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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, 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.

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 = 1.412

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
Total 1.00000000 -0.00563798

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 main page.
• Appendix 1:Total dependent expected return table for an infinite deck.
• Appendix 2a:Dealer probabilities after dealer peeks for blackjack.
• Appendix 2b:Dealer probabilities before dealer peeks for blackjack.
• Appendix 3a:Composition dependent exceptions to single deck basic strategy where the dealer stands on soft 17.
• Appendix 3b:Composition dependent exceptions to double deck basic strategy where the dealer stands on soft 17.
• Appendix 3c:Composition dependent exceptions to single deck basic strategy where the dealer hits a soft 17.
• Appendix 4:Details on the standard deviation in blackjack.
• Appendix 5:Infinite deck expected return according to player hand and dealer up card.
• Appendix 6:Fine points of when to surrender.
• Appendix 7:Effect of card removal.
• Appendix 8:Analysis of some popular blackjack side bets includingSuper Sevens, Streak, Royal Match, and a tie.
• Appendix 9:Composition dependent expected returns for 1, 2, 4, 5, 6, and 8 decks.
• Appendix 10:The effect on the house edge of the continuous shuffling machines vs. the cut card.
• Appendix 11: Value and strategy for 678 and 777 bonuses.
• Appendix 12:Risk of ruin statistics.
• Appendix 13:Probabilities in the first four cards. May be used to test for the number of decks in online blackjack.
• Appendix 14:Value of each initial player card.
• Appendix 15:House edge using total dependent vs composition dependent basic strategy
• Appendix 16: Basic strategy when dealer exposes both cards.
• Appendix 17: The Ace-Five Count. Possibly the easiest way to count cards.
• Appendix 18: Basic strategy exceptions for three to six cards.
• Appendix 19: Blackjack splitting strategy when a back-player is betting.
• Appendix 20: Blackjack doubling strategy, when doubling after splitting aces is allowed.
• Appendix 21: Details on the Wizard's Simple Strategy.
• "21" Movie — Truth and Fiction : My comments on the movie "21."
• Australian Blackjack: Rules and odds for blackjack down under.
• Introduction to Card Counting
• Rule Variations: The effect of just about every known blackjack rule change.
• Automatic Winner Charlie Rule in Blackjack.

Written by: Michael Shackleford

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