Standard Deviation Blackjack Calculator

  

Copyright © 2006-2014 www.beatingbonuses.com. In card games we encounter many types of experiments and categories of events. The Hands per Hour input Retrieved. If you account for all the blackjack rules and basic strategy of blackjack, the standard deviation of the game falls at the value of 1.14, in general. This means that in a game with a 0.5% house edge, the standard deviation marks the odds to win and lose on both sides of the bell curve. Here is a free online arithmetic standard deviation calculator to help you solve your statistical questions. This can also be used as a measure of variability or volatility for the given set of data. Enter the set of values in the online SD calculator to calculate the mean, standard deviation, variance and population standard deviation.

Blackjack Bankroll Calculator This screen can be used to calculate your bankroll needs given a desired risk of ruin. Here, risk of ruin is defined as the probability that you will go bankrupt within a specified number of hands. There are five variables. Do not worry if you do not have a pocket calculator with you at the casino, it is straight forward to work out the blackjack standard deviations for different sized sessions in advance and gain an insight into how the average distribution of outcomes affects your chances of either winning or losing certain amounts of cash.

  • Appendices
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Introduction

Standard

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

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.

Blackjack

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.

Net Win when Hitting, Standing, or Surrendering First Action

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

Standard Deviation Blackjack Calculator Value

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

Standard Deviation Calculator With Steps

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 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 Odds Calculator

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

Internal Links

Standard Deviation And Error Calculator


Standard Deviation Blackjack Calculator Solver

Written by: Michael Shackleford