Playing Deviations If this is your first visit to the Blackjack Forum, be sure to check out the FAQ by clicking the link above. You will have to r e g i s t e r (free) before you can post: click the r e g i s t e r link to proceed. Blackjack - HiLo Basic Strategy and Deviations. HiLo Four Deck H17 Indices along with basic strategy for both 6D and 2D. 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.

• Appendices
• Miscellaneous
• External Links

Introduction

Common questions for novice players: What is single deck blackjack? Is it better to play single deck blackjack versus multi-deck variations? To use the basic strategy, look up your hand along the left vertical edge and the dealer's up card along the top. In both cases, an A stands for ace. From top to bottom are the hard totals, soft totals, and splittable hands. For specific tips, we present two charts depending on whether the dealer hits or stands on soft 17.

Other basic strategy rules:

• Never take insurance or 'even money.' The house edge on insurance is 5.9%, based on one deck.
• If there is no row for splitting (fives and tens), then look up your hand as a hard total (10 or 20).
• If you can't split because of a limit on re-splitting, then look up your hand as a hard total, except aces. In the extremely unlikely event you have a pair of aces you can't re-split and drawing to split aces is allowed, then double against a 5 or 6, otherwise hit.

Internal Links

Blackjack main page.

Blackjack Deviations Flashcards

Blackjack Appendix 3A lists exceptions to the single deck, dealer stands on soft 17, basic strategy based on the exact composition of the player's hand.

Blackjack Appendix 3C lists exceptions to the single deck, dealer hits on soft 17, basic strategy based on the exact composition of the player's hand.

Boss Media Appendix 1 has a composition dependent basic strategy for single-deck, dealer stands on soft 17, blackjack.

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

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:

Blackjack Deviations List

• 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.

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

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 Deviations Chart

Internal Links

Blackjack Betting Deviations

Written by: Michael Shackleford