Blackjack Apprenticeship Deviations
- Blackjack Apprenticeship Deviations Chart
- Blackjack Apprenticeship Deviations
- Blackjack Apprenticeship Playing Deviations
There is a True Count value at which a basic blackjack strategy deviation play is mathematically justified. But many of these situations rarely occur and it is not worth the effort to memorize over 175 individual index numbers. But there is a sub group of these index plays called the ILLUSTRAIOUS 18. Learning to beat blackjack through card counting is a simple, though not easy. Use our Blackjack Training Drills to master blackjack basic strategy.
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 reKO Blackjack Strategy (99.44% Pure) The KO Strategy is one of the very easiest Blackjack card counting strategies, contained in the book Knock-Out Blackjack-The Easiest Card Counting System Ever Devised by Olaf Vancura and Ken Fuchs. ReKO (ridiculously easy KO) is a substantially simplified version.
- Appendices
- Miscellaneous
- External Links
Introduction
Rule variations will have an effect on the player's expected return. The numbers below show the effect on the player's return under various rules and after taking into consideration proper basic strategy adjustments. These changes are relative to the following rules: eight decks, dealer stands on soft 17, player may double on anyfirst two cards, player can double after splitting, playermay split to 4 hands.
Blackjack Rule Variations
| Rule | Effect |
|---|---|
| Blackjacks pay 2 to 1 | +2.27% |
| Triple down on any two cards | +1.64% |
| Five cardCharlie1 | +1.46% |
| Optional half win for 5-card Charlie9 | +0.77% |
| Suited blackjacks pay 2 to 1 | +0.57% |
| Player 21-points is automatic winner | +0.54% |
| Single deck | +0.48% |
| Early surrender against ace | +0.39% |
| Player 21 vs. dealer blackjack is a push | +0.35% |
| 5-card win pays 3-2 | +0.33% |
| Blackjack tie pays 3 to 2 | +0.32% |
| Six-card Charlie pays 3 to 2 | +0.31% |
| Early surrender against ten | +0.24% |
| 5-card (or more) 21 automatically pays 2 to 1 | +0.24% |
| Player may double on any number of cards | +0.23% |
| Ace and 10 after splitting aces is a blackjack (paying 3-2) | +0.21% |
| Player may draw to split aces | +0.19% |
| Double deck | +0.19% |
| Six card Charlie1 | +0.16% |
| Player may double, double for less, or stand after splitting aces | +0.15% |
| Tied blackjack pays 1-2 | +0.11% |
| Ace and 10 after splitting aces is a blackjack (paying 6-5) | +0.11% |
| Double Down Rescue | +0.10% |
| 6-card win pays 2-1 | +0.09% |
| Player may resplit aces | +0.08% |
| Player may double or stand after splitting aces | +0.08% |
| Late surrender against ten | +0.07% |
| Four decks | +0.06% |
| 7-card win pays 5-1 | +0.05% |
| 777 pays 3 to 1 automatically | +0.05% |
| Five decks | +0.03% |
| 777 pays 2 to 1 automatically | +0.03% |
| Six decks | +0.02% |
| Seven card Charlie1 | +0.01% |
| Late surrender against ace | +0.00% |
| Dealer must stop with six cards | +0.00% |
| Late surrender after splitting | +0.00% |
| Split to only 3 hands | -0.01% |
| No-peek: ace showing2 | -0.01% |
| BB+13 | -0.01% |
| Dealer secretly goes first5 | -0.02% |
| OBBO4 | -0.03% |
| European no hole card (splitting)7 | -0.03% |
| European no hole card (doubling)7 | -0.08% |
| Player may double on 9-11 only | -0.09% |
| No-peek: ten showing6 | -0.10% |
| Player may not resplit | -0.10% |
| European no hole card7 | -0.11% |
| Player may not double after splitting | -0.14% |
| Player may double on 10,11 only | -0.18% |
| Player may not split aces | -0.18% |
| Dealer hits on soft 17 | -0.22% |
| Blackkjack pays 7-5 | -0.45% |
| Player may not split | -0.57% |
| Red blackjack pays 2-1, Black blackjack pays 3-2, Mixed blackjack pays 1-1 | -0.57% |
| Blackjack pays 6-5 | -1.39% |
| Player may not double | -1.48% |
| Player loses 17 ties | -1.87% |
| Player may not double nor split | -1.91% |
| Blackjacks pay 1 to 1 | -2.27% |
| Player loses 17,18 ties | -3.58% |
| Player loses 17-19 ties | -5.30% |
| Dealer bust on 22 is a push8 | -6.91% |
| Player loses 17-20 ties | -8.38% |
| Player loses 17-21 ties | -8.86% |
Notes
- A 'Charlie' is an automatic winner. For example the five-card Charlie rule means the player automatically wins with 5 cards, as long as he didn't bust.
- Dealer does not peek for blackjack with an ace up. If dealer has a ace-up blackjack, player loses all bets made, including from doubling and splitting. A 10-up blackjack will be revealed immediately after peeking, and the player will lose only his original wager, except a blackjack tie will push.
- BB+1 refers to an Australian rule, in which the player will lose all busted bets, plus one unit, if the dealer gets a blackjack. For example, if the player splits 8's to three hands of a 20, (doubled) 19, and a busted hand, and the dealer get a blackjack, then the player will lose 2 units, one for the busted hand, and one from the remaining 3 units on the table. There is a great deal of confusion for the term for his rule, some sources calling it 'OBBO,' for Original and Busted Bets Only.
- OBBO stands for Original Bets and Busted Only. 'Original Bets' means each original bet per hand. So, if the player bet $5 and split to three hands, he would have three original bets of $5 each. As far as I know, the OBBO rule is confined to some parts of Australia and Malaysia. In the example above, the player would lose 3 units under this rule, one for the busted hand and two for the number of unbusted hands. The term is discussed in depth in the bookThe Pro's Guide to Spanish 21 and Australian Pontoon by Katarina Walker.
- This strange rule is actually followed in Pennsylvania, on Shufflemaster TMS-300 video multi-player blackjack games. State law prohibits one player taking the cards of another player. The way they implement this rule is to give every player, and the dealer, a separate shoe. The dealer plays out his hand first, secretly, and the results stored in memory. Then the cards used by the dealer are removed for each player shoe. When all players have finished acting, the game shows what cards the dealer drew already. According to Shufflemaster, the effect of this rule is 0.02% in favor of the dealer. Shufflemaster wishes to add this rule is part of a patent-pending technology.
- Dealer does not peek for blackjack with a 10 up. If dealer has a 10-up blackjack, player loses all bets made, including from doubling and splitting. An ace-up blackjack will be revealed immediately after peeking, and the player will lose only his original wager, except a blackjack tie will push.
- Dealer does not take a hole card, or never peeks at it if he does. If the dealer has a blackjack, then player loses all bets made, including from doubling and splitting, except a blackjack tie will push. The cost of this rule to the player is 0.08% when doubling, and 0.03% when splitting. This breakdown is important in the case of Galaxy casinos in Macau, where the player only loses the original bet when doubling, but everything when splitting, if the dealer gets a blackjack.
- The push on a dealer 22 rule is a legally protected rule. The rights belong to the owner of Blackjack Switch.
- This rule can be found at the Pharaoh's Palace in Macau. The player may invoke the rule before the dealer checks for blackjack. The effect shown is based on the dealer not taking a hole card.
Written by: Michael Shackleford
- 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 |
Blackjack Apprenticeship Deviations Chart
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 |
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
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.
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 |
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
Blackjack Apprenticeship Deviations
Internal Links
Blackjack Apprenticeship Playing Deviations
Written by: Michael Shackleford