God of Gambling - Black Jack

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Strategies for playing blackjack

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Provided by The Wizard of Odds

Black Jack Appendix 9f
The following tables indicate the composition dependent expected rules under for blackjack under the following rules:

Four decks
Dealer hits on soft 17
Player may double on any first two cards
Player may resplit to four hands
Numbers are presented both ways for whether the player may or may not double after a split

Expected return of game:

Player may not double after split: -0.007022819
Player may double after split: -0.00558043

Player has two different cards

The following table indicates the return for hands where the player may not split.

Dealer	Player	Stand	Hit	Double	Probability
A	2,A	-0.596474	-0.101172	-0.581196	0.00059556
A	3,A	-0.595965	-0.135655	-0.580368	0.00059556
A	3,2	-0.595457	-0.318446	-1.190914	0.00063526
A	4,A	-0.595416	-0.173615	-0.585335	0.00059556
A	4,2	-0.594949	-0.346188	-1.177880	0.00063526
A	4,3	-0.594461	-0.355533	-1.083119	0.00063526
A	5,A	-0.595655	-0.212274	-0.594241	0.00059556
A	5,2	-0.595170	-0.353184	-1.084618	0.00063526
A	5,3	-0.594650	-0.267129	-0.828680	0.00063526
A	5,4	-0.594091	-0.124178	-0.448239	0.00063526
A	6,A	-0.513575	-0.222214	-0.546802	0.00059556
A	6,2	-0.594186	-0.267109	-0.830184	0.00063526
A	6,3	-0.593668	-0.124752	-0.450110	0.00063526
A	6,4	-0.593115	0.035513	-0.020386	0.00063526
A	6,5	-0.593241	0.111710	0.134492	0.00063526
A	7,A	-0.225191	-0.161674	-0.417624	0.00059556
A	7,2	-0.592793	-0.128143	-0.452358	0.00063526
A	7,3	-0.592278	0.034387	-0.023485	0.00063526
A	7,4	-0.591680	0.110619	0.131911	0.00063526
A	7,5	-0.591886	-0.389279	-0.837167	0.00063526
A	7,6	-0.590945	-0.437537	-0.911017	0.00063526
A	8,A	0.190271	-0.065708	-0.228736	0.00059556
A	8,2	-0.594420	0.034369	-0.027004	0.00063526
A	8,3	-0.593863	0.109865	0.126656	0.00063526
A	8,4	-0.593314	-0.388890	-0.839859	0.00063526
A	8,5	-0.593521	-0.435648	-0.909178	0.00063526
A	8,6	-0.592581	-0.470803	-0.961917	0.00063526
A	8,7	-0.591245	-0.501146	-1.010099	0.00063526
A	9,A	0.607410	0.029283	-0.042944	0.00059556
A	9,2	-0.595990	0.109244	0.121847	0.00063526
A	9,3	-0.595483	-0.389079	-0.843659	0.00063526
A	9,4	-0.594935	-0.423791	-0.887279	0.00063526
A	9,5	-0.595142	-0.468465	-0.979737	0.00063526
A	9,6	-0.594189	-0.509542	-1.027457	0.00063526
A	9,7	-0.592862	-0.536913	-1.073826	0.00063526
A	9,8	-0.509705	-0.575986	-1.151972	0.00063526
A	10,A	1.500000	0.105024	0.111257	0.00239914
A	10,2	-0.599391	-0.383001	-0.832454	0.00255908
A	10,3	-0.598886	-0.426448	-0.893557	0.00255908
A	10,4	-0.598343	-0.468814	-0.958468	0.00255908
A	10,5	-0.598541	-0.510904	-1.029171	0.00255908
A	10,6	-0.597634	-0.538953	-1.077906	0.00255908
A	10,7	-0.512810	-0.578724	-1.157449	0.00255908
A	10,8	-0.222188	-0.640412	-1.280824	0.00255908
A	10,9	0.193983	-0.734426	-1.468852	0.00255908
2	2,A	-0.284357	0.045095	-0.060966	0.00086589
2	3,A	-0.284272	0.021735	-0.061684	0.00092361
2	3,2	-0.285926	-0.126132	-0.571853	0.00086589
2	4,A	-0.284141	-0.001763	-0.067212	0.00092361
2	4,2	-0.285749	-0.139859	-0.551276	0.00086589
2	4,3	-0.285663	-0.111691	-0.428461	0.00092361
2	5,A	-0.280296	-0.022271	-0.070719	0.00092361
2	5,2	-0.281886	-0.107068	-0.423230	0.00086589
2	5,3	-0.281800	-0.022559	-0.196717	0.00092361
2	5,4	-0.281617	0.077524	0.072626	0.00092361
2	6,A	-0.150884	0.000433	-0.003095	0.00092361
2	6,2	-0.282627	-0.022065	-0.196309	0.00086589
2	6,3	-0.282538	0.076868	0.071671	0.00092361
2	6,4	-0.282309	0.189665	0.375075	0.00092361
2	6,5	-0.278471	0.248657	0.493353	0.00092361
2	7,A	0.114542	0.060729	0.117057	0.00092361
2	7,2	-0.283393	0.074734	0.072461	0.00086589
2	7,3	-0.283259	0.188722	0.373259	0.00092361
2	7,4	-0.283077	0.246002	0.487602	0.00092361
2	7,5	-0.279241	-0.253535	-0.507070	0.00092361
2	7,6	-0.279988	-0.314431	-0.628862	0.00092361
2	8,A	0.382107	0.120840	0.237347	0.00092361
2	8,2	-0.284070	0.187995	0.372141	0.00086589
2	8,3	-0.283983	0.244672	0.485011	0.00092361
2	8,4	-0.283803	-0.255220	-0.510439	0.00092361
2	8,5	-0.279969	-0.314244	-0.628489	0.00092361
2	8,6	-0.280710	-0.363686	-0.727371	0.00092361
2	8,7	-0.281537	-0.412240	-0.824480	0.00092361
2	9,A	0.638969	0.182844	0.361686	0.00092361
2	9,2	-0.285101	0.243453	0.482637	0.00086589
2	9,3	-0.285014	-0.256863	-0.513727	0.00092361
2	9,4	-0.284835	-0.304881	-0.609761	0.00092361
2	9,5	-0.280999	-0.362697	-0.725395	0.00092361
2	9,6	-0.281801	-0.421677	-0.843354	0.00092361
2	9,7	-0.282572	-0.469673	-0.939345	0.00092361
2	9,8	-0.152453	-0.536788	-1.073577	0.00092361
2	10,A	1.500000	0.239713	0.473824	0.00369444
2	10,2	-0.290878	-0.251443	-0.502887	0.00346354
2	10,3	-0.290792	-0.307652	-0.615303	0.00369444
2	10,4	-0.290614	-0.364794	-0.729587	0.00369444
2	10,5	-0.286832	-0.423019	-0.846038	0.00369444
2	10,6	-0.287576	-0.471853	-0.943706	0.00369444
2	10,7	-0.157582	-0.538641	-1.077282	0.00369444
2	10,8	0.109414	-0.625997	-1.251995	0.00369444
2	10,9	0.377810	-0.734499	-1.468997	0.00369444
3	2,A	-0.243738	0.073581	0.005080	0.00092361
3	3,A	-0.243614	0.049907	0.001226	0.00086589
3	3,2	-0.245133	-0.093277	-0.490266	0.00086589
3	4,A	-0.239733	0.028628	-0.001187	0.00092361
3	4,2	-0.241230	-0.103957	-0.463292	0.00092361
3	4,3	-0.241056	-0.077103	-0.346213	0.00086589
3	5,A	-0.239109	0.007407	-0.006943	0.00092361
3	5,2	-0.240610	-0.072797	-0.343447	0.00092361
3	5,3	-0.240434	0.009252	-0.123873	0.00086589
3	5,4	-0.236505	0.109220	0.137766	0.00092361
3	6,A	-0.114018	0.029863	0.058843	0.00092361
3	6,2	-0.241330	0.009754	-0.123541	0.00092361
3	6,3	-0.241108	0.108573	0.135757	0.00086589
3	6,4	-0.237227	0.215503	0.430006	0.00092361
3	6,5	-0.236606	0.272474	0.544064	0.00092361
3	7,A	0.143418	0.088592	0.176300	0.00092361
3	7,2	-0.242047	0.106454	0.136505	0.00092361
3	7,3	-0.241872	0.214502	0.427622	0.00086589
3	7,4	-0.237993	0.270365	0.539846	0.00092361
3	7,5	-0.237375	-0.231244	-0.462489	0.00092361
3	7,6	-0.238096	-0.295172	-0.590345	0.00092361
3	8,A	0.401226	0.152991	0.304809	0.00092361
3	8,2	-0.243071	0.213200	0.425153	0.00092361
3	8,3	-0.242897	0.268070	0.534888	0.00086589
3	8,4	-0.239022	-0.232977	-0.465953	0.00092361
3	8,5	-0.238399	-0.295395	-0.590791	0.00092361
3	8,6	-0.239183	-0.349513	-0.699027	0.00092361
3	8,7	-0.239949	-0.401918	-0.803836	0.00092361
3	9,A	0.644935	0.202765	0.404101	0.00092361
3	9,2	-0.249021	0.262106	0.522783	0.00092361
3	9,3	-0.248848	-0.238994	-0.477988	0.00086589
3	9,4	-0.244970	-0.289064	-0.578127	0.00092361
3	9,5	-0.244406	-0.351338	-0.702676	0.00092361
3	9,6	-0.245131	-0.413800	-0.827601	0.00092361
3	9,7	-0.245899	-0.464469	-0.928938	0.00092361
3	9,8	-0.120503	-0.534231	-1.068463	0.00092361
3	10,A	1.500000	0.262244	0.521784	0.00369444
3	10,2	-0.250114	-0.230500	-0.461000	0.00369444
3	10,3	-0.249943	-0.289818	-0.579635	0.00346354
3	10,4	-0.246120	-0.351286	-0.702572	0.00369444
3	10,5	-0.245496	-0.413237	-0.826474	0.00369444
3	10,6	-0.246223	-0.465092	-0.930185	0.00369444
3	10,7	-0.120516	-0.534554	-1.069107	0.00369444
3	10,8	0.137755	-0.624273	-1.248545	0.00369444
3	10,9	0.392518	-0.724746	-1.449491	0.00369444
4	2,A	-0.199743	0.105013	0.075410	0.00092361
4	3,A	-0.195867	0.083700	0.074056	0.00092361
4	3,2	-0.197210	-0.053618	-0.394420	0.00092361
4	4,A	-0.195168	0.060928	0.067459	0.00086589
4	4,2	-0.196511	-0.065491	-0.373989	0.00086589
4	4,3	-0.192612	-0.036176	-0.253886	0.00086589
4	5,A	-0.194637	0.040776	0.061847	0.00092361
4	5,2	-0.195983	-0.034700	-0.257969	0.00092361
4	5,3	-0.192060	0.048507	-0.042578	0.00092361
4	5,4	-0.191356	0.141932	0.207128	0.00086589
4	6,A	-0.072750	0.063127	0.126253	0.00092361
4	6,2	-0.196642	0.047189	-0.046843	0.00092361
4	6,3	-0.192766	0.141224	0.205691	0.00092361
4	6,4	-0.192065	0.244349	0.488698	0.00086589
4	6,5	-0.191538	0.298997	0.597993	0.00092361
4	7,A	0.173450	0.125459	0.250918	0.00092361
4	7,2	-0.197724	0.136946	0.201533	0.00092361
4	7,3	-0.193851	0.242809	0.485617	0.00092361
4	7,4	-0.193153	0.296913	0.593827	0.00086589
4	7,5	-0.192622	-0.208860	-0.417720	0.00092361
4	7,6	-0.193390	-0.275811	-0.551622	0.00092361
4	8,A	0.415146	0.176956	0.353913	0.00092361
4	8,2	-0.203664	0.235210	0.470420	0.00092361
4	8,3	-0.199794	0.290066	0.580131	0.00092361
4	8,4	-0.199092	-0.214012	-0.428025	0.00086589
4	8,5	-0.198620	-0.279167	-0.558334	0.00092361
4	8,6	-0.199330	-0.336760	-0.673519	0.00092361
4	8,7	-0.200418	-0.393452	-0.786904	0.00092361
4	9,A	0.655298	0.229357	0.458714	0.00092361
4	9,2	-0.204880	0.286916	0.573832	0.00092361
4	9,3	-0.201009	-0.216647	-0.433293	0.00092361
4	9,4	-0.200364	-0.270141	-0.540282	0.00086589
4	9,5	-0.199835	-0.336288	-0.672576	0.00092361
4	9,6	-0.200545	-0.403005	-0.806010	0.00092361
4	9,7	-0.201635	-0.457452	-0.914905	0.00092361
4	9,8	-0.084264	-0.531893	-1.063787	0.00092361
4	10,A	1.500000	0.286307	0.572613	0.00369444
4	10,2	-0.205947	-0.209063	-0.418127	0.00369444
4	10,3	-0.202129	-0.270878	-0.541756	0.00369444
4	10,4	-0.201428	-0.336319	-0.672639	0.00346354
4	10,5	-0.200897	-0.402544	-0.805088	0.00369444
4	10,6	-0.201609	-0.457810	-0.915620	0.00369444
4	10,7	-0.079505	-0.530627	-1.061254	0.00369444
4	10,8	0.163389	-0.613514	-1.227029	0.00369444
4	10,9	0.411428	-0.723708	-1.447417	0.00369444
5	2,A	-0.153288	0.139648	0.148011	0.00092361
5	3,A	-0.152642	0.118117	0.145904	0.00092361
5	3,2	-0.150084	-0.012095	-0.300168	0.00092361
5	4,A	-0.152036	0.096157	0.138832	0.00092361
5	4,2	-0.149480	-0.023294	-0.281007	0.00092361
5	4,3	-0.148807	0.005990	-0.168453	0.00092361
5	5,A	-0.151511	0.075888	0.131876	0.00086589
5	5,2	-0.148930	0.007799	-0.170525	0.00086589
5	5,3	-0.148280	0.086370	0.037764	0.00086589
5	5,4	-0.147671	0.177950	0.282001	0.00086589
5	6,A	-0.034467	0.102635	0.205270	0.00092361
5	6,2	-0.150003	0.084068	0.032397	0.00092361
5	6,3	-0.149354	0.175448	0.276933	0.00092361
5	6,4	-0.148748	0.275170	0.550340	0.00092361
5	6,5	-0.148219	0.327316	0.654632	0.00086589
5	7,A	0.201050	0.153409	0.306817	0.00092361
5	7,2	-0.155998	0.166475	0.263608	0.00092361
5	7,3	-0.155351	0.268166	0.536332	0.00092361
5	7,4	-0.154742	0.320172	0.640345	0.00092361
5	7,5	-0.154272	-0.189740	-0.379479	0.00086589
5	7,6	-0.155349	-0.260473	-0.520946	0.00092361
5	8,A	0.441785	0.208204	0.416408	0.00092361
5	8,2	-0.157233	0.265445	0.530890	0.00092361
5	8,3	-0.156583	0.317348	0.634696	0.00092361
5	8,4	-0.156032	-0.191864	-0.383728	0.00092361
5	8,5	-0.155504	-0.260786	-0.521573	0.00086589
5	8,6	-0.156582	-0.321029	-0.642057	0.00092361
5	8,7	-0.162578	-0.383177	-0.766354	0.00092361
5	9,A	0.671436	0.258672	0.517344	0.00092361
5	9,2	-0.158278	0.314817	0.629635	0.00092361
5	9,3	-0.157687	-0.194343	-0.388686	0.00092361
5	9,4	-0.157078	-0.251758	-0.503515	0.00092361
5	9,5	-0.156550	-0.320256	-0.640512	0.00086589
5	9,6	-0.157629	-0.390370	-0.780739	0.00092361
5	9,7	-0.163626	-0.450989	-0.901978	0.00092361
5	9,8	-0.046053	-0.516508	-1.033015	0.00092361
5	10,A	1.500000	0.313008	0.626016	0.00369444
5	10,2	-0.159310	-0.186390	-0.372780	0.00369444
5	10,3	-0.158660	-0.252450	-0.504900	0.00369444
5	10,4	-0.158051	-0.320331	-0.640663	0.00369444
5	10,5	-0.157523	-0.389656	-0.779313	0.00346354
5	10,6	-0.158603	-0.449333	-0.898667	0.00369444
5	10,7	-0.045816	-0.516496	-1.032993	0.00369444
5	10,8	0.195823	-0.609827	-1.219655	0.00369444
5	10,9	0.438165	-0.722070	-1.444140	0.00369444
6	2,A	-0.112532	0.167174	0.208324	0.00092361
6	3,A	-0.111935	0.146848	0.206477	0.00092361
6	3,2	-0.112255	0.021325	-0.224509	0.00092361
6	4,A	-0.111383	0.126567	0.201535	0.00092361
6	4,2	-0.111676	0.012471	-0.205841	0.00092361
6	4,3	-0.111075	0.038569	-0.097207	0.00092361
6	5,A	-0.111221	0.112080	0.205032	0.00092361
6	5,2	-0.111557	0.040496	-0.098602	0.00092361
6	5,3	-0.110956	0.115305	0.100214	0.00092361
6	5,4	-0.110400	0.202961	0.334470	0.00092361
6	6,A	-0.004442	0.126798	0.253595	0.00086589
6	6,2	-0.117554	0.108168	0.085294	0.00086589
6	6,3	-0.116955	0.195592	0.319694	0.00086589
6	6,4	-0.116397	0.293011	0.586022	0.00086589
6	6,5	-0.116337	0.343758	0.687516	0.00086589
6	7,A	0.222101	0.178870	0.357740	0.00092361
6	7,2	-0.118793	0.190690	0.314474	0.00092361
6	7,3	-0.118192	0.289896	0.579793	0.00092361
6	7,4	-0.117691	0.340736	0.681471	0.00092361
6	7,5	-0.117572	-0.173630	-0.347259	0.00092361
6	7,6	-0.123570	-0.250102	-0.500204	0.00086589
6	8,A	0.451752	0.231294	0.462588	0.00092361
6	8,2	-0.119828	0.287230	0.574460	0.00092361
6	8,3	-0.119285	0.337956	0.675913	0.00092361
6	8,4	-0.118726	-0.175220	-0.350439	0.00092361
6	8,5	-0.118607	-0.247268	-0.494537	0.00092361
6	8,6	-0.124606	-0.312919	-0.625837	0.00086589
6	8,7	-0.125846	-0.374576	-0.749152	0.00092361
6	9,A	0.677213	0.280966	0.561933	0.00092361
6	9,2	-0.120900	0.335424	0.670847	0.00092361
6	9,3	-0.120299	-0.177686	-0.355373	0.00092361
6	9,4	-0.119741	-0.237889	-0.475778	0.00092361
6	9,5	-0.119622	-0.309449	-0.618897	0.00092361
6	9,6	-0.125622	-0.383780	-0.767560	0.00086589
6	9,7	-0.126781	-0.434864	-0.869727	0.00092361
6	9,8	-0.011876	-0.513210	-1.026419	0.00092361
6	10,A	1.500000	0.334133	0.668266	0.00369444
6	10,2	-0.121695	-0.169633	-0.339266	0.00369444
6	10,3	-0.121095	-0.238517	-0.477033	0.00369444
6	10,4	-0.120537	-0.309137	-0.618274	0.00369444
6	10,5	-0.120418	-0.380669	-0.761338	0.00369444
6	10,6	-0.126338	-0.434795	-0.869590	0.00346354
6	10,7	-0.011673	-0.513194	-1.026389	0.00369444
6	10,8	0.218639	-0.608353	-1.216705	0.00369444
6	10,9	0.449198	-0.721432	-1.442864	0.00369444
7	2,A	-0.472081	0.119029	-0.177283	0.00092361
7	3,A	-0.471601	0.075026	-0.181438	0.00092361
7	3,2	-0.473744	-0.119489	-0.947488	0.00092361
7	4,A	-0.471066	0.036105	-0.173744	0.00092361
7	4,2	-0.473230	-0.154678	-0.888294	0.00092361
7	4,3	-0.472747	-0.069274	-0.578029	0.00092361
7	5,A	-0.473288	-0.009400	-0.184635	0.00092361
7	5,2	-0.475452	-0.068561	-0.583473	0.00092361
7	5,3	-0.474970	0.084676	-0.174035	0.00092361
7	5,4	-0.474451	0.178579	0.125159	0.00092361
7	6,A	-0.102349	0.055160	-0.006438	0.00092361
7	6,2	-0.476595	0.084790	-0.177547	0.00092361
7	6,3	-0.476108	0.178192	0.121837	0.00092361
7	6,4	-0.475650	0.263931	0.412852	0.00092361
7	6,5	-0.477867	0.293989	0.473101	0.00092361
7	7,A	0.402983	0.172399	0.225480	0.00086589
7	7,2	-0.477635	0.175421	0.118269	0.00086589
7	7,3	-0.477207	0.262042	0.405886	0.00086589
7	7,4	-0.476692	0.293167	0.469940	0.00086589
7	7,5	-0.478909	-0.223059	-0.524096	0.00086589
7	7,6	-0.480052	-0.283238	-0.616348	0.00086589
7	8,A	0.615612	0.221109	0.320950	0.00092361
7	8,2	-0.478446	0.259428	0.398326	0.00092361
7	8,3	-0.477960	0.292163	0.465353	0.00092361
7	8,4	-0.477445	-0.220527	-0.520166	0.00092361
7	8,5	-0.479663	-0.282775	-0.615300	0.00092361
7	8,6	-0.480814	-0.327645	-0.683560	0.00092361
7	8,7	-0.481778	-0.358883	-0.729046	0.00086589
7	9,A	0.773288	0.253501	0.382647	0.00092361
7	9,2	-0.479752	0.291519	0.461327	0.00092361
7	9,3	-0.479268	-0.220984	-0.523830	0.00092361
7	9,4	-0.478752	-0.270153	-0.590256	0.00092361
7	9,5	-0.480975	-0.327523	-0.681924	0.00092361
7	9,6	-0.482037	-0.368223	-0.747781	0.00092361
7	9,7	-0.483097	-0.405216	-0.810432	0.00086589
7	9,8	-0.111160	-0.475319	-0.950637	0.00092361
7	10,A	1.500000	0.290734	0.464189	0.00369444
7	10,2	-0.474716	-0.212542	-0.503932	0.00369444
7	10,3	-0.474233	-0.269367	-0.586268	0.00369444
7	10,4	-0.473721	-0.326238	-0.677342	0.00369444
7	10,5	-0.475899	-0.368503	-0.745856	0.00369444
7	10,6	-0.477058	-0.405505	-0.811011	0.00369444
7	10,7	-0.109970	-0.475803	-0.951607	0.00346354
7	10,8	0.396838	-0.585413	-1.170826	0.00369444
7	10,9	0.614575	-0.711392	-1.422783	0.00369444
8	2,A	-0.509765	0.050688	-0.313928	0.00092361
8	3,A	-0.509230	0.018138	-0.300276	0.00092361
8	3,2	-0.511100	-0.186664	-1.022199	0.00092361
8	4,A	-0.511398	-0.029088	-0.314247	0.00092361
8	4,2	-0.513268	-0.221006	-1.006429	0.00092361
8	4,3	-0.512753	-0.214655	-0.847489	0.00092361
8	5,A	-0.511319	-0.070735	-0.318548	0.00092361
8	5,2	-0.513188	-0.212223	-0.848329	0.00092361
8	5,3	-0.512670	-0.059228	-0.446067	0.00092361
8	5,4	-0.514892	0.100702	-0.019341	0.00092361
8	6,A	-0.382888	-0.071144	-0.249396	0.00092361
8	6,2	-0.514137	-0.058875	-0.448845	0.00092361
8	6,3	-0.513679	0.102668	-0.016898	0.00092361
8	6,4	-0.515843	0.200444	0.294105	0.00092361
8	6,5	-0.515758	0.230237	0.354804	0.00092361
8	7,A	0.109163	0.041697	-0.027229	0.00092361
8	7,2	-0.515041	0.100447	-0.019183	0.00092361
8	7,3	-0.514524	0.202193	0.295685	0.00092361
8	7,4	-0.516689	0.228163	0.348281	0.00092361
8	7,5	-0.516605	-0.283101	-0.638037	0.00092361
8	7,6	-0.517566	-0.340317	-0.726862	0.00092361
8	8,A	0.596957	0.153259	0.193419	0.00086589
8	8,2	-0.516352	0.199835	0.288079	0.00086589
8	8,3	-0.515836	0.227823	0.345738	0.00086589
8	8,4	-0.518002	-0.282865	-0.640233	0.00086589
8	8,5	-0.517922	-0.338532	-0.723484	0.00086589
8	8,6	-0.518794	-0.371252	-0.768217	0.00086589
8	8,7	-0.519657	-0.407971	-0.826337	0.00086589
8	9,A	0.790078	0.191569	0.272846	0.00092361
8	9,2	-0.511321	0.226418	0.345105	0.00092361
8	9,3	-0.510806	-0.282234	-0.636741	0.00092361
8	9,4	-0.512975	-0.327234	-0.698857	0.00092361
8	9,5	-0.512851	-0.371382	-0.765021	0.00092361
8	9,6	-0.513819	-0.417733	-0.844886	0.00092361
8	9,7	-0.514682	-0.451179	-0.902358	0.00092361
8	9,8	-0.390252	-0.498754	-0.997508	0.00086589
8	10,A	1.500000	0.227782	0.346321	0.00369444
8	10,2	-0.512196	-0.272223	-0.618185	0.00369444
8	10,3	-0.511684	-0.324828	-0.694547	0.00369444
8	10,4	-0.513809	-0.368434	-0.759805	0.00369444
8	10,5	-0.513733	-0.416983	-0.842012	0.00369444
8	10,6	-0.514701	-0.450434	-0.900867	0.00369444
8	10,7	-0.385343	-0.498411	-0.996822	0.00369444
8	10,8	0.103131	-0.584919	-1.169839	0.00346354
8	10,9	0.589698	-0.709749	-1.419499	0.00369444
9	2,A	-0.537051	-0.032240	-0.436487	0.00092361
9	3,A	-0.539227	-0.071521	-0.441293	0.00092361
9	3,2	-0.540768	-0.265667	-1.081536	0.00092361
9	4,A	-0.539104	-0.112536	-0.448202	0.00092361
9	4,2	-0.540645	-0.295486	-1.062540	0.00092361
9	4,3	-0.542817	-0.290016	-0.954433	0.00092361
9	5,A	-0.538949	-0.152841	-0.455455	0.00092361
9	5,2	-0.540486	-0.285462	-0.949837	0.00092361
9	5,3	-0.542716	-0.212011	-0.714383	0.00092361
9	5,4	-0.542587	-0.052189	-0.295603	0.00092361
9	6,A	-0.419274	-0.146009	-0.387684	0.00092361
9	6,2	-0.541476	-0.209728	-0.711897	0.00092361
9	6,3	-0.543647	-0.051952	-0.297836	0.00092361
9	6,4	-0.543520	0.116835	0.149459	0.00092361
9	6,5	-0.543360	0.156958	0.231125	0.00092361
9	7,A	-0.182351	-0.097328	-0.282050	0.00092361
9	7,2	-0.542697	-0.052173	-0.294949	0.00092361
9	7,3	-0.544869	0.116772	0.146517	0.00092361
9	7,4	-0.544742	0.156165	0.226910	0.00092361
9	7,5	-0.544586	-0.351710	-0.758593	0.00092361
9	7,6	-0.545439	-0.394601	-0.822584	0.00092361
9	8,A	0.287634	0.007206	-0.070222	0.00092361
9	8,2	-0.537670	0.117102	0.150972	0.00092361
9	8,3	-0.539843	0.153842	0.224717	0.00092361
9	8,4	-0.539720	-0.353168	-0.760375	0.00092361
9	8,5	-0.539520	-0.393419	-0.817534	0.00092361
9	8,6	-0.540469	-0.432288	-0.880220	0.00092361
9	8,7	-0.541703	-0.464749	-0.935196	0.00092361
9	9,A	0.760068	0.111528	0.135641	0.00086589
9	9,2	-0.538549	0.154592	0.224297	0.00086589
9	9,3	-0.540726	-0.352393	-0.761887	0.00086589
9	9,4	-0.540559	-0.381891	-0.796071	0.00086589
9	9,5	-0.540406	-0.429775	-0.873961	0.00086589
9	9,6	-0.541355	-0.473445	-0.952638	0.00086589
9	9,7	-0.542589	-0.502797	-1.005594	0.00086589
9	9,8	-0.420344	-0.548382	-1.096764	0.00086589
9	10,A	1.500000	0.155893	0.224949	0.00369444
9	10,2	-0.539371	-0.341050	-0.739516	0.00369444
9	10,3	-0.541503	-0.380451	-0.794521	0.00369444
9	10,4	-0.541384	-0.426843	-0.869023	0.00369444
9	10,5	-0.541232	-0.472492	-0.949345	0.00369444
9	10,6	-0.542181	-0.502126	-1.004252	0.00369444
9	10,7	-0.421644	-0.547224	-1.094448	0.00369444
9	10,8	-0.186228	-0.610958	-1.221915	0.00369444
9	10,9	0.282025	-0.711335	-1.422671	0.00346354
10	2,A	-0.540060	-0.101005	-0.505785	0.00342412
10	3,A	-0.539920	-0.135860	-0.506008	0.00342412
10	3,2	-0.542021	-0.312335	-1.084042	0.00340610
10	4,A	-0.539817	-0.172939	-0.509750	0.00342412
10	4,2	-0.541913	-0.339672	-1.064902	0.00340610
10	4,3	-0.541832	-0.322763	-0.951751	0.00340610
10	5,A	-0.539637	-0.211321	-0.519456	0.00342412
10	5,2	-0.541797	-0.319734	-0.951709	0.00340610
10	5,3	-0.541651	-0.249688	-0.744354	0.00340610
10	5,4	-0.541541	-0.149368	-0.459610	0.00340610
10	6,A	-0.419454	-0.194959	-0.452642	0.00342412
10	6,2	-0.543112	-0.249702	-0.746678	0.00340610
10	6,3	-0.542967	-0.150299	-0.461865	0.00340610
10	6,4	-0.542858	0.027559	-0.004039	0.00340610
10	6,5	-0.542681	0.118138	0.177799	0.00340610
10	7,A	-0.180201	-0.142446	-0.340658	0.00342412
10	7,2	-0.537660	-0.153134	-0.459236	0.00340610
10	7,3	-0.537516	0.026491	-0.003201	0.00340610
10	7,4	-0.537411	0.117111	0.177694	0.00340610
10	7,5	-0.537186	-0.381902	-0.795549	0.00340610
10	7,6	-0.538516	-0.431265	-0.877595	0.00340610
10	8,A	0.063422	-0.087360	-0.232658	0.00342412
10	8,2	-0.538613	0.026652	-0.005191	0.00340610
10	8,3	-0.538473	0.117046	0.175515	0.00340610
10	8,4	-0.538321	-0.382253	-0.799070	0.00340610
10	8,5	-0.538148	-0.430314	-0.877566	0.00340610
10	8,6	-0.539478	-0.465986	-0.937945	0.00340610
10	8,7	-0.534086	-0.497373	-0.994746	0.00340610
10	9,A	0.554561	0.021363	-0.018877	0.00342412
10	9,2	-0.539505	0.115827	0.170985	0.00340610
10	9,3	-0.539317	-0.382961	-0.802777	0.00340610
10	9,4	-0.539216	-0.418939	-0.855830	0.00340610
10	9,5	-0.539043	-0.463099	-0.931619	0.00340610
10	9,6	-0.540373	-0.505801	-1.012641	0.00340610
10	9,7	-0.534982	-0.533145	-1.066291	0.00340610
10	9,8	-0.412670	-0.578154	-1.156309	0.00340610
10	10,A	1.500000	0.115859	0.169912	0.01348247
10	10,2	-0.540283	-0.372900	-0.783223	0.01341151
10	10,3	-0.540147	-0.417688	-0.854216	0.01341151
10	10,4	-0.540046	-0.461443	-0.928883	0.01341151
10	10,5	-0.539873	-0.503624	-1.007248	0.01341151
10	10,6	-0.541203	-0.532053	-1.064107	0.01341151
10	10,7	-0.417830	-0.577834	-1.155668	0.01341151
10	10,8	-0.172653	-0.641750	-1.283500	0.01341151
10	10,9	0.072642	-0.724834	-1.449668	0.01341151

Players has two equal cards

The following table displays the expected returns when the player may split. The right two columns indicate the return by splitting according to whether the player may double after a split (DAS).

Dealer	Player	Stand	Hit	Double	DAS: no	DAS: yes	Probability
A	A,A	-0.596977	-0.065388	-0.582574	0.129936	0.129936	0.00026056
A	2,2	-0.595967	-0.292870	-1.191935	-0.523195	-0.520176	0.00029778
A	3,3	-0.594987	-0.346177	-1.177947	-0.567464	-0.563587	0.00029778
A	4,4	-0.593902	-0.264116	-0.827063	-0.638449	-0.633573	0.00029778
A	5,5	-0.594290	0.035263	-0.020254	-0.756718	-0.751508	0.00029778
A	6,6	-0.592290	-0.388892	-0.836346	-0.726878	-0.721704	0.00029778
A	7,7	-0.589617	-0.481972	-0.984407	-0.728168	-0.723472	0.00029778
A	8,8	-0.592875	-0.536768	-1.073536	-0.513126	-0.509865	0.00029778
A	9,9	-0.216334	-0.638282	-1.276564	-0.238548	-0.236700	0.00029778
A	10,10	0.599679	-0.860952	-1.721905	-0.250831	-0.250831	0.00507366
2	A,A	-0.282726	0.083156	-0.055862	0.493092	0.493092	0.00043294
2	2,2	-0.285960	-0.112376	-0.571921	-0.151294	-0.074782	0.00037882
2	3,3	-0.285842	-0.139940	-0.551452	-0.204753	-0.127417	0.00043294
2	4,4	-0.285483	-0.021159	-0.199796	-0.256987	-0.179178	0.00043294
2	5,5	-0.277730	0.191055	0.378266	-0.312614	-0.267077	0.00043294
2	6,6	-0.279215	-0.253537	-0.507075	-0.262551	-0.183477	0.00043294
2	7,7	-0.280755	-0.374122	-0.748245	-0.194569	-0.115025	0.00043294
2	8,8	-0.282263	-0.469361	-0.938721	0.002476	0.077654	0.00043294
2	9,9	0.114027	-0.624788	-1.249576	0.154920	0.195379	0.00043294
2	10,10	0.631826	-0.853306	-1.706611	0.290251	0.290251	0.00727344
3	A,A	-0.242201	0.105864	0.010606	0.540506	0.540506	0.00043294
3	2,2	-0.245303	-0.079865	-0.490606	-0.093882	-0.001961	0.00043294
3	3,3	-0.244961	-0.106527	-0.470379	-0.134526	-0.043167	0.00037882
3	4,4	-0.237152	0.011189	-0.123541	-0.176421	-0.082191	0.00043294
3	5,5	-0.235882	0.216556	0.432104	-0.237246	-0.180725	0.00043294
3	6,6	-0.237332	-0.231295	-0.462590	-0.185354	-0.084502	0.00043294
3	7,7	-0.238920	-0.359510	-0.719019	-0.122140	-0.022688	0.00043294
3	8,8	-0.240979	-0.462481	-0.924963	0.066914	0.151261	0.00043294
3	9,9	0.133040	-0.625189	-1.250378	0.201582	0.247859	0.00043294
3	10,10	0.642519	-0.853018	-1.706035	0.369875	0.369875	0.00727344
4	A,A	-0.198317	0.130565	0.080030	0.590708	0.590708	0.00043294
4	2,2	-0.201106	-0.043137	-0.402213	-0.028908	0.078060	0.00043294
4	3,3	-0.193311	-0.063477	-0.368335	-0.046051	0.063111	0.00043294
4	4,4	-0.191887	0.051068	-0.042119	-0.091427	0.020377	0.00037882
4	5,5	-0.190827	0.244820	0.489639	-0.151381	-0.079921	0.00043294
4	6,6	-0.192244	-0.208472	-0.416945	-0.100981	0.021163	0.00043294
4	7,7	-0.194479	-0.344370	-0.688740	-0.041183	0.082117	0.00043294
4	8,8	-0.206360	-0.459113	-0.918225	0.126004	0.220050	0.00043294
4	9,9	0.163507	-0.613494	-1.226987	0.267444	0.318420	0.00043294
4	10,10	0.652914	-0.852794	-1.705588	0.454153	0.454153	0.00727344
5	A,A	-0.155794	0.162955	0.149183	0.643381	0.643381	0.00043294
5	2,2	-0.150729	0.000025	-0.301459	0.060606	0.186867	0.00043294
5	3,3	-0.149439	-0.023324	-0.280561	0.037170	0.168707	0.00043294
5	4,4	-0.148198	0.089391	0.039393	-0.008387	0.127230	0.00043294
5	5,5	-0.147145	0.276981	0.553963	-0.061772	0.028405	0.00037882
5	6,6	-0.149356	-0.186242	-0.372484	-0.017658	0.128319	0.00043294
5	7,7	-0.161344	-0.333648	-0.667296	0.024302	0.167357	0.00043294
5	8,8	-0.163814	-0.451043	-0.902086	0.200780	0.305136	0.00043294
5	9,9	0.195823	-0.609801	-1.219603	0.339023	0.395029	0.00043294
5	10,10	0.669203	-0.852399	-1.704799	0.546225	0.546225	0.00727344
6	A,A	-0.112187	0.190740	0.211466	0.687177	0.687177	0.00043294
6	2,2	-0.112850	0.032260	-0.225700	0.127860	0.272914	0.00043294
6	3,3	-0.111632	0.012463	-0.205750	0.104880	0.253005	0.00043294
6	4,4	-0.110518	0.118208	0.101627	0.061419	0.213307	0.00043294
6	5,5	-0.110282	0.299512	0.599024	0.005685	0.107862	0.00043294
6	6,6	-0.122332	-0.176676	-0.353352	0.034145	0.193330	0.00037882
6	7,7	-0.124808	-0.323281	-0.646562	0.089435	0.250066	0.00043294
6	8,8	-0.126803	-0.434843	-0.869686	0.258316	0.372287	0.00043294
6	9,9	0.218471	-0.608375	-1.216749	0.389918	0.449632	0.00043294
6	10,10	0.675762	-0.852223	-1.704446	0.619060	0.619060	0.00727344
7	A,A	-0.469898	0.163757	-0.172621	0.481997	0.481997	0.00043294
7	2,2	-0.474248	-0.089107	-0.948496	-0.048711	0.008892	0.00043294
7	3,3	-0.473262	-0.154760	-0.888339	-0.108674	-0.050921	0.00043294
7	4,4	-0.472234	0.088868	-0.168701	-0.229377	-0.170028	0.00043294
7	5,5	-0.476727	0.262157	0.410917	-0.331931	-0.298132	0.00043294
7	6,6	-0.479008	-0.224556	-0.527096	-0.307274	-0.247094	0.00043294
7	7,7	-0.481104	-0.337152	-0.704102	-0.104458	-0.047696	0.00037882
7	8,8	-0.482548	-0.405174	-0.810348	0.263135	0.317829	0.00043294
7	9,9	0.399617	-0.585162	-1.170324	0.334178	0.363443	0.00043294
7	10,10	0.771374	-0.849732	-1.699465	0.448488	0.448488	0.00727344
8	A,A	-0.507875	0.094344	-0.310003	0.364258	0.364258	0.00043294
8	2,2	-0.511613	-0.155451	-1.023226	-0.215469	-0.177332	0.00043294
8	3,3	-0.510585	-0.220202	-1.001065	-0.269545	-0.230560	0.00043294
8	4,4	-0.514918	-0.058680	-0.450579	-0.364754	-0.324723	0.00043294
8	5,5	-0.514807	0.200331	0.296188	-0.483867	-0.460215	0.00043294
8	6,6	-0.516710	-0.283253	-0.638127	-0.452066	-0.411611	0.00043294
8	7,7	-0.518333	-0.380634	-0.788316	-0.427940	-0.389402	0.00043294
8	8,8	-0.520981	-0.450837	-0.901675	-0.066948	-0.031681	0.00037882
8	9,9	0.095884	-0.585162	-1.170324	0.207154	0.227909	0.00043294
8	10,10	0.789718	-0.849395	-1.698790	0.211256	0.211256	0.00727344
9	A,A	-0.535511	-0.000541	-0.448071	0.242792	0.242792	0.00043294
9	2,2	-0.538592	-0.236442	-1.077184	-0.403000	-0.383304	0.00043294
9	3,3	-0.542943	-0.296613	-1.067142	-0.447515	-0.428380	0.00043294
9	4,4	-0.542748	-0.209639	-0.714277	-0.525773	-0.505957	0.00043294
9	5,5	-0.542426	0.116751	0.151645	-0.652721	-0.638490	0.00043294
9	6,6	-0.544301	-0.350806	-0.756276	-0.616990	-0.596861	0.00043294
9	7,7	-0.546673	-0.441447	-0.902100	-0.588271	-0.570144	0.00043294
9	8,8	-0.536732	-0.503887	-1.007774	-0.408872	-0.389563	0.00043294
9	9,9	-0.186289	-0.611291	-1.222581	-0.094641	-0.082797	0.00037882
9	10,10	0.754917	-0.848731	-1.697462	-0.062776	-0.062776	0.00727344
10	A,A	-0.537981	-0.064423	-0.502927	0.183155	0.183155	0.00161350
10	2,2	-0.542162	-0.286004	-1.084324	-0.507055	-0.496764	0.00159661
10	3,3	-0.541877	-0.339159	-1.064826	-0.552659	-0.541764	0.00159661
10	4,4	-0.541723	-0.247273	-0.744401	-0.626348	-0.615051	0.00159661
10	5,5	-0.541359	0.028133	-0.001792	-0.738825	-0.727567	0.00159661
10	6,6	-0.543907	-0.382407	-0.798973	-0.713569	-0.702475	0.00159661
10	7,7	-0.533125	-0.477970	-0.961397	-0.659718	-0.648161	0.00159661
10	8,8	-0.535048	-0.533098	-1.066195	-0.482603	-0.471782	0.00159661
10	9,9	-0.167428	-0.642116	-1.284232	-0.315709	-0.306088	0.00159661
10	10,10	0.561473	-0.846180	-1.692361	-0.211025	-0.211025	0.02598479

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