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Expected Returns in Infinite-Deck Blackjack

Introduction

Expected Returns in Infinite-Deck Blackjack

The following tables display expected returns for any play in blackjack based on the following rules: dealer stands on a soft 17, an infinite deck, the player may double after a split, split up to three times except for aces, and draw only one card to split aces. Based on these rules, the player's expected value is -0.511734%.

To use this table, look up the returns for any given play, the one with the greatest return is the best play. For example suppose you have two 8's and the dealer has a 10. The return by standing is -0.5404, by hitting is -0.5398, doubling is -1.0797, and by splitting is -0.4807. So splitting 8's you stand to lose the least, 48.07 cents per original dollar bet, and is thus the best play.

Stand

Player's Expected Return by StandingExpand

Player's
Hand		 Dealer's Up Card
2 3 4 5 6 7 8 9 10 Ace
0-16 -0.292784 -0.252250 -0.211063 -0.167193 -0.153699 -0.475375 -0.510518 -0.543150 -0.540430 -0.666951
17 -0.152975 -0.117216 -0.080573 -0.044941 0.011739 -0.106809 -0.381951 -0.423154 -0.419721 -0.478033
18 0.121742 0.148300 0.175854 0.199561 0.283444 0.399554 0.105951 -0.183163 -0.178301 -0.100199
19 0.386305 0.404363 0.423179 0.439512 0.495977 0.615976 0.593854 0.287597 0.063118 0.277636
20 0.639987 0.650272 0.661050 0.670360 0.703959 0.773227 0.791815 0.758357 0.554538 0.655470
21 0.882007 0.885300 0.888767 0.891754 0.902837 0.925926 0.930605 0.939176 0.962624 0.922194

Hit

Player's Expected Return by HittingExpand

Player's
Hand		 Dealer's Up Card
2 3 4 5 6 7 8 9 10 Ace
4 -0.114913 -0.082613 -0.049367 -0.012380 0.011130 -0.088279 -0.159334 -0.240666 -0.289198 -0.253077
5 -0.128216 -0.095310 -0.061479 -0.023979 -0.001186 -0.119447 -0.188093 -0.266615 -0.313412 -0.278575
6 -0.140759 -0.107291 -0.072917 -0.034916 -0.013006 -0.151933 -0.217242 -0.292641 -0.337749 -0.304147
7 -0.109183 -0.076583 -0.043022 -0.007271 0.029185 -0.068808 -0.210605 -0.285365 -0.319055 -0.310072
8 -0.021798 0.008005 0.038784 0.070805 0.114960 0.082207 -0.059898 -0.210186 -0.249375 -0.197029
9 0.074446 0.101265 0.128981 0.158032 0.196019 0.171868 0.098376 -0.052178 -0.152953 -0.065681
10 0.182500 0.206088 0.230470 0.256259 0.287795 0.256909 0.197954 0.116530 0.025309 0.081450
11 0.238351 0.260325 0.283020 0.307350 0.333690 0.292147 0.229982 0.158257 0.119482 0.143001
12 -0.253390 -0.233691 -0.213537 -0.193271 -0.170526 -0.212848 -0.271575 -0.340013 -0.381043 -0.350540
13 -0.307791 -0.291210 -0.274224 -0.257333 -0.235626 -0.269073 -0.323605 -0.387155 -0.425254 -0.396930
14 -0.362192 -0.348729 -0.334911 -0.321395 -0.300726 -0.321282 -0.371919 -0.430930 -0.466307 -0.440007
15 -0.416594 -0.406249 -0.395599 -0.385457 -0.365826 -0.369762 -0.416782 -0.471578 -0.504428 -0.480006
16 -0.470995 -0.463768 -0.456286 -0.449520 -0.430927 -0.414779 -0.458440 -0.509322 -0.539826 -0.517149
17 -0.536151 -0.531674 -0.527011 -0.522986 -0.508753 -0.483486 -0.505983 -0.553695 -0.584463 -0.557300
18 -0.622439 -0.620005 -0.617462 -0.615260 -0.607479 -0.591144 -0.591056 -0.616528 -0.647671 -0.626515
19 -0.729077 -0.728033 -0.726937 -0.725991 -0.722554 -0.715450 -0.713660 -0.715574 -0.729449 -0.724795
20 -0.855230 -0.854977 -0.854710 -0.854480 -0.853628 -0.851852 -0.851492 -0.850833 -0.849029 -0.852139
Soft 12 0.081836 0.103507 0.126596 0.156482 0.185954 0.165473 0.095115 0.000066 -0.070002 -0.020478
Soft 13 0.046636 0.074119 0.102477 0.133363 0.161693 0.122386 0.054057 -0.037695 -0.104851 -0.057308
Soft 14 0.022392 0.050807 0.080081 0.111894 0.139165 0.079507 0.013277 -0.075163 -0.139467 -0.093874
Soft 15 -0.000121 0.029160 0.059285 0.091960 0.118246 0.037028 -0.027055 -0.112189 -0.173704 -0.130027
Soft 16 -0.021025 0.009059 0.039975 0.073449 0.098821 -0.004890 -0.066795 -0.148644 -0.207441 -0.165637
Soft 17 -0.000491 0.028975 0.059326 0.091189 0.128052 0.053823 -0.072915 -0.149787 -0.196867 -0.179569
Soft 18 0.062905 0.090248 0.118502 0.147613 0.190753 0.170676 0.039677 -0.100744 -0.143808 -0.092935
Soft 19 0.123958 0.149340 0.175577 0.202986 0.239799 0.220620 0.152270 0.007893 -0.088096 -0.005743
Soft 20 0.182500 0.206088 0.230470 0.256259 0.287795 0.256909 0.197954 0.116530 0.025309 0.081450
Soft 21 0.238351 0.260325 0.283020 0.307350 0.333690 0.292147 0.229982 0.158257 0.119482 0.143001

Double

Player's Expected Return by DoublingExpand

Player's
Hand		 Dealer's Up Card
2 3 4 5 6 7 8 9 10 Ace
Hard 4 -0.585567 -0.504500 -0.422126 -0.334385 -0.307398 -0.950750 -1.021035 -1.086299 -1.080861 -1.333902
Hard 5 -0.585567 -0.504500 -0.422126 -0.334385 -0.307398 -0.950750 -1.021035 -1.086299 -1.080861 -1.333902
Hard 6 -0.564058 -0.483726 -0.402051 -0.315577 -0.281946 -0.894048 -1.001256 -1.067839 -1.062290 -1.304837
Hard 7 -0.435758 -0.359779 -0.282299 -0.202730 -0.138337 -0.589336 -0.847076 -0.957074 -0.950866 -1.130452
Hard 8 -0.204491 -0.136216 -0.066372 0.003456 0.087015 -0.187730 -0.451987 -0.718501 -0.746588 -0.810746
Hard 9 0.061119 0.120816 0.181949 0.243057 0.317055 0.104250 -0.026442 -0.300996 -0.466707 -0.432911
Hard 10 0.358939 0.409321 0.460940 0.512517 0.575590 0.392412 0.286636 0.144328 -0.008659 -0.014042
Hard 11 0.470641 0.517795 0.566041 0.614699 0.667380 0.462889 0.350693 0.227783 0.179689 0.109061
Hard 12 -0.506780 -0.467382 -0.427073 -0.386542 -0.341052 -0.506712 -0.615661 -0.737506 -0.796841 -0.829344
Hard 13 -0.615582 -0.582420 -0.548448 -0.514667 -0.471253 -0.587423 -0.690966 -0.807790 -0.867544 -0.880582
Hard 14 -0.724385 -0.697459 -0.669823 -0.642791 -0.601453 -0.668135 -0.766271 -0.878075 -0.938247 -0.931821
Hard 15 -0.833187 -0.812497 -0.791198 -0.770915 -0.731653 -0.748846 -0.841576 -0.948360 -1.008950 -0.983059
Hard 16 -0.941990 -0.927536 -0.912573 -0.899039 -0.861853 -0.829558 -0.916881 -1.018644 -1.079653 -1.034297
Hard 17 -1.072302 -1.063348 -1.054023 -1.045971 -1.017505 -0.966972 -1.011965 -1.107390 -1.168926 -1.114600
Hard 18 -1.244877 -1.240010 -1.234924 -1.230519 -1.214958 -1.182288 -1.182112 -1.233057 -1.295342 -1.253031
Hard 19 -1.458155 -1.456066 -1.453874 -1.451983 -1.445108 -1.430899 -1.427320 -1.431149 -1.458898 -1.449590
Hard 20 -1.710461 -1.709954 -1.709420 -1.708961 -1.707256 -1.703704 -1.702984 -1.701665 -1.698058 -1.704278
Soft 12 -0.071570 -0.007228 0.058427 0.125954 0.179748 -0.183866 -0.314441 -0.456367 -0.514028 -0.624391
Soft 13 -0.071570 -0.007228 0.058427 0.125954 0.179748 -0.183866 -0.314441 -0.456367 -0.514028 -0.624391
Soft 14 -0.071570 -0.007228 0.058427 0.125954 0.179748 -0.183866 -0.314441 -0.456367 -0.514028 -0.624391
Soft 15 -0.071570 -0.007228 0.058427 0.125954 0.179748 -0.183866 -0.314441 -0.456367 -0.514028 -0.624391
Soft 16 -0.071570 -0.007228 0.058427 0.125954 0.179748 -0.183866 -0.314441 -0.456367 -0.514028 -0.624391
Soft 17 -0.007043 0.055095 0.118653 0.182378 0.256104 -0.013758 -0.255102 -0.400984 -0.458316 -0.537198
Soft 18 0.119750 0.177641 0.237004 0.295225 0.381506 0.219948 -0.029917 -0.290219 -0.346892 -0.362813
Soft 19 0.241855 0.295824 0.351154 0.405972 0.479599 0.319835 0.195269 -0.072946 -0.235468 -0.188428
Soft 20 0.358939 0.409321 0.460940 0.512517 0.575590 0.392412 0.286636 0.144328 -0.008659 -0.014042
Soft 21 0.470641 0.517795 0.566041 0.614699 0.667380 0.462889 0.350693 0.227783 0.179689 0.109061

Split

Player's Expected Return by SplittingExpand

Player's
Hand		 Dealer's Up Card
2 3 4 5 6 7 8 9 10 Ace
2,2 -0.084336 -0.015650 0.059088 0.151665 0.226890 0.006743 -0.176693 -0.386883 -0.507175 -0.433570
3,3 -0.137710 -0.056273 0.029932 0.126284 0.201318 -0.053043 -0.231843 -0.436607 -0.553507 -0.482405
4,4 -0.192325 -0.108712 -0.020395 0.081913 0.151377 -0.166452 -0.326068 -0.511152 -0.625044 -0.560206
5,5 -0.290154 -0.208718 -0.119335 -0.019231 0.045404 -0.293928 -0.454237 -0.634113 -0.729969 -0.668811
6,6 -0.212560 -0.119715 -0.021320 0.080912 0.153668 -0.264427 -0.425122 -0.610576 -0.716103 -0.653362
7,7 -0.131478 -0.043733 0.049255 0.146678 0.247385 -0.050148 -0.391981 -0.577584 -0.657268 -0.651641
8,8 0.073852 0.146187 0.220849 0.297475 0.409329 0.321042 -0.022736 -0.387228 -0.480686 -0.372535
9,9 0.195625 0.258548 0.323474 0.391987 0.471339 0.364837 0.234447 -0.078010 -0.317336 -0.136810
10,10 0.134774 0.212836 0.293403 0.380367 0.468117 0.296633 0.064443 -0.206733 -0.371278 -0.249494
A,A 0.470641 0.517795 0.566041 0.614699 0.667380 0.462889 0.350693 0.227783 0.179689 0.109061

 

Here are basic strategy tables for infinite decks.

The only differences between infinite and four decks is to hit soft 13 vs. 5, and soft 15 vs 4 only when the dealer stands on soft 17.

I have had a lot of requests for my actual spreadsheet through the years. It is available to the public at Google docs. Note that this document allows for infinite re-splitting, while the tables above are based on a maximum of three splits (except aces).

Written by: Michael Shackleford