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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.

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).

When blackjack deviations concern how one bets at the blackjack table, these are known as betting deviations. Sometimes a change to the betting scheme can be a simple aspect of money management. Knowing how to properly use a betting bankroll is very important if one decides to make blackjack a serious endeavor.

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:

Blackjack Apprenticeship Playing Deviations

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

Blackjack Apprenticeship Deviations