Gambling Guide-
Appendices
- 1: Total dependent expected return table for an infinite deck
- 2a: Probabilities of dealer blackjack after peek
- 2b: Probabilities of dealer blackjack before peek
- 3a: Exceptions to single-deck S17 basic strategy
- 3b: Exceptions to double-deck S17 basic strategy
- 3c: Exceptions to single-deck H17 basic strategy
- 4: Blackjack standard deviation details
- 5: Infinite deck expected return by player hand and dealer upcard
- 6: Fine points of when to surrender
- 7: Effect of card removal
- 8: Analysis of blackjack side bets
- 9: Composition-dependent expected returns for 1 to 8 decks
- 10: Continuous shuffling machine vs. cut card
- 11: Value and strategy for 678 and 777 bonuses
- 12: Risk of ruin statistics
- 13: Probabilities in the first four cards
- 14: Value of each initial player card
- 15: House edge using total-dependent vs. composition-dependent basic strategy
- 16: Basic strategy when dealer exposes both cards
- 17: The Ace-Five card counting method
- 18: Basic strategy exceptions for three to six cards
- 19: Blackjack splitting strategy when a back-player is betting
- 20: Blackjack doubling strategy when doubling after splitting aces is allowed
- 21: Details on the Wizard's Simple Strategy
- Miscellaneous
- External Links
Expected Returns in Infinite-Deck Blackjack
Introduction
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
