Hard 12 vs 2 or 3

I know that 31% (rounded up) of the deck are 10s (10, J, Q & K). If dealer has a 2 showing & a 10 in the hole, we know all things are equal. That leaves 69% of the time, the dealer is drawing to a 2+A - 9. I always hit hard 12 v 2 or 3 but so many other player do not. Please show the math to explain advantage / disadvantage in this situation.

If you don't hit, you only win if the dealer busts. The chances of that happening are about 35% on the two and a bit more with the three.
By hitting,you have a 69% chance of improving your hand.

More common sense than math, but I hope it helps.

deleted double post

Infinite deck numbers: also see https://wizardofodds.com/games/blackjack/expected-return-infinite-deck/ . Essentially your EV for standing is "EV 16-12" and for infinite decks you can calculate your chances when hitting as the your chances (EV) for each card you could draw divided by 13 (technically with finite decks it depends on the cards that have gone). Remember you're bound to lose if you draw a picture, so it's (4*-1 + EV21 + ... + EV13)/13.

Note: the decision with a 4 is quite close, so sometimes you should consider hitting 12, however for 2 and 3 the difference is larger.

EV	EV	EV	EV	EV	EV	Dealer	EV
21	20	19	18	17	16-12	Upcard	Hits 12
.888 575 661	.434 957 754	-.018 660 154	-.241 508 831	-.464 357 508	-.575 781 847	10	-.438 778 497
.939 176 156	.758 356 871	.287 596 757	-.183 163 357	-.423 154 240	-.543 149 681	9	-.368 752 811
.930 605 053	.791 815 160	.593 853 668	.105 951 349	-.381 950 971	-.510 517 515	8	-.307 830 446
.925 926 296	.773 227 227	.615 976 496	.399 554 167	-.106 808 989	-.475 375 183	7	-.253 355 811
.902 836 744	.703 958 570	.495 977 074	.283 443 916	.011 739 161	-.153 699 016	6	-.170 526 200
.891 753 827	.670 359 691	.439 512 104	.199 561 195	-.044 941 376	-.167 192 661	5	-.193 271 169
.888 767 293	.661 049 962	.423 178 925	.175 854 437	-.080 573 373	-.211 063 109	4	-.213 536 553
.885 300 357	.650 272 094	.404 362 937	.148 300 073	-.117 216 241	-.252 250 229	3	-.233 690 900
.882 006 515	.639 986 575	.386 304 686	.121 741 902	-.152 974 588	-.292 783 727	2	-.253 389 986
.638 441 869	.146 094 839	-.115 482 960	-.377 060 760	-.638 638 560	-.769 427 459	Ace	-.571 104 262

These are UK "no peek" EV, so are different to wizard's numbers for Aces and Tens.

Quote: billryan

If you don't hit, you only win if the dealer busts. The chances of that happening are about 35% on the two and a bit more with the three.
By hitting,you have a 69% chance of improving your hand.

More common sense than math, but I hope it helps.
link to original post

You have a 38% chance of improving your hand by hitting a 12 vs 2. Only 5-9 improve it

So why not hit a 16 vs 2? Also a 38% chance of improvement

Quote: Ace2

Quote: billryan

If you don't hit, you only win if the dealer busts. The chances of that happening are about 35% on the two and a bit more with the three.
By hitting,you have a 69% chance of improving your hand.

More common sense than math, but I hope it helps.
link to original post

You have a 38% chance of improving your hand by hitting a 12 vs 2. Only 5-9 improve it

So why not hit a 16 vs 2? Also a 38% chance of improvement
link to original post

You are right that the chances of improving to a 17-21 total are almost identical for hitting a 12 total or a 16 total. (For infinite deck, they are identical. The cards in your hand change the probability by a tiny bit for finite decks.) However, you have extra outs hitting to a 12, because Ace, 2, 3, 4 result in a hand where you stand and can win with a dealer bust. Hitting 16, you either make 17-21 or lose no matter what the dealer does.

I play a lot online. This is my pet peeve hand. I swear I bust more than 50% of the time. I usually hit it against 4’s and 5’s, that way I can complain more. Of course I have to hit against a 6 too cause they always flip an ace. It’s extra fun when I split 2’s three times and bust on every one. Every now and then when I’m betting low I’ll actually double it and it busts everytime. But if you hit a 13 you always get a 5.