Blackjack Appendix 4: Standard deviation in blackjack

Last update: May 22, 2007

This appendix presents information pertinent to the standard deviation in blackjack. It is based on 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

Net Win in Blackjack
Net win	Total	Probability	Return
8	1079	0.00000063	0.00000506
7	10440	0.00000612	0.00004287
6	64099	0.00003761	0.00022563
5	247638	0.00014528	0.00072642
4	1307719	0.00076721	0.00306885
3	4437365	0.00260331	0.00780994
2	99686181	0.05848386	0.11696773
1.5	77147473	0.04526086	0.06789129
1	540233094	0.31694382	0.31694382
0	144520347	0.08478716	0
-0.5	76163623	0.04468366	-0.02234183
-1	684733650	0.40171937	-0.40171937
-2	71380000	0.0418772	-0.0837544
-3	3559202	0.00208811	-0.00626434
-4	828010	0.00048578	-0.00194311
-5	152687	0.00008958	-0.00044789
-6	30536	0.00001791	-0.00010749
-7	3972	0.00000233	-0.00001631
-8	305	0.00000018	-0.00000143
Total	1704507420	1	-0.00291455

This table reflects a standard deviation of 1.1418.

Here is a summary, which answers the frequently asked question, what is the probabiity of a net win, loss, and push.

Summarized Net Win in Blackjack
Event	Avg. win	Total	Probability	Return
Net win	1.210803	723135088	0.424249	0.513682
Net push	0	144520347	0.084787	0
Net loss	-1.052208	836851985	0.490964	-0.516596
Total		1704507420	1	-0.002915

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
Net win	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 in Blackjack when Doubling First Action
Net win	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 in Blackjack when Splitting First Action
Net win	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	Std. 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

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