Last Updated: January 21, 2019

# Variance in Blackjack

## 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

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

Event Probability
Win 42.43%
Push 8.48%
Loss 49.09%

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

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