2346x v. two three card draws. 2-7

[dibbs]

I know its a bitch with all the perms and all that, but is this hand a fav. vs two three card draws? would you gain equity by dropping the six?

[timprov]

I'm sure it has to be, especially if one or both of the 3-card draws aren't 27. You might be in trouble if you're against 27xxx and 2777x.

Dropping the six isn't going to do you any good.

[dibbs]

thanks Tim, I was thinking it was a decent advantage, obviously as long as im not facing nightmare scenarios like 277xx etc, but wanted to make sure I aint crazy.

[Chris Daddy Cool]

ditching the 6 is probably a bad idea since your opponents drawing that many are likely to be mass hording the key cards, mainly the 7.

[MarkGritter]

[ QUOTE ]
I know its a bitch with all the perms and all that, but is this hand a fav. vs two three card

draws? would you gain equity by dropping the six?

[/ QUOTE ]

Let's say you have 2346A and villians have 27JQK and 27TJQ, so two sevens are dead. And that you discard a 9 on the last draw, which is nearly always incorrect strategy.

Villians cannot make a smoother hand than you can, so your T would be better than their T, for example. I'll try to approximate your hot + cold equity by modelling the draw as three events. First you draw, then villian 1 draws with whatever cards are left over after you make your hand, then villian 2 is treated like a repeat of villian 1. (So villian's probability is dependent on you, but villian 2 is not dependent on villian 1--- again, this is flawed, but it simplifies analysis immensely and is probably not too horribly far from the truth.)

Surprisingly, the equity for the two villians using this simple model is 61%! (One villian has 47% equity with the same dead cards.)

<font class="small">Code:</font><hr /><pre>
2346A 72QJT Villain's
Hero's hand Percentage dead cards 72KQJ Equity x2 trials Weighted equity
7 5 4 3 2 0.0157872 7 5 0.0171004 0.033908376 0.000535318
7 6 4 3 2 0.12471 7 0.05500025 0.106975472 0.013340911
8 5 4 3 2 0.0315744 8 5 0.151586 0.280193685 0.008846947
8 6 4 3 2 0.249421 8 0.165965 0.304385619 0.075920165
9 5 4 3 2 0.018876 9 5 0.417701 0.660927875 0.012475675
9 6 4 3 2 0.0557701 9 0.417701 0.660927875 0.036860014
T-high 0.0559846 T 0.518035 0.767709739 0.042979923
J-high 0.0559846 J 0.583121 0.826211899 0.046255143
Q-high 0.037323 Q 0.628465 0.861961744 0.032170998
K-high 0.037323 K 0.698452 0.909068804 0.033929175
A-high 0.0559846 A 0.747094 0.936038555 0.052403744
2's 0.0186615 2 0.822057 0.968336289 0.018070608
3's 0.0559846 3 0.815681 0.966026506 0.054082608
4's 0.0559846 4 0.834246 0.972525611 0.054446457
5's 0.0144144 5 5 0.859236 0.980185496 0.014128786
6's 0.0418275 6 0.88956 0.987803006 0.04131733
straight 0.0743887 5 0.999998 1 0.0743887
0.612152502
</pre><hr />

Why is this?

You are only 13% to make a 7, and 28% to make an 8. Against the non-wheel hands a single villian still has 5-15% equity, and the presence of two villians (again, in a model where they are not competing for cards) almost doubles this. That's a big chunk of your equity.

A 9, although it would be a favorite against two players drawing one, is not a favorite as the 'hand to beat' in a hot and cold model, and of course things get worse from there.

Note that your chance of ending up with something truly horrific is quite high (&gt;15%), again because of the weakness of the model--- the simulation doesn't have you stand pat with a 9, T, or J where appropriate. So your true hot-and-cold equity is likely to be quite a bit higher, because you may be able to stand pat more roughly. Villians are also competing for the same set of cards, so modelling their draws as independent events is incorrect and this will reduce villians' equity as well. But it is worth noting that even this poor draw has a 15% chance of beating your 8.

If I ever get my full hot+cold simulation working it will be interesting to compare the numbers.

[dibbs]

Very interesting, thanks a lot Mark. I've been playing hands like this pretty hard and whiffing on the end so was starting to wonder. Visualizing an 8 out draw that comes in around 40% of the time (less obviously) and two 3 card draws that pull in around 20% (right?) each the 61% number doesn't come as a total shock, I was kind of guessing from the gut hero would be around 55% or something.

As an educated guess (unless punching the numbers in the sim is real quick, which I dont think it is) would you say 2346A v. a two card draw and a three card draw would benefit from dropping the six and drawing smoother than villans, or shoot straight for the one card draw?

[randomstumbl]

[ QUOTE ]
Very interesting, thanks a lot Mark. I've been playing hands like this pretty hard and whiffing on the end so was starting to wonder. Visualizing an 8 out draw that comes in around 40% of the time (less obviously) and two 3 card draws that pull in around 20% (right?) each the 61% number doesn't come as a total shock, I was kind of guessing from the gut hero would be around 55% or something.

As an educated guess (unless punching the numbers in the sim is real quick, which I dont think it is) would you say 2346A v. a two card draw and a three card draw would benefit from dropping the six and drawing smoother than villans, or shoot straight for the one card draw?

[/ QUOTE ]

You will never benefit from dropping the 6 in an actual game. You might be able to create a scenario where the dead cards favor it against several opponents, but even then it would be tough.

Dropping the 6 is like starting with aces full of 2's in high draw and throwing away the 2's.

[MarkGritter]

[ QUOTE ]

You will never benefit from dropping the 6 in an actual game. You might be able to create a scenario where the dead cards favor it against several opponents, but even then it would be tough.

[/ QUOTE ]

I agree. If you hit a 7 or an 8 your hand is already strong and it will be relatively infrequently that #5 will win where #6 doesn't. Having a 1-card draw also lets you drive the betting which can create some fold equity. (You have a small chance of picking up a 5 along the way as well.)

If you keep your 6 the chance of making an 86 or better is about 44% with no 7s dead, 40% with one 7 dead, or 35% with two dead.

If you dump your 6 these probabilities go to 28-26-25. This is too much to pay for the slight increase in made hand strength.

[dibbs]

Sounds good, thanks guys.

[SumZero]

Droping the 6 is almost as big a mistake as standing pat.

From pokenum:

Drop a6:

pokenum -mc 500000 -l27 2h 7s / kd qd jd - 2s 7h / kh qh jh - 2d 3c 4c / 6d ah
5-card Draw 2-7 Lowball: 500000 sampled outcomes
cards win %win lose %lose tie %tie EV
7s 2h 157769 31.55 341650 68.33 581 0.12 0.316
2s 7h 157036 31.41 342399 68.48 565 0.11 0.315
4c 3c 2d 184505 36.90 315261 63.05 234 0.05 0.369

Drop none:

pokenum -mc 500000 -l27 2h 7s / kd qd jd - 2s 7h / kh qh jh - 2c 3c 4c 6d ah
5-card Draw 2-7 Lowball: 500000 sampled outcomes
cards win %win lose %lose tie %tie EV
7s 2h 176716 35.34 322836 64.57 448 0.09 0.354
2s 7h 176088 35.22 323464 64.69 448 0.09 0.353
4c 3c 2d 6d Ah 146748 29.35 353252 70.65 0 0.00 0.293

Drop just the A:

pokenum -mc 500000 -l27 2h 7s / kd qd jd - 2s 7h / kh qh jh - 2d 3c 4c 6d / ah
5-card Draw 2-7 Lowball: 500000 sampled outcomes
cards win %win lose %lose tie %tie EV
7s 2h 134507 26.90 365062 73.01 431 0.09 0.269
2s 7h 134634 26.93 364932 72.99 434 0.09 0.270
4c 3c 6d 2d 230338 46.07 269485 53.90 177 0.04 0.461

Even for more pathological cases (against 4 7s with a 2 and 4 8s with a 2), keeping the 6 is what you want:

drop A:

pokenum -mc 500000 -l27 2h 7s / 7d 7h 7c - 2s 8h / 8d 8s 8c - 2d 3c 4c 6d / ah
5-card Draw 2-7 Lowball: 500000 sampled outcomes
cards win %win lose %lose tie %tie EV
7s 2h 148033 29.61 351967 70.39 0 0.00 0.296
2s 8h 144550 28.91 355450 71.09 0 0.00 0.289
4c 3c 6d 2d 207417 41.48 292583 58.52 0 0.00 0.415

Drop 6A:

pokenum -mc 500000 -l27 2h 7s / 7d 7h 7c - 2s 8h / 8d 8s 8c - 2d 3c 4c / 6d ah
5-card Draw 2-7 Lowball: 500000 sampled outcomes
cards win %win lose %lose tie %tie EV
7s 2h 175797 35.16 324203 64.84 0 0.00 0.352
2s 8h 171221 34.24 328779 65.76 0 0.00 0.342
4c 3c 2d 152982 30.60 347018 69.40 0 0.00 0.306

srop none:

pokenum -mc 500000 -l27 2h 7s / 7d 7h 7c - 2s 8h / 8d 8s 8c - 2d 3c 4c 6d ah
5-card Draw 2-7 Lowball: 500000 sampled outcomes
cards win %win lose %lose tie %tie EV
7s 2h 199816 39.96 300184 60.04 0 0.00 0.400
2s 8h 194094 38.82 305906 61.18 0 0.00 0.388
4c 3c 6d 2d Ah 106090 21.22 393910 78.78 0 0.00 0.212

[timprov]

[ QUOTE ]
Droping the 6 is almost as big a mistake as standing pat.

From pokenum:


[/ QUOTE ]

I'm 99% certain this is a triple draw question.

[dibbs]

100% /images/graemlins/smirk.gif