a mistake in basic strategy of 77 VS 10, single deck?

a mistake in basic strategy of 77 VS 10, single deck?

the reason for asking this is that i am learning Combinatorial Analysis and i finished now bac and blackjack and there is some difference with published results.

the rule, I do it by dealer's peek. other rules are not important by this hand.

the published strategy for 77 VS 10, single deck is to stand. but i found:
1) if only to consider cards 77, no 10, it is better to stand. It is the same with published strategy.
2) if to consider all cards 77 and 10, it is better to hit, not to stand. It is different with published strategy.

I will try to show the steps one by one.
by no cards dealt, here is the dealer's possibility:
dealer's total by up card 10 percentage
17 0.124155911
18 0.122485681
19 0.124420908
20 0.35686881
21 0.039569635
bust 0.232499056
total 1

it is the same with others, i just add some digits.

now the hand 77V10 is dealt out.
we only consider 77 are dealt out (no 10), here is dealer's possibility:
dealer's final by up card 10 percentage
17 0.087174884
18 0.12980531
19 0.128219226
20 0.370620409
21 0.039049847
bust 0.245130325
total 1

for player's hand, if we hit, we set it by 3 cards followed, the combinations to be 50*49*48=117600
stand at 15
score combinations percentage expected value
15 9408 0.08 -0.040779148
16 9408 0.08 -0.040779148
17 9408 0.08 -0.033805157
18 9408 0.08 -0.016446742
19 9408 0.08 0.004195221
20 9408 0.08 0.044102392
21 4704 0.04 0.038438006
bust 56448 0.48 -0.48
total 117600 1 -0.525074576
stand at 16
score combinations percentage expected value
16 9984 0.084897959 -0.043275831
17 10176 0.086530612 -0.036564762
18 10176 0.086530612 -0.017789333
19 10176 0.086530612 0.004537688
20 10176 0.086530612 0.047702587
21 5472 0.046530612 0.044713599
bust 61440 0.52244898 -0.52244898
total 117600 1 -0.523125031
77 stand at 17
score combinations percentage expected value
16 0 0 0
17 10968 0.093265306 -0.039410604
18 10800 0.091836735 -0.018880188
19 10992 0.093469388 0.004901559
20 10992 0.093469388 0.051527794
21 6288 0.053469388 0.051381416
bust 67560 0.574489796 -0.574489796
total 117600 1 -0.524969819
this is the result considering 3 up tables.
player's 77 vs dealer 10(10 not considered) expected value
stand -0.509739351
to hit and later to stand at 15 -0.525074576
to hit and later to stand at 16 -0.523125031
to hit and later to stand at 17 -0.524969819
we can see, it is better to stand.

Now we only consider 7710 are dealt out, here is dealer's possibility:
dealer's final by up card 10 percentage
17 0.090223316
18 0.133784945
19 0.132135123
20 0.357270422
21 0.040927529
Bust 0.245658664
total 1
for player's hand, if we hit, we set it by 3 cards followed, the combinations to be 49*48*47= 110,544
stand at 15
score combinations percentage ev
15 9024 0.081632653 -0.041525116
16 9024 0.081632653 -0.041525116
17 9024 0.081632653 -0.034159947
18 9024 0.081632653 -0.015873559
19 9024 0.081632653 0.005834202
20 9024 0.081632653 0.045785675
21 4512 0.040816327 0.039145815
bust 51888 0.469387755 -0.469387755
total 110544 1 -0.511705801

stand at 16
score combinations percentage expected value
16 9588 0.086734694 -0.044120436
17 9776 0.088435374 -0.03700661
18 9776 0.088435374 -0.017196355
19 9776 0.088435374 0.006320386
20 9776 0.088435374 0.049601148
21 5264 0.047619048 0.045670118
bust 56588 0.511904762 -0.511904762
total 110544 1 -0.508636511

stand at 17
score combinations percentage expected value
16 0 0 0
17 10552 0.095455203 -0.039944123
18 10388 0.093971631 -0.018272886
19 10576 0.095672311 0.006837602
20 10576 0.095672311 0.053660162
21 6064 0.054855985 0.052610865
bust 62388 0.564372558 -0.564372558
total 110544 1 -0.509480938
this is the result considering 3 up tables.

player's 77 vs dealer 10(10 considered) expected value
stand -0.508682671
to hit and later to stand at 15 -0.511705801
to hit and later to stand at 16 -0.508636511
to hit and later to stand at 17 -0.509480938
we can see, it is better to hit 77 VS 10.


Now it can be either my mistake or other’s mistake.
The reason for my mistake is that I am too new.
The reason for other’s mistake can be they are comparing it with standing 17, not 15,16,17.

it seems no one is interested in this topic.

here i found some more info:
by this famous link of CA,http://www.bjstrat.net/cgi-bin/cdca.cgi
i found the dealer's possiblity after considering 77 and after considering 7710 is the exactly the same with mine:
if to choose player's card 77, the dealer's possiblity of A2345678910 is based on A2345678910 are not removed. if we want to consider dealer's removed card, we need to decrease one appointed card from Shoe Composition. then if we decrease 10 by 1, we can find that 'hit' is the best option.

isn't it a proof that basic stragy of this hand is wrong? my cacluation is right? i don't know. because the strategy is the same with me, but the numbers are different.

is there any one here who can help me? if so, i will show him how to play other hands better. it is big money in real casino, for me, it is 0 money, because i have no place to play. also, by chance if i am right, i will add his name on when i publish this article in China.

You might have an error somewhere. https://wizardofodds.com/games/blackjack/hand-calculator/ has standing on 77v 10 as -.509739 and hitting as -.514818 . These numbers also match the one in your link.

thanks for your reply.
when facing them, i firstly, like u, thought i 99% had an error.

so i tried to find mistakes in myself.
the dealer's possblity is some difficulty. but they are the same number.
the player's 77, it is easy to put all possbility in excel. so i did.

the left caculation is easy. to show it, this is the google shared file of excel.
https://docs.google.com/spreadsheets/d/1XTLH10z1Ziw7YUYeAWRmvRYvdFB3r5hNjIVhyKHkF_E/edit?ouid=117228603529916828384&usp=docs_home&ths=true
it is my first to use this function,not sure if it is workable by others.

this file can lead to the CA of all games, guessed, and i am sure that this way works for baccarat, which originally from Eliot. all main bets and side bets can be done by this way, then everyone can do counting systyem by himself.so taking some time to check this will self-help a lot.

I’ll have a look.

I could quote several legitimate blackjack authors who say the right play with 7,7 vs. 10 is to stand. The reason is half the sevens are already removed from the the deck.

Quote: tomchina123

the published strategy for 77 VS 10, single deck is to stand. but i found:
1) if only to consider cards 77, no 10, it is better to stand. It is the same with published strategy.

why would one consider only player 77
and nothing the dealer has?

Is this a BJ variant or just common method of calculating?

Quote: Wizard

I could quote several legitimate blackjack authors who say the right play with 7,7 vs. 10 is to stand. The reason is half the sevens are already removed from the the deck.

The other reason is the dealer is less likely to get 17, thus reducing the advantage of getting low scores such as 18 and winning against 17.
Note I haven't double checked these figures but they confirm that you don't hit 77 vs 10 in single deck but would do if the deck was formed of five suits (i.e. 1.25 decks).
Interestingly you stand on 772 or 7711 on finite number of decks.

Hand	Win	Tie	Lose	EV Stand	EV Hit
77	2 266 583 232	0	6 979 858 368	-0.509 739	-0.514 886
771	2 040 642 744	0	6 242 627 856	-0.507 286	-0.476 721
772	1 960 584 336	0	6 138 613 584	-0.515 857	-0.526 637
773	1 963 722 312	691 707 288	5 443 768 320	-0.429 678
774	2 656 826 584	1 054 080 960	4 388 290 376	-0.213 782
775	3 722 009 800	1 035 020 016	3 342 168 104	0.046 899
776	4 759 542 296	3 043 870 280	295 785 344	0.551 136
777	7 785 858 256	313 339 664	0	0.961 312
7711	1 832 643 120	0	5 569 428 480	-0.504 830	-0.522 302
7712	1 761 116 364	597 860 676	4 878 604 080	-0.430 736
7713	2 366 084 784	933 285 528	3 938 210 808	-0.217 217
7714	3 310 891 080	922 749 408	3 003 940 632	0.042 411
7715	4 247 258 736	2 725 817 472	264 504 912	0.550 288
7716	6 971 376 720	266 204 400	0	0.963 219
77111	1 641 548 160	532 070 448	4 423 879 992	-0.421 725
77112	2 096 446 932	830 232 780	3 524 207 808	-0.221 328
77113	2 935 946 532	821 805 804	2 693 135 184	0.037 640
77114	3 770 022 816	2 422 263 600	258 601 104	0.544 332
77115	6 213 283 992	237 603 528	0	0.963 167
7721	1 761 116 364	597 860 676	4 878 604 080	-0.430 736
7722	2 271 999 792	896 370 096	3 904 720 752	-0.230 836
7723	3 178 215 900	910 475 460	2 984 399 280	0.027 402
7724	4 102 707 420	2 699 692 932	270 690 288	0.541 774
7725	6 825 815 676	247 274 964	0	0.965 040

Quote: Wizard

I could quote several legitimate blackjack authors who say the right play with 7,7 vs. 10 is to stand. The reason is half the sevens are already removed from the the deck.

What about surrendering?

1 surrendering is only -50%, surely better if the option is possible.

2 by 16 Vs 10, staying is a lot of times to be better. the reason can be that card 5, makes 16+5, 10+5, 15+5, the same with other small card, so it is bad or good here and there.

3 the 77Vs10. yes, we have only 2 sevens left, when i firstly saw strategy, it was easy to think player is hard to get another 7 from left cards, so better to stay. but by some hands, we cannot only think by this way, like the 16. so can be 77vs10 be like this?

4 that i specially care for this hand is my data after CA is a lot of small difference with the published data. it can be the reason of total dependent or composition or 3 cards or 4 cards.., then it is hard to question. so i pick this hand.

5 all the experts are powerful, so when in doubt, i did 2 datas, 77 removed and 7710 removed. also, when 7710 removed, i found the mistake can be from that they may consider to stay at 17 when coding rather than to stay at 16.

maybe i make some careless mistake, i am using excel to put all combinations. not coding. it is a different way, very slow and brutal force.

Quote: tomchina123

consider to stay at 17 when coding rather than to stay at 16

As you probably know the technically correct method is to work out what to do with all ways of getting to 17's, then 16's etc. In this case the only 16's to worry about are 772 and 7711 but they need to know about your chances with 772x and 7711x. Similarly the only 15 is 771 and this needs 771x (including 7711).

Hand EV Stand EV Hit
772 yours -0.515 857 -0.526 637
mine: dealer's 10 not considered.
-0.515857161 -0.534281989
mine: dealer's 10 considered.
-0.514246374 -0.521199017


7711 yours -0.504 830 -0.522 302
mine: dealer's 10 not considered.
-0.50482967 -0.529991852
mine: dealer's 10 considered.
-0.503658146 -0.516650066

77111 yours -0.421 725
mine: dealer's 10 not considered.
-0.421725263
mine: dealer's 10 considered.
-0.41775807

7721 yours -0.430 736
mine: dealer's 10 not considered.
-0.430736135
mine: dealer's 10 considered.
-0.426209636

the reslut is the same, standing 16 is better.but for the numbers, when standing, it seems yours is the same with me when not considering 10.

omg how many statistics)

it is by full combinations. so full theorictical statistics. or 14,658,134,400 hands.

i am also not sure of the calcluation, as you can see, it is a question mark.

let's be so.

i am doing sth on poker. it seems there is a need by different aspects. i finished basic things. i will check how excel performs vs coding.

for 77 vs 10 in single deck, removing the 10 and both sevens from the deck, I calculate the dealer probabilities (via Excel spreadsheet) to be:

17 0.0872
18 0.1298
19 0.1282
20 0.3706
21 0.0390
Bust 0.2451

Stand= -0.5097
Hit = -0.5148

When calculating the Hit EV, I do include that the card or cards that are drawn to 7-7 are therefore unavailable to the dealer, thus changing the above dealer probabilities. For instance, when HIT and drawing a 5 to make a 19, I factor in that that 5 is now unavailable to the dealer (to make a 15, for example.)

The EV calculation for HIT also takes into account that we know that the dealer's facedown card is not an Ace, thereby elevating the chances of the player drawing an Ace slightly and reducing the probabilities of drawing all other ranks.

Quote: gordonm888

...The EV calculation for HIT also takes into account that we know that the dealer's facedown card is not an Ace, thereby elevating the chances of the player drawing an Ace slightly and reducing the probabilities of drawing all other ranks.

I think I initially used the same method, working out the stand chances on a variety of card holdings for the player and then working out the chances of each dealer hand given the cards that were missing.

The part I haven't yet worked out properly is (say) starting with 77 then your chances when hitting are Pr(A)*f(771)+Pr(2)*f(772)...Pr(7)*f(777)+Pr(8)*Bust..Pr(10)*Bust. I've only looked at what's left in a fresh deck with 77X missing, but appreciate that one has to take into account you know the dealer hasn't an Ace. It's probably only a small factor so guess that's why my hit numbers are just slightly different from those on the wizard tool.

One method maybe that you just consider all the different starting two-card hands for the dealer, so against an K2 your 77 options etc.

thanks for your attention. gordonm888.

for your data,

17 0.0872
18 0.1298
19 0.1282
20 0.3706
21 0.0390
Bust 0.2451

are u sure it is 10-removed? because it is the same with my 10-not-reomoved.
also you may check this link, http://www.bjstrat.net/cgi-bin/cdca.cgi ( this website is recommended by WIZARD here doing EOR,,https://wizardofodds.com/games/blackjack/appendix/7/, which i may read most of the time in the world.)
you may see, you numbers are the same with 10 not removed.


for stand, Stand= -0.5097, it is the same with my 10-not removed.

for Hit = -0.5148, i cannot see this number in my caculation. do you set it to stand at 15 or 16 or 17 after hitting?

quote:''When calculating the Hit EV, I do include that the card or cards that are drawn to 7-7 are therefore unavailable to the dealer, thus changing the above dealer probabilities. For instance, when HIT and drawing a 5 to make a 19, I factor in that that 5 is now unavailable to the dealer (to make a 15, for example.)''.

by first thinking, i though i had mistake because of this point, the "5". but i found i didn't.

77 is surely 'yes'. but if to removed possible future instanced card 5, we don't really need to remove it. because after getting 5, the stragey to be faced is 14+5 VS 10, only in this hand, 5 is removed. (so) we consider all possible cards but have no need to remove it. don't u think so?

quote: "The EV calculation for HIT also takes into account that we know that the dealer's facedown card is not an Ace, thereby elevating the chances of the player drawing an Ace slightly and reducing the probabilities of drawing all other ranks.''

well noted, i am doing it like this.

While 7,7 against a 10 is a stand due to half the key cards to reach a 21 for a player being removed there is a small strategy error on wiz of odds basic strategy and that's a player 9,9 against a dealer Ace to split with hit on soft 17 in single deck. It's slightly better to stand according to the hand calculator on site (which is very helpful thank you!)

I Wanted to correct my own mistake some time ago. now i am doing it.

the right thing to do: by 77V10 in single deck is so stand. i tried to find why i made this mistake, but to timecosting.

again, wizard is always right.