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Standard deviation in blackjack
Introduction
This appendix presents information pertinent to the standard deviation in blackjack. It is based on a 30 billion hand simulation of the following rules and totaldependent basic strategy.
 Six decks
 Dealer stands on soft 17
 Double on any first two cards
 Double after split allowed
 Late surrender allowed
 Resplit aces allowed
 Player may resplit to four hands
 Cut card placed after 4.5 decks.
Net Win in Blackjack
Net win  Total  Probability  Return 

8  19,400  0.00000065  0.00000517 
7  184,258  0.00000614  0.00004299 
6  1,133,977  0.00003780  0.00022678 
5  4,354,362  0.00014514  0.00072568 
4  22,919,054  0.00076391  0.00305566 
3  77,926,000  0.00259735  0.00779205 
2  1,754,285,814  0.05847205  0.11694410 
1.5  1,358,059,775  0.04526545  0.06789817 
1  9,509,186,226  0.31695040  0.31695040 
0  2,544,674,396  0.08481647  0.00000000 
0.5  1,340,412,558  0.04467725  0.02233863 
1  12,052,824,995  0.40173235  0.40173235 
2  1,255,856,242  0.04185891  0.08371781 
3  62,471,923  0.00208225  0.00624675 
4  14,509,961  0.00048363  0.00193452 
5  2,693,092  0.00008976  0.00044882 
6  538,465  0.00001795  0.00010769 
7  70,847  0.00000236  0.00001653 
8  5,667  0.00000019  0.00000151 
Total  30,002,127,012  1.00000000  0.00290361 
This table reflects a standard deviation of 1.1417.
Here is a summary, which answers the frequently asked question, what is the probability of a net win, loss, and push. This answers one of the most frequent questions I get, which is what is the probability of a net win in blackjack. This shows it is 42.42%. If we ignore ties, then it is 46.36%.
Summarized Net Win in Blackjack
Event  Probability 

Win  42.42% 
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 
Following is the standard deviation per individual hand, playing flat betting 1 to 3 hands at the same time. These numbers are courtesy of Marsha Ness, who used Blackjack Audit simulation software. The rules are 6 decks, dealer stands on soft 17, double any two cards, double after split, no surrender, split up to 4 hands, no resplitting aces, no drawing to split aces.
Multiple Hand Standard Deviation
Hands  Standard Dev. 

1  1.15514 
2  1.34942 
3  1.51957 
For example, the total standard deviation of three hands of $100 each, played at the same time, would be $100 × sqr(3) × 1.51957 = $263.20. By comparison, the standard deviation of a single hand of $300 would be $300 × 1.15514 = $346.54.
According to Professional Blackjack by Stanford Wong (page 203), the variance for similar rules is 1.32 and the covariance is 0.48. The total variance of n hands would be 1.32*n + 0.48*n*(n1). Take the final square root to get the standard deviation.
The next table is a practical application of the standard deviation. It is useful if you wish to know the probability of a large net loss or win after a session of flat betting. The left column represents the number of hands in the session. The top row represents the probability that the result, after adjusting for the house edge, will exceed the table value. The body of the table represents the number of units won or lost, after adjusting for the house edge.For example suppose a blackjack player loses 100 units over a session of 1000 bets. Assuming an 0.4% house edge, 4 of the losses are expected due to the house edge and 96 are the result of bad luck. The player wishes to know the probability of a loss of this magnitude. The table shows the probability of a loss of 95 units to be 0.5%. Thus the player can expect to lose 95 units or more about 1 session in 200.
Probability of Loss Table
Number of Hands 
10%  5%  2.5%  1%  0.5%  0.25%  0.1%  0.05%  0.01% 

100  15  19  23  27  30  33  36  39  43 
200  21  27  32  39  43  46  51  55  60 
300  26  33  40  47  52  57  63  67  74 
400  30  38  46  54  60  66  73  77  85 
500  33  43  51  61  67  73  81  86  95 
600  37  47  56  67  74  80  89  95  105 
700  40  51  61  72  80  87  96  102  113 
800  42  54  65  77  85  93  103  109  121 
900  45  58  69  82  91  99  109  116  128 
1000  47  61  72  86  95  104  115  122  135 
2000  67  86  103  122  135  147  162  173  191 
3000  82  105  126  149  165  180  199  211  234 
4000  95  122  145  172  191  208  229  244  270 
5000  106  136  162  193  213  232  256  273  302 
6000  116  149  178  211  234  255  281  299  331 
7000  125  161  192  228  252  275  303  323  357 
8000  134  172  205  244  270  294  324  345  382 
9000  142  183  217  259  286  312  344  366  405 
10000  150  192  229  272  302  329  363  386  427 
20000  212  272  324  385  427  465  513  546  604 
30000  259  333  397  472  523  569  628  668  739 
40000  299  385  458  545  603  657  725  772  854 
50000  335  430  513  609  675  735  811  863  955 
60000  367  471  561  667  739  805  888  945  1046 
70000  396  509  606  721  798  869  959  1021  1129 
80000  423  544  648  771  853  930  1025  1092  1207 
90000  449  577  688  817  905  986  1088  1158  1281 
100000  473  608  725  862  954  1039  1146  1220  1350 
200000  669  860  1025  1219  1349  1470  1621  1726  1909 
300000  820  1054  1256  1493  1653  1800  1986  2114  2338 
400000  947  1217  1450  1723  1908  2078  2293  2441  2700 
500000  1059  1360  1621  1927  2134  2324  2564  2729  3018 
600000  1160  1490  1776  2111  2337  2546  2808  2990  3307 
700000  1252  1610  1918  2280  2525  2750  3033  3229  3572 
800000  1339  1721  2050  2437  2699  2939  3243  3452  3818 
900000  1420  1825  2175  2585  2863  3118  3439  3661  4050 
1000000  1497  1924  2292  2725  3017  3286  3626  3859  4269 
RTG Rules
Following is a win/loss table under the following "Real Time Gaming" rules. It should be noted that Real Time Gaming rules are configurable, so this is just for one possible case.
 Six decks.
 Dealer hits soft 17.
 Shuffle after every hand.
 Double after split allowed.
 Surrender not allowed.
 Aces get one card each (no resplitting).
 All other pairs may be resplit to three hands maximum.
Real Time Gaming Rules
Net win  Count  Probability  Return 

6  203,699  0.000015  0.000092 
5  1,479,973  0.000111  0.000557 
4  9,988,320  0.000752  0.003007 
3  33,201,243  0.002499  0.007497 
2  818,955,513  0.061640  0.123281 
1.5  602,151,074  0.045322  0.067983 
1  4,281,302,973  0.322242  0.322242 
0  1,156,576,253  0.087052  0.000000 
1  5,753,875,365  0.433078  0.433078 
2  592,497,718  0.044596  0.089191 
3  28,590,629  0.002152  0.006456 
4  6,289,071  0.000473  0.001893 
5  809,116  0.000061  0.000304 
6  79,053  0.000006  0.000036 
Total  13,286,000,000  1.000000  0.006300 
Here are some more bits of information about the RTG rules simulation.
Others Facts and Figures
Question  Answer 

Standard deviation  1.161350 
Probability win  43.26% 
Probability push  8.71% 
Probability loss  48.04% 
Probability win given bet resolved  47.38% 
Probability player bust  15.73% 
Probability dealer bust  24.36% 
Internal Links
 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 AceFive 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 backplayer 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