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Black Jack Appendix 4
This appendix presents information pertinent to the standard deviation in blackjack. It should be noted my numbers come in a little higher than those using Stanford Wong's Blackjack Count Analyzer. My standard deviations come in about 0.016 higher. That said, the following table indicates the possible outcomes based on a 200 million hand simulation. Atlantic City rules (8 decks, double any first two cards, double after a split, split up to four hands, no surrender, dealer stands on soft 17) were used.

Final outcome in blackjack
Net win Number Probability
-8 48 0.00000024
-7 653 0.00000327
-6 4654 0.00002327
-5 22415 0.00011208
-4 114254 0.00057127
-3 465229 0.00232615
-2 9248859 0.04624430
-1 85919012 0.42959500
0 17594957 0.08797480
1 64221574 0.32110800
1.5 9052402 0.04526200
2 12616269 0.06308130
3 525109 0.00262555
4 169229 0.00084615
5 33994 0.00016997
6 9610 0.00004805
7 1576 0.00000788
8 156 0.00000078

The table shows the following information:

  • Probability of net win = 43.31%
  • Probability of tie = 8.80%
  • Probability of net loss = 47.89%
  • Average loss = 1.11
  • Average win = 1.22
  • Standard deviation = 1.17

The next table shows the stardard variation under other rule variations. Note that the most important variable is whether the player is allowed to double on soft totals, followed by being allowed to double after a split.

Standard Deviation in Blackjack
Number
of Decks		 Maximum
Splits		 Double
after
Split		 Soft
Doubling
Allowed		 Standard
Deviation
1 1 No Yes 1.1450
1 2 Yes Yes 1.1582
2 2 Yes Yes 1.1638
4 2 Yes Yes 1.1666
6 2 Yes Yes 1.1676
8 1 Yes Yes 1.1611
8 2 Yes Yes 1.1680
8 3 Yes Yes 1.1695
8 3 No Yes 1.1468
8 3 Yes No 1.1417

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
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