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Last Updated: June 14, 2013

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 total-dependent 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*(n-1). 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 re-splitting).
  • All other pairs may be re-split 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 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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