.00096531 8 6,4 -0.533077 0.207506 0.317153 0.00096531 8 6,5 -0.533297 0.229714 0.365672 0.00096531 8 7,A 0.120931 0.047500 -0.015314 0.00096531 8 7,2 -0.528340 0.107594 0.007362 0.00096531 8 7,3 -0.526077 0.217139 0.326863 0.00096531 8 7,4 -0.535471 0.221712 0.340213 0.00096531 8 7,5 -0.535706 -0.320983 -0.711203 0.00096531 8 7,6 -0.538333 -0.394373 -0.842908 0.00096531 8 8,A 0.607840 0.157747 0.190197 0.00072398 8 8,2 -0.534064 0.207288 0.294474 0.00072398 8 8,3 -0.531845 0.220266 0.329975 0.00072398 8 8,4 -0.541217 -0.319226 -0.719326 0.00072398 8 8,5 -0.541529 -0.386852 -0.829134 0.00072398 8 8,6 -0.542601 -0.369092 -0.774545 0.00072398 8 8,7 -0.545308 -0.379554 -0.777379 0.00072398 8 9,A 0.784814 0.171549 0.229785 0.00096531 8 9,2 -0.513558 0.215256 0.327731 0.00096531 8 9,3 -0.511343 -0.316143 -0.703700 0.00096531 8 9,4 -0.520792 -0.338911 -0.724110 0.00096531 8 9,5 -0.520326 -0.370139 -0.761928 0.00096531 8 9,6 -0.523099 -0.420733 -0.855589 0.00096531 8 9,7 -0.525806 -0.427790 -0.855581 0.00096531 8 9,8 -0.414899 -0.475296 -0.950593 0.00072398 8 10,A 1.500000 0.221207 0.332681 0.00386124 8 10,2 -0.517357 -0.274473 -0.625855 0.00386124 8 10,3 -0.515179 -0.328235 -0.705640 0.00386124 8 10,4 -0.523850 -0.357370 -0.739262 0.00386124 8 10,5 -0.524234 -0.417965 -0.844149 0.00386124 8 10,6 -0.527007 -0.424823 -0.849645 0.00386124 8 10,7 -0.394240 -0.473803 -0.947605 0.00386124 8 10,8 0.095530 -0.565137 -1.130274 0.00289593 8 10,9 0.576828 -0.697331 -1.394661 0.00386124 9 2,A -0.517198 -0.013718 -0.372682 0.00096531 9 3,A -0.526764 -0.059660 -0.393525 0.00096531 9 3,2 -0.532954 -0.262096 -1.065907 0.00096531 9 4,A -0.526348 -0.113295 -0.421784 0.00096531 9 4,2 -0.532529 -0.303726 -1.045309 0.00096531 9 4,3 -0.542010 -0.303648 -0.946164 0.00096531 9 5,A -0.525481 -0.166412 -0.452038 0.00096531 9 5,2 -0.531595 -0.283812 -0.926277 0.00096531 9 5,3 -0.542147 -0.216991 -0.702933 0.00096531 9 5,4 -0.541653 -0.050633 -0.278961 0.00096531 9 6,A -0.407070 -0.134674 -0.345244 0.00096531 9 6,2 -0.536503 -0.208317 -0.691586 0.00096531 9 6,3 -0.545956 -0.051070 -0.289689 0.00096531 9 6,4 -0.545472 0.117492 0.164633 0.00096531 9 6,5 -0.544508 0.151960 0.239911 0.00096531 9 7,A -0.178832 -0.086958 -0.254497 0.00096531 9 7,2 -0.540546 -0.052384 -0.274062 0.00096531 9 7,3 -0.550044 0.117569 0.153705 0.00096531 9 7,4 -0.549545 0.149505 0.224729 0.00096531 9 7,5 -0.548658 -0.389559 -0.826051 0.00096531 9 7,6 -0.551096 -0.418484 -0.869390 0.00096531 9 8,A 0.287946 0.005020 -0.060207 0.00096531 9 8,2 -0.520050 0.120803 0.174428 0.00096531 9 8,3 -0.529562 0.140061 0.215208 0.00096531 9 8,4 -0.529140 -0.393813 -0.833734 0.00096531 9 8,5 -0.527475 -0.412582 -0.847602 0.00096531 9 8,6 -0.531614 -0.437187 -0.887881 0.00096531 9 8,7 -0.535909 -0.443182 -0.893558 0.00096531 9 9,A 0.765635 0.096533 0.110516 0.00072398 9 9,2 -0.523864 0.142601 0.213840 0.00072398 9 9,3 -0.533417 -0.392171 -0.840364 0.00072398 9 9,4 -0.532217 -0.365184 -0.758677 0.00072398 9 9,5 -0.531402 -0.425858 -0.859833 0.00072398 9 9,6 -0.535541 -0.479661 -0.966804 0.00072398 9 9,7 -0.539836 -0.482037 -0.964075 0.00072398 9 9,8 -0.411646 -0.531882 -1.063764 0.00072398 9 10,A 1.500000 0.148864 0.216490 0.00386124 9 10,2 -0.527482 -0.344351 -0.745531 0.00386124 9 10,3 -0.536257 -0.358832 -0.752138 0.00386124 9 10,4 -0.535908 -0.413520 -0.839394 0.00386124 9 10,5 -0.535093 -0.475256 -0.951924 0.00386124 9 10,6 -0.539232 -0.479306 -0.958613 0.00386124 9 10,7 -0.416111 -0.526459 -1.052918 0.00386124 9 10,8 -0.196136 -0.593227 -1.186454 0.00386124 9 10,9 0.264279 -0.697795 -1.395590 0.00289593 10 2,A -0.538705 -0.088817 -0.478997 0.00362483 10 3,A -0.538043 -0.123446 -0.479655 0.00362483 10 3,2 -0.546375 -0.307757 -1.092750 0.00354603 10 4,A -0.538314 -0.170424 -0.496264 0.00362483 10 4,2 -0.546578 -0.345111 -1.073122 0.00354603 10 4,3 -0.547042 -0.334779 -0.955483 0.00354603 10 5,A -0.537311 -0.223337 -0.536694 0.00362483 10 5,2 -0.546774 -0.322021 -0.955173 0.00354603 10 5,3 -0.546016 -0.250838 -0.738023 0.00354603 10 5,4 -0.546165 -0.137522 -0.438370 0.00354603 10 6,A -0.417783 -0.188464 -0.432804 0.00362483 10 6,2 -0.551019 -0.250053 -0.745151 0.00354603 10 6,3 -0.550310 -0.140495 -0.444546 0.00354603 10 6,4 -0.550442 0.035322 0.011219 0.00354603 10 6,5 -0.549500 0.113832 0.170737 0.00354603 10 7,A -0.186187 -0.138665 -0.321341 0.00362483 10 7,2 -0.528701 -0.154043 -0.435521 0.00354603 10 7,3 -0.528008 0.030281 0.014484 0.00354603 10 7,4 -0.528224 0.109571 0.170991 0.00354603 10 7,5 -0.526434 -0.384545 -0.790615 0.00354603 10 7,6 -0.530953 -0.450337 -0.908270 0.00354603 10 8,A 0.064313 -0.085536 -0.224373 0.00362483 10 8,2 -0.532853 0.031008 0.005950 0.00354603 10 8,3 -0.532205 0.109187 0.162257 0.00354603 10 8,4 -0.531574 -0.385856 -0.805908 0.00354603 10 8,5 -0.530710 -0.446362 -0.909614 0.00354603 10 8,6 -0.535229 -0.464761 -0.936341 0.00354603 10 8,7 -0.514015 -0.474794 -0.949587 0.00354603 10 9,A 0.554551 0.008647 -0.052039 0.00362483 10 9,2 -0.536793 0.103847 0.142776 0.00354603 10 9,3 -0.535298 -0.388543 -0.821117 0.00354603 10 9,4 -0.535592 -0.398702 -0.818901 0.00354603 10 9,5 -0.534729 -0.453066 -0.911135 0.00354603 10 9,6 -0.539248 -0.509827 -1.023930 0.00354603 10 9,7 -0.518034 -0.512009 -1.024017 0.00354603 10 9,8 -0.390691 -0.558165 -1.116330 0.00354603 10 10,A 1.500000 0.104682 0.139083 0.01359313 10 10,2 -0.539571 -0.346618 -0.739114 0.01329763 10 10,3 -0.539002 -0.393091 -0.811084 0.01329763 10 10,4 -0.539296 -0.445788 -0.899256 0.01329763 10 10,5 -0.538433 -0.501091 -1.002183 0.01329763 10 10,6 -0.542952 -0.506929 -1.013859 0.01329763 10 10,7 -0.412341 -0.556704 -1.113407 0.01329763 10 10,8 -0.155192 -0.622794 -1.245589 0.01329763 10 10,9 0.102517 -0.710037 -1.420073 0.01329763 Blackjack Appendix 9 - 1 deck, dealer stands on soft 17 Dealer Player Stand Hit Double Split DAS-No Split DAS-Yes Probability A A,A -0.663142 -0.030635 -0.598755 0.223931 0.223931 0.00012189 A 2,2 -0.660049 -0.258901 -1.320098 -0.441608 -0.434211 0.00024379 A 3,3 -0.660147 -0.333963 -1.298180 -0.480539 -0.470279 0.00024379 A 4,4 -0.655759 -0.208972 -0.796495 -0.576512 -0.562934 0.00024379 A 5,5 -0.653287 0.090561 0.055149 -0.714734 -0.697709 0.00024379 A 6,6 -0.617010 -0.386204 -0.834602 -0.657391 -0.638018 0.00024379 A 7,7 -0.630502 -0.494721 -1.029343 -0.610436 -0.597411 0.00024379 A 8,8 -0.643532 -0.494905 -0.989811 -0.323759 -0.316498 0.00024379 A 9,9 -0.055174 -0.624843 -1.249686 -0.070608 -0.069038 0.00024379 A 10,10 0.650097 -0.883183 -1.766366 -0.299410 -0.299410 0.00517130 2 A,A -0.274327 0.094777 -0.019360 0.565704 0.565704 0.00036199 2 2,2 -0.290803 -0.113174 -0.581605 -0.128956 -0.035789 0.00018100 2 3,3 -0.294866 -0.152948 -0.567470 -0.200303 -0.107639 0.00036199 2 4,4 -0.290133 -0.012616 -0.184577 -0.255913 -0.161808 0.00036199 2 5,5 -0.255915 0.223862 0.446442 -0.267488 -0.206910 0.00036199 2 6,6 -0.261714 -0.252671 -0.505342 -0.211108 -0.104559 0.00036199 2 7,7 -0.268309 -0.406388 -0.812776 -0.148796 -0.047919 0.00036199 2 8,8 -0.274814 -0.454093 -0.908186 0.044815 0.123747 0.00036199 2 9,9 0.137057 -0.627497 -1.254994 0.175980 0.211065 0.00036199 2 10,10 0.627226 -0.846661 -1.693322 -0.043444 -0.043444 0.00723982 3 A,A -0.232311 0.120586 0.054882 0.612856 0.612856 0.00036199 3 2,2 -0.250722 -0.081767 -0.501443 -0.068689 0.039858 0.00036199 3 3,3 -0.246186 -0.118136 -0.472052 -0.127153 -0.022805 0.00018100 3 4,4 -0.213730 0.028836 -0.082465 -0.131818 -0.015657 0.00036199 3 5,5 -0.206172 0.254782 0.509564 -0.176326 -0.102942 0.00036199 3 6,6 -0.211033 -0.222134 -0.444267 -0.120497 0.008941 0.00036199 3 7,7 -0.219400 -0.388265 -0.776530 -0.062670 0.056070 0.00036199 3 8,8 -0.228354 -0.449931 -0.899863 0.111595 0.198718 0.00036199 3 9,9 0.122553 -0.638281 -1.276562 0.174122 0.218237 0.00036199 3 10,10 0.636134 -0.846301 -1.692602 0.005250 0.005250 0.00723982 4 A,A -0.178249 0.145730 0.136650 0.668582 0.668582 0.00036199 4 2,2 -0.191999 -0.034842 -0.383998 0.006268 0.131291 0.00036199 4 3,3 -0.159691 -0.047439 -0.302167 0.016377 0.154244 0.00036199 4 4,4 -0.151694 0.097855 0.044128 -0.017034 0.125183 0.00018100 4 5,5 -0.146291 0.294911 0.589822 -0.064209 0.026796 0.00036199 4 6,6 -0.151883 -0.190113 -0.380227 -0.012894 0.140454 0.00036199 4 7,7 -0.163937 -0.368803 -0.737605 0.034983 0.187843 0.00036199 4 8,8 -0.215266 -0.461146 -0.922291 0.132831 0.230561 0.00036199 4 9,9 0.166978 -0.597001 -1.194002 0.258432 0.308760 0.00036199 4 10,10 0.644848 -0.846273 -1.692545 0.063866 0.063866 0.00723982 5 A,A -0.130086 0.182014 0.215727 0.732160 0.732160 0.00036199 5 2,2 -0.107014 0.035944 -0.214028 0.137126 0.300407 0.00036199 5 3,3 -0.101455 0.008318 -0.183982 0.128780 0.304545 0.00036199 5 4,4 -0.094897 0.153926 0.162314 0.091925 0.266161 0.00036199 5 5,5 -0.090048 0.347346 0.694691 0.057835 0.185562 0.00018100 5 6,6 -0.102166 -0.162360 -0.324720 0.086307 0.277969 0.00036199 5 7,7 -0.155509 -0.370327 -0.740653 0.055550 0.223685 0.00036199 5 8,8 -0.165443 -0.452874 -0.905749 0.217988 0.329648 0.00036199 5 9,9 0.202893 -0.590310 -1.180620 0.349438 0.406878 0.00036199 5 10,10 0.673675 -0.845597 -1.691194 0.125864 0.125864 0.00723982 6 A,A -0.103505 0.199607 0.247914 0.758276 0.758276 0.00036199 6 2,2 -0.124502 0.032055 -0.249003 0.135806 0.306525 0.00036199 6 3,3 -0.118893 0.013896 -0.214511 0.123137 0.302028 0.00036199 6 4,4 -0.114034 0.175290 0.193184 0.077621 0.250343 0.00036199 6 5,5 -0.111669 0.361823 0.723645 0.031578 0.155292 0.00036199 6 6,6 -0.165187 -0.193568 -0.387137 -0.003913 0.169401 0.00018100 6 7,7 -0.174225 -0.366941 -0.733882 0.072477 0.243170 0.00036199 6 8,8 -0.178171 -0.396692 -0.793384 0.269350 0.385156 0.00036199 6 9,9 0.265195 -0.586713 -1.173426 0.365645 0.423942 0.00036199 6 10,10 0.697403 -0.845027 -1.690053 0.139930 0.139930 0.00723982 7 A,A -0.452479 0.158489 -0.136975 0.540712 0.540712 0.00036199 7 2,2 -0.471027 -0.091472 -0.942054 -0.050453 0.016482 0.00036199 7 3,3 -0.467213 -0.164040 -0.871217 -0.107550 -0.041923 0.00036199 7 4,4 -0.462717 0.111349 -0.108353 -0.252865 -0.178019 0.00036199 7 5,5 -0.484469 0.279059 0.466340 -0.352877 -0.310701 0.00036199 7 6,6 -0.493437 -0.264854 -0.598513 -0.267925 -0.192858 0.00036199 7 7,7 -0.501956 -0.389227 -0.823012 -0.110429 -0.051412 0.00018100 7 8,8 -0.502514 -0.373561 -0.747122 0.251647 0.303621 0.00036199 7 9,9 0.401060 -0.566049 -1.132097 0.335235 0.358752 0.00036199 7 10,10 0.764677 -0.843026 -1.686052 0.055594 0.055594 0.00723982 8 A,A -0.499700 0.093060 -0.295648 0.406468 0.406468 0.00036199 8 2,2 -0.514917 -0.140878 -1.029833 -0.218289 -0.178911 0.00036199 8 3,3 -0.510419 -0.230703 -0.999660 -0.264924 -0.221195 0.00036199 8 4,4 -0.529261 -0.054359 -0.447062 -0.369395 -0.322222 0.00036199 8 5,5 -0.530853 0.207838 0.322885 -0.509576 -0.481765 0.00036199 8 6,6 -0.535771 -0.321707 -0.711353 -0.418848 -0.369478 0.00036199 8 7,7 -0.539340 -0.407893 -0.857865 -0.422623 -0.380884 0.00036199 8 8,8 -0.551276 -0.426315 -0.852630 -0.087081 -0.059689 0.00018100 8 9,9 0.064518 -0.566265 -1.132529 0.190184 0.207350 0.00036199 8 10,10 0.783251 -0.842727 -1.685455 -0.095365 -0.095365 0.00723982 9 A,A -0.510996 -0.002488 -0.420608 0.289770 0.289770 0.00036199 9 2,2 -0.523455 -0.221986 -1.046910 -0.395441 -0.368892 0.00036199 9 3,3 -0.542453 -0.309366 -1.065288 -0.424234 -0.400308 0.00036199 9 4,4 -0.542638 -0.204341 -0.701087 -0.514284 -0.488027 0.00036199 9 5,5 -0.540668 0.120345 0.174553 -0.667997 -0.649823 0.00036199 9 6,6 -0.548502 -0.386242 -0.817364 -0.584405 -0.556836 0.00036199 9 7,7 -0.555391 -0.474654 -0.978238 -0.569340 -0.548789 0.00036199 9 8,8 -0.516426 -0.487124 -0.974248 -0.427146 -0.401442 0.00036199 9 9,9 -0.196372 -0.594732 -1.189465 -0.108930 -0.099162 0.00018100 9 10,10 0.743970 -0.842055 -1.684111 -0.254578 -0.254578 0.00723982 10 A,A -0.530674 -0.046842 -0.468319 0.194251 0.194251 0.00138886 10 2,2 -0.547137 -0.275101 -1.094274 -0.474492 -0.465966 0.00132976 10 3,3 -0.545672 -0.343585 -1.071267 -0.516190 -0.505094 0.00132976 10 4,4 -0.547191 -0.241025 -0.738996 -0.598618 -0.586290 0.00132976 10 5,5 -0.545139 0.038319 0.018006 -0.734630 -0.722816 0.00132976 10 6,6 -0.552167 -0.386149 -0.803852 -0.669574 -0.658131 0.00132976 10 7,7 -0.509739 -0.514818 -1.034724 -0.633557 -0.619017 0.00132976 10 8,8 -0.518291 -0.511755 -1.023510 -0.457453 -0.446436 0.00132976 10 9,9 -0.133285 -0.624465 -1.248929 -0.277030 -0.271554 0.00132976 10 10,10 0.583154 -0.836969 -1.673938 -0.263251 -0.263251 0.02327085 For similar tables under other rules visit my blackjack appendix 9. 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