Blackjack Deviations
Many of the deviations from basic strategy during various high or low counts make sense i.e. Doubling on hard 8 vs. When teaching my girlfriend basic strategy i have found it sensible to explain the reason for each decision. Now if any of you have an understanding of some of the harder to grasp deviations i would love an explanation. Top 20 blackjack deviations At timestamp 38:42 of the same video linked above, Munchkin suggests you should get your running count accurate to 1 mistake (by a count of 1) per 3-4 shoes (about 20 decks).
- Appendices
- Miscellaneous
- External Links
Introduction
This appendix presents information pertinent to the standard deviation in blackjack. It assumes the player is following basic strategy in a cut card game. Each table is the product of a separate simulation of about ten billion hands played. As a reminder, the total variance playing x hands at once is the variance plus covariance × (x-1).
The following table is the product of many simulations and a lot of programming work. It shows the variance and covariance for various sets of rules.
Summary Table
| Decks | Soft 17 | Double After Split | Surrender Allowed | Re-split Aces Allowed | Expected Value | Variance | Covariance |
|---|---|---|---|---|---|---|---|
| 6 | Stand | Yes | Yes | Yes | -0.00281 | 1.303 | 0.479 |
| 6 | Stand | No | No | No | -0.00573 | 1.295 | 0.478 |
| 6 | Hit | Yes | Yes | Yes | -0.00473 | 1.312 | 0.487 |
| 6 | Hit | No | No | No | -0.00787 | 1.308 | 0.488 |
| 6 | Hit | Yes | No | No | -0.00628 | 1.346 | 0.499 |
| 6 | Hit | No | Yes | No | -0.00699 | 1.272 | 0.475 |
| 6 | Hit | No | No | Yes | -0.00717 | 1.311 | 0.488 |
| 8 | Hit | No | No | No | -0.00812 | 1.309 | 0.489 |
| 2 | Hit | Yes | No | No | -0.00398 | 1.341 | 0.495 |
By way of comparison, Stanford Wong, in his book Professional Blackjack (page 203) says the variance is 1.28 and the covariance 0.47 for his Benchmark Rules, which are six decks, dealer stands on soft 17, no double after split, no re-splitting aces, no surrender. The second row of my table shows that for the same rules I get 1.295 and 0.478 respectively, which is close enough for me.
Effect on Variance of Rule Changes
The next table shows the effect on the expected value, variance and covariance of various rule changes compared to the Wong Benchmark Rules.
Effect of Rule Variation
| Rule | Expected Value | Variance | Covariance |
|---|---|---|---|
| Stand on soft 17 | 0.00191 | -0.00838 | -0.00764 |
| Double after split allowed | 0.00159 | 0.03753 | 0.01091 |
| Surrender allowed | 0.00088 | -0.03629 | -0.01247 |
| Re-split aces allowed | 0.00070 | 0.00207 | 0.00037 |
| Eight decks | -0.00025 | 0.00071 | 0.00063 |
| Two decks | 0.00230 | -0.00530 | -0.00422 |
What follows are tables showing the probability of the net win for one to three hands under the Liberal Strip Rules, defined above.
Liberal Strip Rules — Playing One Hand at a Time
The first table shows the probability of each net outcome playing a single hand under what I call 'liberal strip rules,' which are as follows:
- Six decks
- Dealer stands on soft 17 (S17)
- Double on any first two cards (DA2)
- Double after split allowed (DAS)
- Late surrender allowed (LS)
- Re-split aces allowed (RSA)
- Player may re-split up to three times (P3X)
6 Decks S17 DA2 DAS LS RSA P3X — One Hand
| Net win | Probability | Return |
|---|---|---|
| -8 | 0.00000019 | -0.00000154 |
| -7 | 0.00000235 | -0.00001643 |
| -6 | 0.00001785 | -0.00010709 |
| -5 | 0.00008947 | -0.00044736 |
| -4 | 0.00048248 | -0.00192993 |
| -3 | 0.00207909 | -0.00623728 |
| -2 | 0.04180923 | -0.08361847 |
| -1 | 0.40171191 | -0.40171191 |
| -0.5 | 0.04470705 | -0.02235353 |
| 0 | 0.08483290 | 0.00000000 |
| 1 | 0.31697909 | 0.31697909 |
| 1.5 | 0.04529632 | 0.06794448 |
| 2 | 0.05844299 | 0.11688598 |
| 3 | 0.00259645 | 0.00778935 |
| 4 | 0.00076323 | 0.00305292 |
| 5 | 0.00014491 | 0.00072453 |
| 6 | 0.00003774 | 0.00022646 |
| 7 | 0.00000609 | 0.00004263 |
| 8 | 0.00000066 | 0.00000526 |
| Total | 1.00000000 | -0.00277282 |
The table above reflects the following:
- House edge = 0.28%
- Variance = 1.303
- Standard deviation = 1.142
Probability of Net Win
I'm frequently asked about the probability of a net win in blackjack. The following table answers that question.
Summarized Net Win in Blackjack
The next three tables break down the possible events by whether the first action was to hit, stand, or surrender; double; or split.
Net Win when Hitting, Standing, or Surrendering First Action
| Event | Total | Probability | Return |
|---|---|---|---|
| 1.5 | 77147473 | 0.05144768 | 0.07717152 |
| 1 | 537410636 | 0.35838544 | 0.35838544 |
| 0 | 127597398 | 0.08509145 | 0 |
| -0.5 | 76163623 | 0.05079158 | -0.02539579 |
| -1 | 681213441 | 0.45428386 | -0.45428386 |
| Total | 1499532571 | 1 | -0.04412269 |
Net Win when Doubling First Action
| Event | Total | Probability | Return |
|---|---|---|---|
| 2 | 89463603 | 0.54980265 | 1.09960529 |
| 0 | 11301274 | 0.06945249 | 0 |
| -2 | 61954607 | 0.38074486 | -0.76148972 |
| Total | 162719484 | 1 | 0.33811558 |
Net Win when Splitting First Action
| Event | Total | Probability | Return |
|---|---|---|---|
| 8 | 1079 | 0.00002554 | 0.00020428 |
| 7 | 10440 | 0.00024707 | 0.00172948 |
| 6 | 64099 | 0.00151694 | 0.00910166 |
| 5 | 247638 | 0.00586051 | 0.02930255 |
| 4 | 1307719 | 0.030948 | 0.123792 |
| 3 | 4437365 | 0.10501306 | 0.31503917 |
| 2 | 10222578 | 0.24192379 | 0.48384758 |
| 1 | 2822458 | 0.06679526 | 0.06679526 |
| 0 | 5621675 | 0.1330405 | 0 |
| -1 | 3520209 | 0.08330798 | -0.08330798 |
| -2 | 9425393 | 0.2230579 | -0.4461158 |
| -3 | 3559202 | 0.08423077 | -0.25269231 |
| -4 | 828010 | 0.01959538 | -0.07838153 |
| -5 | 152687 | 0.00361343 | -0.01806717 |
| -6 | 30536 | 0.00072265 | -0.00433592 |
| -7 | 3972 | 0.000094 | -0.000658 |
| -8 | 305 | 0.00000722 | -0.00005774 |
| Total | 42255365 | 1 | 0.14619552 |
Liberal Strip Rules — Playing Two Hands at a Time
The following table shows the net result playing two hands at a time under the Liberal Strip Rules, explained above. The Return column shows the net win between the two hands.
6 Decks S17 DA2 DAS LS RSA P3X — Two Hands
| Net win | Probability | Return |
|---|---|---|
| -14 | 0.00000000 | 0.00000000 |
| -13 | 0.00000000 | -0.00000001 |
| -12 | 0.00000001 | -0.00000006 |
| -11 | 0.00000003 | -0.00000035 |
| -10 | 0.00000023 | -0.00000228 |
| -9 | 0.00000163 | -0.00001464 |
| -8 | 0.00001040 | -0.00008324 |
| -7.5 | 0.00000000 | -0.00000003 |
| -7 | 0.00005327 | -0.00037288 |
| -6.5 | 0.00000009 | -0.00000061 |
| -6 | 0.00024527 | -0.00147159 |
| -5.5 | 0.00000114 | -0.00000629 |
| -5 | 0.00106847 | -0.00534234 |
| -4.5 | 0.00000967 | -0.00004352 |
| -4 | 0.00654661 | -0.02618644 |
| -3.5 | 0.00005733 | -0.00020065 |
| -3 | 0.04607814 | -0.13823442 |
| -2.5 | 0.00214887 | -0.00537218 |
| -2 | 0.23285866 | -0.46571732 |
| -1.5 | 0.03547663 | -0.05321495 |
| -1 | 0.09903321 | -0.09903321 |
| -0.5 | 0.01386072 | -0.00693036 |
| 0 | 0.14677504 | 0.00000000 |
| 0.5 | 0.05888290 | 0.02944145 |
| 1 | 0.06026238 | 0.06026238 |
| 1.5 | 0.01030563 | 0.01545845 |
| 2 | 0.17250085 | 0.34500170 |
| 2.5 | 0.03020186 | 0.07550465 |
| 3 | 0.06443204 | 0.19329612 |
| 3.5 | 0.00559850 | 0.01959474 |
| 4 | 0.01072401 | 0.04289604 |
| 4.5 | 0.00024927 | 0.00112171 |
| 5 | 0.00187139 | 0.00935695 |
| 5.5 | 0.00007341 | 0.00040373 |
| 6 | 0.00049405 | 0.00296428 |
| 6.5 | 0.00001414 | 0.00009193 |
| 7 | 0.00012404 | 0.00086825 |
| 7.5 | 0.00000369 | 0.00002767 |
| 8 | 0.00002933 | 0.00023466 |
| 8.5 | 0.00000060 | 0.00000508 |
| 9 | 0.00000543 | 0.00004888 |
| 9.5 | 0.00000007 | 0.00000063 |
| 10 | 0.00000083 | 0.00000834 |
| 11 | 0.00000013 | 0.00000141 |
| 12 | 0.00000002 | 0.00000028 |
| 13 | 0.00000000 | 0.00000005 |
| 14 | 0.00000000 | 0.00000001 |
| Total | 1.00000000 | -0.00563798 |
Blackjack Deviations From Basic Strategy
The table above reflects the following:
- House edge = 0.28%
- Variance per round = 3.565
- Variance per hand = 1.782
- Standard deviation per hand= 1.335
Blackjack Deviations
Liberal Strip Rules — Playing Three Hands at a Time
The following table shows the net result playing three hands at a time under the Liberal Strip Rules, explained above. The Return column shows the net win between the three hands.
Blackjack Deviations
6 Decks S17 DA2 DAS LS RSA P3X — Three Hands
| Net win | Probability | Return |
|---|---|---|
| -16 | 0.00000000 | -0.00000001 |
| -15 | 0.00000000 | -0.00000001 |
| -14 | 0.00000001 | -0.00000007 |
| -13 | 0.00000003 | -0.00000041 |
| -12 | 0.00000018 | -0.00000218 |
| -11 | 0.00000100 | -0.00001099 |
| -10.5 | 0.00000000 | 0.00000000 |
| -10 | 0.00000531 | -0.00005309 |
| -9.5 | 0.00000001 | -0.00000006 |
| -9 | 0.00002581 | -0.00023228 |
| -8.5 | 0.00000005 | -0.00000047 |
| -8 | 0.00011292 | -0.00090339 |
| -7.5 | 0.00000049 | -0.00000370 |
| -7 | 0.00046097 | -0.00322680 |
| -6.5 | 0.00000397 | -0.00002581 |
| -6 | 0.00197390 | -0.01184341 |
| -5.5 | 0.00002622 | -0.00014419 |
| -5 | 0.00969361 | -0.04846807 |
| -4.5 | 0.00022638 | -0.00101870 |
| -4 | 0.04183392 | -0.16733566 |
| -3.5 | 0.00319799 | -0.01119297 |
| -3 | 0.15826947 | -0.47480842 |
| -2.5 | 0.02641456 | -0.06603640 |
| -2 | 0.08893658 | -0.17787317 |
| -1.5 | 0.02183548 | -0.03275322 |
| -1 | 0.09681697 | -0.09681697 |
| -0.5 | 0.04992545 | -0.02496273 |
| 0 | 0.06712076 | 0.00000000 |
| 0.5 | 0.02111145 | 0.01055572 |
| 1 | 0.08978272 | 0.08978272 |
| 1.5 | 0.03789943 | 0.05684914 |
| 2 | 0.04349592 | 0.08699183 |
| 2.5 | 0.01123447 | 0.02808618 |
| 3 | 0.10813504 | 0.32440511 |
| 3.5 | 0.02489093 | 0.08711825 |
| 4 | 0.06196736 | 0.24786943 |
| 4.5 | 0.00906613 | 0.04079759 |
| 5 | 0.01805409 | 0.09027044 |
| 5.5 | 0.00154269 | 0.00848480 |
| 6 | 0.00409323 | 0.02455940 |
| 6.5 | 0.00027059 | 0.00175885 |
| 7 | 0.00107315 | 0.00751203 |
| 7.5 | 0.00007208 | 0.00054062 |
| 8 | 0.00030105 | 0.00240840 |
| 8.5 | 0.00001824 | 0.00015505 |
| 9 | 0.00008014 | 0.00072126 |
| 9.5 | 0.00000431 | 0.00004096 |
| 10 | 0.00001901 | 0.00019010 |
| 10.5 | 0.00000081 | 0.00000846 |
| 11 | 0.00000398 | 0.00004379 |
| 11.5 | 0.00000013 | 0.00000144 |
| 12 | 0.00000078 | 0.00000939 |
| 12.5 | 0.00000002 | 0.00000023 |
| 13 | 0.00000016 | 0.00000214 |
| 13.5 | 0.00000001 | 0.00000008 |
| 14 | 0.00000003 | 0.00000045 |
| 14.5 | 0.00000000 | 0.00000001 |
| 15 | 0.00000001 | 0.00000009 |
| 15.5 | 0.00000000 | 0.00000000 |
| 16 | 0.00000000 | 0.00000002 |
| 17 | 0.00000000 | 0.00000001 |
| Total | 1.00000000 | -0.00854917 |
Blackjack Betting Deviations
The table above reflects the following:
- House edge = 0.285%
- Variance per round = 6.785
- Variance per hand = 2.262
- Standard deviation per hand= 1.504