## Blackjack Standard

• Appendices
• Miscellaneous

Basic strategy was derived from a computer simulation. Somebody taught a computer how to play blackjack and then told it to play several hundred MILLION hands of blackjack and record what happened. Through the computer’s trial and error it figured out which decisions are best for the player, given every possible combination of starting player hands and dealer up card. The most common variation of blackjack with 6 decks, no side bets, limits between 1 and 1500 and normal rules. The basic strategy guide is optimised for this version of the game. Blackjack at 888 Casino allows players to play up to 5 hands simultaneously.

## 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, if the variance of one hand is v, the covariance is c, and the number of hands played at once is n, then the total variance is n×v + n×(n-1)×c.

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

DecksSoft 17Double
After
Split
Surrender
Allowed
Re-split
Aces
Allowed
Expected
Value
VarianceCovariance
6StandYesYesYes-0.002811.3030.479
6StandNoNoNo-0.005731.2950.478
6HitYesYesYes-0.004731.3120.487
6HitNoNoNo-0.007871.3080.488
6HitYesNoNo-0.006281.3460.499
6HitNoYesNo-0.006991.2720.475
6HitNoNoYes-0.007171.3110.488
8HitNoNoNo-0.008121.3090.489
2HitYesNoNo-0.003981.3410.495

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

RuleExpected
Value
VarianceCovariance
Stand on soft 170.00191-0.00838-0.00764
Double after split allowed0.001590.037530.01091
Surrender allowed0.00088-0.03629-0.01247
Re-split aces allowed0.000700.002070.00037
Eight decks-0.000250.000710.00063
Two decks0.00230-0.00530-0.00422

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 winProbabilityReturn
-80.00000019-0.00000154
-70.00000235-0.00001643
-60.00001785-0.00010709
-50.00008947-0.00044736
-40.00048248-0.00192993
-30.00207909-0.00623728
-20.04180923-0.08361847
-10.40171191-0.40171191
-0.50.04470705-0.02235353
00.084832900.00000000
10.316979090.31697909
1.50.045296320.06794448
20.058442990.11688598
30.002596450.00778935
40.000763230.00305292
50.000144910.00072453
60.000037740.00022646
70.000006090.00004263
80.000000660.00000526
Total1.00000000-0.00277282

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.

## Blackjack Standard Rules

EventTotalProbabilityReturn
1.5771474730.051447680.07717152
15374106360.358385440.35838544
01275973980.085091450
-0.5761636230.05079158-0.02539579
-16812134410.45428386-0.45428386
Total14995325711-0.04412269

### Net Win when Doubling First Action

EventTotalProbabilityReturn
2894636030.549802651.09960529
0113012740.069452490
-2619546070.38074486-0.76148972
Total16271948410.33811558

### Net Win when Splitting First Action

EventTotalProbabilityReturn
810790.000025540.00020428
7104400.000247070.00172948
6640990.001516940.00910166
52476380.005860510.02930255
413077190.0309480.123792
344373650.105013060.31503917
2102225780.241923790.48384758
128224580.066795260.06679526
056216750.13304050
-135202090.08330798-0.08330798
-294253930.2230579-0.4461158
-335592020.08423077-0.25269231
-48280100.01959538-0.07838153
-51526870.00361343-0.01806717
-6305360.00072265-0.00433592
-739720.000094-0.000658
-83050.00000722-0.00005774
Total4225536510.14619552

## 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 winProbabilityReturn
-140.000000000.00000000
-130.00000000-0.00000001
-120.00000001-0.00000006
-110.00000003-0.00000035
-100.00000023-0.00000228
-90.00000163-0.00001464
-80.00001040-0.00008324
-7.50.00000000-0.00000003
-70.00005327-0.00037288
-6.50.00000009-0.00000061
-60.00024527-0.00147159
-5.50.00000114-0.00000629
-50.00106847-0.00534234
-4.50.00000967-0.00004352
-40.00654661-0.02618644
-3.50.00005733-0.00020065
-30.04607814-0.13823442
-2.50.00214887-0.00537218
-20.23285866-0.46571732
-1.50.03547663-0.05321495
-10.09903321-0.09903321
-0.50.01386072-0.00693036
00.146775040.00000000
0.50.058882900.02944145
10.060262380.06026238
1.50.010305630.01545845
20.172500850.34500170
2.50.030201860.07550465
30.064432040.19329612
3.50.005598500.01959474
40.010724010.04289604
4.50.000249270.00112171
50.001871390.00935695
5.50.000073410.00040373
60.000494050.00296428
6.50.000014140.00009193
70.000124040.00086825
7.50.000003690.00002767
80.000029330.00023466
8.50.000000600.00000508
90.000005430.00004888
9.50.000000070.00000063
100.000000830.00000834
110.000000130.00000141
120.000000020.00000028
130.000000000.00000005
140.000000000.00000001
Total1.00000000-0.00563798

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.

## Blackjack Standard Strategy Chart

Net winProbabilityReturn
-160.00000000-0.00000001
-150.00000000-0.00000001
-140.00000001-0.00000007
-130.00000003-0.00000041
-120.00000018-0.00000218
-110.00000100-0.00001099
-10.50.000000000.00000000
-100.00000531-0.00005309
-9.50.00000001-0.00000006
-90.00002581-0.00023228
-8.50.00000005-0.00000047
-80.00011292-0.00090339
-7.50.00000049-0.00000370
-70.00046097-0.00322680
-6.50.00000397-0.00002581
-60.00197390-0.01184341
-5.50.00002622-0.00014419
-50.00969361-0.04846807
-4.50.00022638-0.00101870
-40.04183392-0.16733566
-3.50.00319799-0.01119297
-30.15826947-0.47480842
-2.50.02641456-0.06603640
-20.08893658-0.17787317
-1.50.02183548-0.03275322
-10.09681697-0.09681697
-0.50.04992545-0.02496273
00.067120760.00000000
0.50.021111450.01055572
10.089782720.08978272
1.50.037899430.05684914
20.043495920.08699183
2.50.011234470.02808618
30.108135040.32440511
3.50.024890930.08711825
40.061967360.24786943
4.50.009066130.04079759
50.018054090.09027044
5.50.001542690.00848480
60.004093230.02455940
6.50.000270590.00175885
70.001073150.00751203
7.50.000072080.00054062
80.000301050.00240840
8.50.000018240.00015505
90.000080140.00072126
9.50.000004310.00004096
100.000019010.00019010
10.50.000000810.00000846
110.000003980.00004379
11.50.000000130.00000144
120.000000780.00000939
12.50.000000020.00000023
130.000000160.00000214
13.50.000000010.00000008
140.000000030.00000045
14.50.000000000.00000001
150.000000010.00000009
15.50.000000000.00000000
160.000000000.00000002
170.000000000.00000001
Total1.00000000-0.00854917

## Blackjack Standard Game

1974 silver dollar moon landing back value. The table above reflects the following:

## Blackjack Standard Deviation Graph

• House edge = 0.285%
• Variance per round = 6.785
• Variance per hand = 2.262
• Standard deviation per hand= 1.504