Reason number of decks matter in blackjack

I think your article is well analyzed and but only moderately interesting. Its clearly the article you want to write and you have done a workmanlike job of writing it.

BUT . . .

Blackjack is a compilation of 650 different starting hands - 65 different player hands versus 10 dealer upcard possibilities. When you average effects across all 650 hands -which you always seem to do - you may obscure as much -or more -than you show.


Let's look at some of those 65 player hands against any dealer hand (so I'm averaging across 10 dealer upcards, weighted by probabilities) using your set of BJ rules

96 vs X;___ 1 Deck EV = -39.85% 8 Deck EV = -39.62%

93 vs X;___ 1 Deck EV = -33.04% 8 Deck EV = -31.79%


So we see that both of these player hands are higher EV when they are dealt in an 8 deck game. Why? I know why, and you know why. The best chance of improving a 9-6 is to draw a 6 - but you already hold one of those in your hand. Similarly, a player holding 9-3 is most improved by drawing a 9.

Whereas

65 vs X;_____ 1 Deck EV = 34.43% 8 Deck EV = 31.37%

65 is much lower EV when it is dealt in an 8 deck game. Why? Because when you are doubling down or hitting an 11, you are hoping for a Ten-value card, and you really don't want a 5. Drawing a 6 isn't very exciting either. But the "effect of removal" of the 6 and 5 is much smaller with multiple decks.

You might list the 65 player hands and, for each, show these EV numbers and the probability of these hands for 1 and 8 decks, and actually educate people and give them an AHA! moment so they understand effect of removal and probability of occurrence. You could divide Player BJ Hands into those that are improved by multiple decks and those that are not improved. Maybe to add interest you could highlight the hand that is most hurt by multiple decks and the hand that is most improved by multiple decks. That would "teach" and would show information in a different way than what people have seen before.

Indeed, I've always wondered why the WOO site has a blind spot - why it never shows the EV of starting Blackjack hands so that people can understand the patterns and trends and just how awful and how great certain hands are. That kind of information -the EVs of starting hands- is shown on WOO for UTH and for Mississippi Stud - why is it not shown for Blackjack? You could use colors on grids and make it visually interesting.

I do not want to come across as nagging - I am just trying to help. You asked for feedback! And I am in quarantine until I get my covid test results back (after a contact with someone who has now tested positive), so I have nothing else to do.

Quote: gordonm888

I
Indeed, I've always wondered why the WOO site has a blind spot - why it never shows the EV of starting Blackjack hands so that people can understand the patterns and trends and just how awful and how great certain hands are. That kind of information -the EVs of starting hands- is shown on WOO for UTH and for Mississippi Stud - why is it not shown for Blackjack? You could use colors on grids and make it visually interesting.

https://wizardofodds.com/games/blackjack/player-expected-return/

Quote: Wizard

That is one of the essences of a well-designed casino game -- the player doesn't see where the house has the advantage. In blackjack and most poker games it is a player positional disadvantage. The idea to remove the 10's in Spanish 21 was also a good one.

I get asked from time to time by the other side about how to significantly increase the house edge, like by 1%, in blackjack without changing the 3-2 payoff, mandating a side bet, charging a commission on wins, nor change the deck composition in a way the average player won't notice. Nothing good has ever come to mind.

When Dealer and player both have BJ, they go to war. Player makes an equal bet and each gets one more card and hi card wins, but if the dealer wins, the player just gets paid 1-1 instead of 3-2. It would add some excitement while slightly raising the house edge.

Quote: billryan

When Dealer and player both have BJ, they go to war, each gets one more card and hi card wins, but if the dealer wins, the player just gets paid 1-1 instead of 3-2. It would add some excitement while slightly raising the house edge.

I really like this idea but how does it increase the house edge since normally this is a push?

Quote: Hunterhill

How about dealer wins on a push of 17

I have never seen that, but the player would definitely feel that, given that they are used to that being a push.

Quote: gordonm888

IYou might list the 65 player hands and, for each, show these EV numbers and the probability of these hands for 1 and 8 decks, and actually educate people and give them an AHA! moment so they understand effect of removal and probability of occurrence.

I do plan to add such a table, only for all 550 possible starting hands.

Quote:

Indeed, I've always wondered why the WOO site has a blind spot - why it never shows the EV of starting Blackjack hands so that people can understand the patterns and trends and just how awful and how great certain hands are.

It does that have, for 12 different sets of rules: Blackjack Expected Values.

Quote: Wizard

I get asked from time to time by the other side about how to significantly increase the house edge, like by 1%, in blackjack without changing the 3-2 payoff, mandating a side bet, charging a commission on wins, nor change the deck composition in a way the average player won't notice. Nothing good has ever come to mind.

You should tell them BJ is good the way it is, that would be at 3:2, not 6:5. 6:5 isn't blackjack in my opinion.

Take a 6 deck game with standard rules (Vegas rules), it looks like a house edge of only about half a percent. But sit down and watch people play. Few people play like me, or you or i dare say the majority of this forum. Most people are making all the wrong plays, everything from mildly wrong, to things like a no bust strategy. Add it all together and the house has more than 1% advantage.

Furthermore, they have the sidebets on most games. I happen to hate most of them because it slows the game down, but it adds to the house edge significantly. Some of these side bets its like a 7-10% house edge. What more do they want? Seems fair the way it is. Let the suckers play sidebets and play stupidly to a higher house edge, while people that want a fair game and learn how to play play to a half percent disadvantage. Add it all up and it's right where it should be. But of course the casino is always looking to expand it's edge.

Quote: gordonm888

Blackjack is a compilation of 650 different starting hands ...

This table lists the expected value of all 650 for both one and eight decks under the same rules as my simulation. The values are taken from these expected value tables:

Blackjack expected Returns for one deck and dealer hits on soft 17 and
https://wizardofodds.com/games/blackjack/appendix/9/8dh17r4/

Unlike my simulation study, these tables assume perfect combinatorial strategy at every decision point. This helps in single deck much more than eight decks.

The expected value columns are the product of the expected value of the given situation and the probability of it happening. It is listed in order of the expected value difference, in order of what is best for the player in a single-deck game.

PLAYER	DEALER	One Deck EV	Eight Deck EV	Difference
10,6	10	-0.006741	-0.007197	0.000456
10,2	10	-0.004609	-0.005063	0.000454
10,9	10	0.001363	0.000911	0.000452
10,3	10	-0.005227	-0.005660	0.000433
10,4	10	-0.005928	-0.006230	0.000302
10,8	10	-0.002064	-0.002357	0.000293
10,A	2	0.005792	0.005502	0.000290
10,A	3	0.005792	0.005502	0.000290
10,A	4	0.005792	0.005502	0.000290
10,A	5	0.005792	0.005502	0.000290
10,A	6	0.005792	0.005502	0.000290
10,A	7	0.005792	0.005502	0.000290
10,A	8	0.005792	0.005502	0.000290
10,A	9	0.005792	0.005502	0.000290
10,A	10	0.020390	0.020196	0.000194
10,2	2	-0.000706	-0.000898	0.000191
8,8	10	-0.000594	-0.000782	0.000188
10,5	5	-0.000386	-0.000573	0.000186
10,4	4	-0.000540	-0.000724	0.000183
6,4	5	0.000654	0.000487	0.000167
6,5	4	0.000679	0.000533	0.000146
10,7	10	-0.005483	-0.005623	0.000140
6,3	5	0.000377	0.000238	0.000139
9,A	10	0.002010	0.001881	0.000129
10,3	3	-0.000758	-0.000882	0.000124
7,4	5	0.000698	0.000575	0.000123
5,4	6	0.000415	0.000293	0.000122
6,5	3	0.000607	0.000487	0.000121
7,3	4	0.000549	0.000433	0.000116
10,9	5	0.001718	0.001603	0.000114
7,3	5	0.000593	0.000480	0.000113
6,6	10	-0.000513	-0.000626	0.000112
9,9	10	-0.000177	-0.000283	0.000106
2,2	10	-0.000366	-0.000471	0.000106
7,4	3	0.000590	0.000485	0.000106
6,5	2	0.000547	0.000442	0.000105
10,5	10	-0.006663	-0.006769	0.000105
8,3	5	0.000675	0.000573	0.000102
6,3	4	0.000278	0.000177	0.000101
9,7	9	-0.000349	-0.000450	0.000101
10,9	7	0.002356	0.002257	0.000099
6,4	3	0.000482	0.000384	0.000099
3,3	10	-0.000457	-0.000555	0.000098
7,7	10	-0.000678	-0.000774	0.000096
6,A	5	0.000270	0.000177	0.000093
8,2	5	0.000571	0.000478	0.000093
9,7	7	-0.000271	-0.000364	0.000093
8,7	8	-0.000275	-0.000366	0.000092
7,4	6	0.000707	0.000617	0.000090
6,4	7	0.000459	0.000369	0.000090
8,7	7	-0.000235	-0.000324	0.000089
7,2	5	0.000320	0.000232	0.000088
4,4	10	-0.000321	-0.000407	0.000087
7,2	4	0.000261	0.000175	0.000086
9,2	5	0.000656	0.000570	0.000085
10,9	A	0.000570	0.000485	0.000085
7,4	2	0.000523	0.000439	0.000084
8,2	3	0.000465	0.000382	0.000083
6,4	2	0.000416	0.000336	0.000080
2,A	5	0.000206	0.000126	0.000079
8,3	4	0.000603	0.000525	0.000078
7,3	6	0.000602	0.000525	0.000078
7,A	4	0.000299	0.000222	0.000077
9,4	9	-0.000264	-0.000342	0.000077
8,3	2	0.000515	0.000438	0.000077
6,5	A	0.000152	0.000075	0.000077
9,8	9	-0.000298	-0.000375	0.000077
7,3	2	0.000410	0.000335	0.000075
9,5	9	-0.000308	-0.000382	0.000074
9,6	9	-0.000347	-0.000420	0.000072
3,A	5	0.000197	0.000125	0.000072
5,3	6	0.000172	0.000100	0.000072
8,3	6	0.000686	0.000615	0.000071
7,4	A	0.000146	0.000074	0.000071
5,4	3	0.000188	0.000118	0.000070
7,2	3	0.000186	0.000117	0.000069
10,4	5	-0.000524	-0.000592	0.000068
5,4	5	0.000300	0.000233	0.000067
10,9	2	0.001453	0.001386	0.000066
8,6	8	-0.000267	-0.000330	0.000063
7,A	5	0.000336	0.000275	0.000061
8,2	4	0.000486	0.000426	0.000060
8,2	6	0.000582	0.000522	0.000060
8,3	A	0.000132	0.000073	0.000059
10,3	5	-0.000536	-0.000593	0.000057
10,9	8	0.002227	0.002171	0.000057
9,2	4	0.000579	0.000522	0.000057
10,6	5	-0.000536	-0.000593	0.000057
6,2	5	0.000126	0.000071	0.000055
6,5	7	0.000483	0.000429	0.000054
7,5	5	-0.000088	-0.000142	0.000054
6,6	5	0.000101	0.000048	0.000054
8,4	4	-0.000128	-0.000180	0.000052
9,2	6	0.000664	0.000612	0.000052
2,A	6	0.000237	0.000185	0.000052
10,6	7	-0.001453	-0.001504	0.000052
6,6	4	0.000053	0.000002	0.000052
6,6	2	-0.000035	-0.000086	0.000051
8,5	5	-0.000091	-0.000142	0.000051
4,4	5	0.000098	0.000048	0.000050
9,5	5	-0.000094	-0.000143	0.000049
6,6	3	0.000005	-0.000044	0.000049
10,8	5	0.000766	0.000717	0.000049
9,4	4	-0.000132	-0.000180	0.000049
7,3	8	0.000316	0.000267	0.000049
4,3	5	0.000048	-0.000001	0.000048
4,A	5	0.000170	0.000122	0.000048
10,2	5	-0.000547	-0.000594	0.000047
9,2	A	0.000118	0.000071	0.000047
3,A	6	0.000231	0.000185	0.000046
8,8	A	-0.000113	-0.000158	0.000046
9,A	2	0.000630	0.000584	0.000046
7,2	6	0.000329	0.000284	0.000045
3,3	5	0.000111	0.000067	0.000045
9,A	5	0.000658	0.000614	0.000043
5,3	4	0.000082	0.000039	0.000043
9,8	8	-0.000300	-0.000343	0.000043
5,3	5	0.000112	0.000069	0.000043
5,4	4	0.000214	0.000172	0.000042
9,2	3	0.000519	0.000477	0.000042
5,A	6	0.000225	0.000184	0.000041
5,2	5	0.000041	0.000000	0.000041
8,A	5	0.000443	0.000403	0.000041
10,8	7	0.001501	0.001460	0.000041
10,7	4	-0.000259	-0.000299	0.000040
6,4	8	0.000306	0.000266	0.000040
5,4	2	0.000108	0.000069	0.000040
8,2	7	0.000402	0.000363	0.000040
7,5	4	-0.000143	-0.000183	0.000040
7,7	4	0.000068	0.000030	0.000039
6,2	4	0.000077	0.000039	0.000039
6,A	4	0.000150	0.000112	0.000038
3,2	5	0.000022	-0.000016	0.000038
7,7	2	-0.000017	-0.000055	0.000038
6,3	2	0.000106	0.000068	0.000038
9,A	7	0.000746	0.000709	0.000037
5,2	6	0.000070	0.000032	0.000037
3,3	4	0.000059	0.000022	0.000037
7,6	4	-0.000147	-0.000183	0.000036
4,2	5	0.000010	-0.000026	0.000036
6,4	10	0.000125	0.000089	0.000036
8,A	3	0.000401	0.000366	0.000035
7,7	3	0.000021	-0.000014	0.000035
10,5	4	-0.000711	-0.000746	0.000035
5,4	7	0.000195	0.000161	0.000034
2,2	5	0.000110	0.000076	0.000034
10,7	7	-0.000351	-0.000385	0.000034
3,A	4	0.000109	0.000075	0.000033
8,A	2	0.000382	0.000349	0.000033
8,2	9	0.000168	0.000135	0.000033
9,A	8	0.000758	0.000725	0.000032
10,10	A	0.003070	0.003038	0.000032
4,3	6	0.000062	0.000032	0.000031
8,3	7	0.000456	0.000426	0.000031
10,9	6	0.001685	0.001655	0.000030
6,3	7	0.000191	0.000160	0.000030
6,5	8	0.000353	0.000323	0.000030
7,7	8	-0.000138	-0.000167	0.000029
4,A	6	0.000211	0.000182	0.000029
8,A	7	0.000593	0.000565	0.000029
9,A	6	0.000650	0.000621	0.000028
4,2	6	0.000036	0.000008	0.000028
8,8	9	-0.000145	-0.000173	0.000028
9,A	3	0.000619	0.000592	0.000027
2,A	10	-0.000322	-0.000349	0.000027
6,6	7	-0.000070	-0.000097	0.000027
2,2	2	-0.000006	-0.000033	0.000027
8,A	6	0.000450	0.000423	0.000027
10,6	8	-0.001640	-0.001667	0.000027
5,2	4	-0.000009	-0.000036	0.000026
9,A	4	0.000628	0.000602	0.000026
2,2	9	-0.000080	-0.000106	0.000026
10,6	4	-0.000722	-0.000747	0.000025
5,2	2	-0.000071	-0.000096	0.000025
9,3	3	-0.000185	-0.000210	0.000025
7,6	5	-0.000122	-0.000147	0.000024
4,3	4	-0.000011	-0.000035	0.000024
6,4	9	0.000159	0.000135	0.000024
5,4	10	-0.000488	-0.000512	0.000024
6,A	A	-0.000112	-0.000135	0.000024
9,3	9	-0.000284	-0.000307	0.000024
6,3	3	0.000136	0.000113	0.000023
9,9	A	-0.000045	-0.000068	0.000023
8,4	5	-0.000124	-0.000147	0.000023
9,8	10	-0.001385	-0.001408	0.000023
4,2	4	-0.000037	-0.000060	0.000023
7,A	6	0.000350	0.000327	0.000023
3,2	6	0.000039	0.000017	0.000022
10,3	9	-0.001386	-0.001408	0.000022
8,2	10	0.000110	0.000088	0.000022
7,7	9	-0.000172	-0.000194	0.000022
10,6	6	-0.000418	-0.000439	0.000021
6,3	8	0.000113	0.000092	0.000021
6,5	9	0.000232	0.000210	0.000021
9,4	5	-0.000127	-0.000148	0.000021
8,6	5	-0.000127	-0.000147	0.000020
6,A	3	0.000072	0.000052	0.000020
7,A	3	0.000180	0.000160	0.000020
6,6	A	-0.000098	-0.000118	0.000020
7,6	3	-0.000202	-0.000222	0.000020
7,3	10	0.000107	0.000088	0.000020
3,A	10	-0.000447	-0.000467	0.000020
3,A	8	0.000034	0.000014	0.000019
3,2	2	-0.000092	-0.000112	0.000019
4,4	9	-0.000074	-0.000093	0.000019
3,3	2	-0.000039	-0.000058	0.000019
2,2	8	-0.000051	-0.000070	0.000019
3,3	9	-0.000112	-0.000131	0.000019
10,8	3	0.000525	0.000506	0.000019
2,A	9	-0.000013	-0.000032	0.000019
2,2	3	0.000015	-0.000004	0.000019
8,A	10	0.000233	0.000215	0.000018
4,4	6	0.000108	0.000090	0.000018
8,5	3	-0.000204	-0.000222	0.000018
2,2	A	-0.000072	-0.000090	0.000018
7,A	8	0.000117	0.000099	0.000018
7,7	A	-0.000128	-0.000146	0.000018
7,2	2	0.000083	0.000065	0.000018
2,A	4	0.000113	0.000095	0.000018
9,3	5	-0.000130	-0.000148	0.000018
9,6	5	-0.000130	-0.000148	0.000018
3,3	8	-0.000080	-0.000097	0.000017
7,A	2	0.000123	0.000106	0.000017
3,3	A	-0.000088	-0.000105	0.000017
4,2	2	-0.000107	-0.000123	0.000017
8,5	4	-0.000169	-0.000185	0.000016
2,2	4	0.000048	0.000031	0.000016
4,4	A	-0.000065	-0.000081	0.000016
3,3	3	-0.000004	-0.000020	0.000016
9,2	7	0.000440	0.000424	0.000016
9,9	9	-0.000018	-0.000034	0.000016
8,6	3	-0.000208	-0.000223	0.000015
3,3	6	0.000123	0.000108	0.000015
8,4	8	-0.000231	-0.000246	0.000015
3,2	4	-0.000037	-0.000052	0.000015
7,3	9	0.000148	0.000133	0.000015
6,3	10	-0.000498	-0.000513	0.000015
3,A	A	-0.000068	-0.000083	0.000015
6,2	3	0.000022	0.000007	0.000015
9,4	10	-0.001414	-0.001428	0.000015
8,A	4	0.000396	0.000381	0.000014
6,A	2	0.000014	-0.000001	0.000014
8,A	9	0.000278	0.000264	0.000014
8,6	4	-0.000172	-0.000186	0.000014
8,5	8	-0.000280	-0.000294	0.000014
6,6	9	-0.000140	-0.000153	0.000014
7,A	A	-0.000084	-0.000098	0.000014
5,4	8	0.000105	0.000091	0.000013
7,5	3	-0.000200	-0.000213	0.000013
5,3	7	0.000090	0.000077	0.000013
10,3	4	-0.000735	-0.000748	0.000013
3,2	3	-0.000070	-0.000083	0.000013
4,A	A	-0.000092	-0.000105	0.000013
10,9	4	0.001531	0.001518	0.000013
7,2	8	0.000104	0.000091	0.000013
9,5	4	-0.000173	-0.000186	0.000013
6,2	7	0.000089	0.000077	0.000012
8,7	3	-0.000211	-0.000223	0.000012
10,8	2	0.000414	0.000403	0.000011
8,7	10	-0.001684	-0.001695	0.000011
7,7	5	0.000081	0.000070	0.000011
7,7	7	-0.000009	-0.000020	0.000011
8,5	2	-0.000249	-0.000260	0.000011
7,6	2	-0.000249	-0.000260	0.000011
8,8	2	0.000043	0.000032	0.000010
5,2	3	-0.000058	-0.000069	0.000010
9,6	4	-0.000176	-0.000186	0.000010
4,3	3	-0.000058	-0.000068	0.000010
7,6	6	-0.000099	-0.000109	0.000010
3,A	9	-0.000058	-0.000067	0.000010
6,2	2	-0.000011	-0.000021	0.000010
10,9	3	0.001457	0.001448	0.000009
5,A	A	-0.000118	-0.000127	0.000009
8,7	4	-0.000177	-0.000186	0.000009
2,A	A	-0.000053	-0.000061	0.000009
7,4	9	0.000217	0.000208	0.000008
8,6	2	-0.000252	-0.000260	0.000008
8,6	6	-0.000101	-0.000109	0.000008
5,A	4	0.000065	0.000056	0.000008
9,3	4	-0.000178	-0.000187	0.000008
7,4	8	0.000328	0.000320	0.000008
3,3	7	-0.000015	-0.000023	0.000008
6,4	A	0.000029	0.000022	0.000008
6,A	7	0.000058	0.000050	0.000008
4,A	4	0.000064	0.000057	0.000007
5,5	10	0.000051	0.000044	0.000007
9,5	2	-0.000253	-0.000260	0.000007
5,3	3	0.000014	0.000007	0.000007
9,8	2	-0.000135	-0.000142	0.000007
7,A	9	-0.000084	-0.000091	0.000007
8,4	3	-0.000207	-0.000214	0.000007
6,6	8	-0.000116	-0.000123	0.000007
4,4	8	-0.000020	-0.000026	0.000007
7,5	7	-0.000187	-0.000194	0.000007
4,4	3	0.000010	0.000004	0.000006
6,5	5	0.000570	0.000563	0.000006
4,4	4	0.000024	0.000018	0.000006
7,6	7	-0.000239	-0.000245	0.000006
6,A	9	-0.000130	-0.000136	0.000006
9,6	6	-0.000104	-0.000109	0.000006
8,8	3	0.000070	0.000065	0.000006
9,8	A	-0.000319	-0.000324	0.000005
9,7	4	-0.000182	-0.000187	0.000005
4,4	2	-0.000005	-0.000010	0.000005
2,2	6	0.000123	0.000117	0.000005
10,2	4	-0.000750	-0.000755	0.000005
8,7	A	-0.000314	-0.000319	0.000005
8,7	2	-0.000256	-0.000260	0.000005
7,3	A	0.000026	0.000021	0.000004
10,6	9	-0.001851	-0.001855	0.000004
9,7	8	-0.000413	-0.000417	0.000004
8,2	A	0.000025	0.000021	0.000004
10,2	3	-0.000848	-0.000852	0.000004
7,5	6	-0.000106	-0.000109	0.000004
5,3	2	-0.000018	-0.000022	0.000004
9,6	2	-0.000257	-0.000261	0.000003
6,A	8	-0.000063	-0.000066	0.000003
10,8	A	-0.000566	-0.000569	0.000003
9,7	A	-0.000339	-0.000341	0.000003
9,4	A	-0.000267	-0.000270	0.000002
2,2	7	0.000006	0.000004	0.000002
4,4	7	0.000040	0.000038	0.000002
5,5	4	0.000213	0.000211	0.000002
6,4	4	0.000423	0.000421	0.000002
8,4	6	-0.000108	-0.000110	0.000002
7,4	10	0.000606	0.000605	0.000002
8,7	9	-0.000428	-0.000429	0.000001
8,5	6	-0.000109	-0.000110	0.000001
2,A	3	0.000069	0.000068	0.000001
5,5	A	0.000011	0.000010	0.000001
8,8	8	-0.000011	-0.000012	0.000001
9,7	2	-0.000260	-0.000261	0.000001
6,5	10	0.000605	0.000605	0.000001
6,2	8	-0.000054	-0.000054	0.000000
A,A	9	0.000105	0.000104	0.000000
8,3	9	0.000208	0.000207	0.000000
5,3	8	-0.000055	-0.000055	0.000000
5,4	A	-0.000079	-0.000079	0.000000
9,7	10	-0.001816	-0.001815	0.000000
4,A	7	0.000033	0.000034	-0.000001
5,4	9	-0.000049	-0.000048	-0.000001
6,3	9	-0.000049	-0.000048	-0.000002
9,8	5	-0.000044	-0.000042	-0.000002
5,2	7	-0.000065	-0.000063	-0.000002
10,7	5	-0.000171	-0.000169	-0.000002
9,4	6	-0.000112	-0.000110	-0.000002
5,5	6	0.000256	0.000258	-0.000002
6,3	A	-0.000081	-0.000079	-0.000002
9,5	6	-0.000113	-0.000110	-0.000002
5,5	2	0.000161	0.000163	-0.000002
4,A	3	0.000024	0.000027	-0.000003
3,2	8	-0.000174	-0.000172	-0.000003
5,5	9	0.000063	0.000066	-0.000003
5,5	3	0.000184	0.000186	-0.000003
9,2	8	0.000316	0.000319	-0.000003
7,2	9	-0.000051	-0.000048	-0.000003
8,7	5	-0.000153	-0.000150	-0.000003
3,A	2	0.000018	0.000020	-0.000003
4,2	3	-0.000099	-0.000096	-0.000003
10,10	5	0.004864	0.004868	-0.000004
9,3	6	-0.000115	-0.000111	-0.000004
4,3	7	-0.000068	-0.000063	-0.000004
9,5	3	-0.000230	-0.000225	-0.000005
3,2	7	-0.000115	-0.000110	-0.000005
9,7	5	-0.000156	-0.000151	-0.000005
4,A	9	-0.000109	-0.000103	-0.000006
9,4	3	-0.000232	-0.000225	-0.000007
2,A	7	0.000104	0.000111	-0.000007
9,6	3	-0.000233	-0.000225	-0.000007
5,A	5	0.000108	0.000115	-0.000007
9,8	4	-0.000084	-0.000077	-0.000007
A,A	A	0.000026	0.000033	-0.000007
6,2	6	0.000087	0.000094	-0.000008
9,5	A	-0.000304	-0.000297	-0.000008
5,A	3	0.000000	0.000008	-0.000008
9,8	3	-0.000119	-0.000110	-0.000008
9,4	2	-0.000270	-0.000262	-0.000008
4,A	8	-0.000034	-0.000026	-0.000009
6,2	9	-0.000201	-0.000193	-0.000009
3,2	A	-0.000210	-0.000201	-0.000009
3,2	9	-0.000253	-0.000244	-0.000009
A,A	2	0.000205	0.000214	-0.000009
5,A	2	-0.000029	-0.000019	-0.000009
4,A	2	-0.000010	0.000000	-0.000009
5,5	7	0.000169	0.000178	-0.000010
10,8	4	0.000594	0.000605	-0.000010
2,A	8	0.000038	0.000048	-0.000010
9,7	3	-0.000236	-0.000226	-0.000010
9,9	2	0.000075	0.000086	-0.000011
7,5	2	-0.000244	-0.000233	-0.000011
A,A	8	0.000147	0.000159	-0.000012
2,A	2	0.000029	0.000041	-0.000012
7,2	A	-0.000092	-0.000080	-0.000012
5,2	9	-0.000274	-0.000262	-0.000012
9,6	7	-0.000351	-0.000338	-0.000012
3,A	3	0.000033	0.000045	-0.000012
8,6	A	-0.000310	-0.000297	-0.000012
3,A	7	0.000058	0.000071	-0.000013
5,5	8	0.000117	0.000129	-0.000013
6,2	A	-0.000181	-0.000168	-0.000013
5,3	A	-0.000181	-0.000168	-0.000013
A,A	3	0.000222	0.000235	-0.000013
6,3	6	0.000264	0.000277	-0.000013
A,A	7	0.000196	0.000210	-0.000014
A,A	5	0.000265	0.000279	-0.000014
8,8	5	0.000119	0.000133	-0.000015
10,4	6	-0.000458	-0.000443	-0.000015
5,2	A	-0.000237	-0.000222	-0.000015
A,A	4	0.000242	0.000257	-0.000015
8,8	4	0.000082	0.000097	-0.000015
9,6	A	-0.000337	-0.000322	-0.000015
7,3	3	0.000356	0.000371	-0.000015
5,3	9	-0.000209	-0.000194	-0.000016
5,2	8	-0.000210	-0.000194	-0.000016
5,A	7	-0.000023	-0.000007	-0.000016
4,3	2	-0.000118	-0.000101	-0.000016
9,5	8	-0.000357	-0.000341	-0.000017
10,5	6	-0.000459	-0.000443	-0.000017
7,A	10	-0.000503	-0.000486	-0.000017
4,2	7	-0.000158	-0.000141	-0.000017
9,4	7	-0.000265	-0.000247	-0.000017
8,2	2	0.000306	0.000324	-0.000018
8,4	A	-0.000262	-0.000245	-0.000018
8,4	2	-0.000251	-0.000233	-0.000018
7,5	A	-0.000263	-0.000245	-0.000018
5,A	8	-0.000081	-0.000063	-0.000018
6,A	10	-0.000683	-0.000665	-0.000018
4,2	A	-0.000236	-0.000217	-0.000018
7,4	4	0.000496	0.000515	-0.000019
9,8	7	-0.000119	-0.000100	-0.000019
9,3	A	-0.000263	-0.000245	-0.000019
8,A	A	0.000096	0.000116	-0.000020
4,3	A	-0.000244	-0.000223	-0.000021
7,2	7	0.000133	0.000154	-0.000021
9,8	6	-0.000030	-0.000008	-0.000022
10,4	8	-0.001380	-0.001358	-0.000022
5,A	9	-0.000161	-0.000138	-0.000022
9,3	2	-0.000257	-0.000234	-0.000023
9,6	8	-0.000406	-0.000383	-0.000024
4,2	9	-0.000293	-0.000270	-0.000024
10,4	9	-0.001597	-0.001573	-0.000024
10,3	6	-0.000468	-0.000444	-0.000024
7,7	6	0.000087	0.000111	-0.000024
4,2	8	-0.000226	-0.000201	-0.000025
4,3	8	-0.000220	-0.000195	-0.000025
8,3	10	0.000575	0.000601	-0.000026
10,7	3	-0.000468	-0.000442	-0.000026
8,7	6	-0.000139	-0.000113	-0.000026
8,5	A	-0.000299	-0.000273	-0.000026
8,6	9	-0.000422	-0.000396	-0.000026
10,8	6	0.000784	0.000811	-0.000026
A,A	6	0.000274	0.000300	-0.000027
9,9	5	0.000147	0.000174	-0.000028
9,9	8	0.000075	0.000103	-0.000028
7,2	10	-0.000546	-0.000518	-0.000028
9,7	6	-0.000142	-0.000114	-0.000028
9,4	8	-0.000327	-0.000298	-0.000029
A,A	10	0.000270	0.000299	-0.000029
4,3	9	-0.000293	-0.000264	-0.000029
7,6	A	-0.000303	-0.000274	-0.000029
4,A	10	-0.000618	-0.000588	-0.000030
9,9	4	0.000111	0.000142	-0.000031
10,5	3	-0.000934	-0.000903	-0.000032
8,8	7	0.000110	0.000142	-0.000032
9,9	7	0.000145	0.000177	-0.000032
3,2	10	-0.001091	-0.001059	-0.000033
7,3	7	0.000322	0.000355	-0.000033
6,A	6	0.000191	0.000223	-0.000033
9,9	3	0.000078	0.000112	-0.000033
10,2	6	-0.000479	-0.000445	-0.000033
9,5	10	-0.001607	-0.001572	-0.000034
8,4	7	-0.000237	-0.000199	-0.000038
9,5	7	-0.000336	-0.000297	-0.000039
10,2	7	-0.000819	-0.000780	-0.000039
8,6	7	-0.000336	-0.000297	-0.000039
6,4	6	0.000472	0.000511	-0.000039
8,8	6	0.000127	0.000167	-0.000040
9,3	7	-0.000239	-0.000199	-0.000040
8,3	3	0.000427	0.000467	-0.000040
10,6	3	-0.000944	-0.000904	-0.000040
8,5	9	-0.000398	-0.000358	-0.000040
8,2	8	0.000213	0.000255	-0.000042
6,2	10	-0.000887	-0.000844	-0.000042
10,3	A	-0.001128	-0.001085	-0.000043
6,6	6	0.000036	0.000079	-0.000043
10,4	3	-0.000947	-0.000904	-0.000043
9,2	2	0.000380	0.000423	-0.000043
10,7	A	-0.001350	-0.001306	-0.000044
10,2	A	-0.001018	-0.000974	-0.000044
10,6	A	-0.001417	-0.001373	-0.000044
5,3	10	-0.000889	-0.000844	-0.000045
7,6	9	-0.000404	-0.000358	-0.000046
9,2	9	0.000155	0.000201	-0.000046
10,7	2	-0.000622	-0.000576	-0.000046
9,3	8	-0.000305	-0.000254	-0.000051
6,5	6	0.000549	0.000601	-0.000052
10,5	7	-0.001407	-0.001354	-0.000054
9,9	6	0.000148	0.000202	-0.000054
7,5	8	-0.000310	-0.000254	-0.000056
10,3	7	-0.001044	-0.000987	-0.000057
10,7	9	-0.001607	-0.001549	-0.000057
10,5	2	-0.001110	-0.001052	-0.000058
7,A	7	0.000298	0.000356	-0.000058
7,5	9	-0.000376	-0.000317	-0.000059
5,2	10	-0.001142	-0.001081	-0.000061
7,4	7	0.000353	0.000414	-0.000061
8,4	9	-0.000380	-0.000318	-0.000062
10,2	8	-0.001060	-0.000997	-0.000063
8,5	7	-0.000316	-0.000253	-0.000063
10,4	A	-0.001256	-0.001191	-0.000065
9,A	A	0.000305	0.000371	-0.000066
10,6	2	-0.001121	-0.001053	-0.000068
8,3	8	0.000239	0.000309	-0.000070
8,6	10	-0.001648	-0.001577	-0.000071
7,5	10	-0.001364	-0.001291	-0.000073
7,6	8	-0.000381	-0.000304	-0.000076
8,4	10	-0.001368	-0.001291	-0.000077
4,2	10	-0.001224	-0.001146	-0.000077
10,3	8	-0.001267	-0.001189	-0.000078
10,8	9	-0.000757	-0.000677	-0.000080
10,2	9	-0.001330	-0.001249	-0.000081
10,10	8	0.005671	0.005755	-0.000085
10,5	8	-0.001614	-0.001529	-0.000085
10,10	7	0.005536	0.005621	-0.000085
10,7	6	-0.000118	-0.000033	-0.000085
9,3	10	-0.001378	-0.001293	-0.000085
9,2	10	0.000506	0.000593	-0.000087
8,A	8	0.000440	0.000529	-0.000089
10,10	6	0.004838	0.004927	-0.000089
10,8	8	0.000277	0.000371	-0.000095
5,5	5	0.000126	0.000222	-0.000096
9,6	10	-0.001808	-0.001709	-0.000099
5,A	10	-0.000810	-0.000711	-0.000099
4,3	10	-0.001187	-0.001086	-0.000101
10,5	9	-0.001835	-0.001731	-0.000104
10,10	2	0.004503	0.004610	-0.000108
10,10	3	0.004577	0.004689	-0.000111
10,5	A	-0.001404	-0.001292	-0.000111
10,7	8	-0.001522	-0.001407	-0.000115
10,4	2	-0.001174	-0.001058	-0.000116
10,3	2	-0.001177	-0.001059	-0.000119
9,A	9	0.000554	0.000674	-0.000120
10,10	9	0.005386	0.005507	-0.000121
10,10	4	0.004635	0.004767	-0.000132
10,4	7	-0.001321	-0.001187	-0.000134
8,5	10	-0.001583	-0.001447	-0.000135
7,6	10	-0.001597	-0.001449	-0.000148
10,9	9	0.000765	0.001012	-0.000247
10,A	A	0.003014	0.003691	-0.000677
10,10	10	0.013570	0.014752	-0.001181

I'm thinking of making a table that considers just the total of the player's initial hand and based on simulation results.

Quote: Ace2

https://wizardofodds.com/games/blackjack/player-expected-return/

Thanks for pointing that out - I had never found it. I'm not crazy about it being infinite deck, and the colors and graphical format are very dated.

In my previous post I was speaking from the heart because I believe the appearance and content of WOO could be improved to make it more attractive and interesting (its not my site but I do care about it and I think it has great bones). That table should be brought up to modern graphical standards and made more "central" or at least easier to find.

Quote: Wizard

This table lists the expected value of all 650 for both one and eight decks under the same rules as my simulation. The values are taken from these expected value tables:

Blackjack expected Returns for one deck and dealer hits on soft 17 and
https://wizardofodds.com/games/blackjack/appendix/9/8dh17r4/

Unlike my simulation study, these tables assume perfect combinatorial strategy at every decision point. This helps in single deck much more than eight decks.

The expected value columns are the product of the expected value of the given situation and the probability of it happening. It is listed in order of the expected value difference, in order of what is best for the player in a single-deck game.

Note: TABLE DELETED for the sake of brevity


I'm thinking of making a table that considers just the total of the player's initial hand and based on simulation results.

MAGNIFICENT! Personally, I love it! Its something new on a topic (Blackjack) that has been very worked over, and its interesting. I may not be your average WOO user, but I think its great!

Edit: If you make a table based on players hand total, I think you may be throwing out the baby with the bathwater. A T,3 vs T will be very different than a 9,4 vs T and a 7-6 is always different than T-3. Indeed, T-3 vs T and 7-6 vs T are at polar opposite ends of the table.