Should I have kept 87?

DealPete · #1 07-27-2007, 01:00 AM

I was near the FT bubble in Stars' $10 nightly 2-7 tourney. Nothing really known about CO. When he stood pat I assumed he had 8 or better, and the fact that I had two 8s made it relatively more likely he had a 7. However if a 9 is in his range he is more likely to have a 9. Is it better to keep the 87 or chuck both the 8s?

Triple Draw 2-7 Lowball Limit ($400/$800) (converter)

SB ($1680), Seat 4
Hero ($1675), Seat 5
UTG ($28500), Seat 6
CO ($1065), Seat 2
Button ($6630), Seat 3

Button at Seat 3.
SB posts $200. Hero posts $400.

Hand: [8h Jh Jc 2c 4d]

Round 1: (1.50 SB)

UTG folds. CO raises. Button folds. SB folds. Hero raises. CO calls (all-in). 
Hero discards Jh Jc. CO stands pat.

Hand: [8h 2c 4d 7d 8d]

Round 2: (5.83 SB)

No action, HU w/all-in.
Hero discards 8h 8d. CO stands pat.

Hand: [2c 4d 7d 5c Ks]

Round 3: (2.91 BB)

No action, HU w/all-in.
Hero discards Ks. CO stands pat.

Hand: [2c 4d 7d 5c 4c]

Round 4: (2.91 BB)

No action, HU w/all-in.

Total pot: (2.91 BB - $2330)

Phat Mack · #2 07-27-2007, 02:27 AM

I'm not sure about your round 1 raise, but leaving that aside, note that he is all-in and has seen you draw two in front of him. He's going to stand pat here with a lot of hands when he sees you drawing two, not just 7s.

Also note that you kept the 8 on the first draw. If you kept the 8 thinking an 8 was good, then tossed the 8 on round 2 because you saw another 8, and decided he has a 7--no, I don't think that line of reasoning is justified.

Another thing to consider is that if you draw 2 cards again on round two, he will remain pat again. You are making his decisions easy. If you discard one, he might make a drawing mistake there or on round three.

jmo

MarkGritter · #3 07-27-2007, 03:21 AM

You're correct that the second 8 makes him more likely to have a 7, but this is significant only if you believe he wouldn't pat a 9 here.

If he would pat any 9 or better then the chance that he has a 7 (based on seeing 24788JJ) is just 9.6%. (Of course he can also have an 85 or 86.)

If he only pats 8s or better then the chance that he has a 7 is more like 32%. More importantly, you are drawing dead against about 59% of his range.

The question is, do you actually improve your chances of winning in this case?

Suppose your opponent has a 7 or 8 and you draw one. Then over two draws you have about a 14% chance of beating him (according to the TD2-7 dead out calculator.) You have 32% chance when drawing live (87432 or worse) which is consistent with drawing live about 40% of the time.

If you draw to 742 and don't keep an 8 on the last draw either, then your draw has a distribution something like this: (hand, probability, cumulative probability)

Code:<hr /><pre>
7 5 4 3 2 0.0515454 0.0515454
7 6 4 3 2 0.0515454 0.103091
7 6 5 4 2 0.0515454 0.154636
8 7 4 3 2 0.0107341 0.16537
8 7 5 4 2 0.0107341 0.176104
8 7 6 4 2 0.0107341 0.186838
</pre><hr />

But that assumes all your outs are live, which they aren't. Let's work through a few sample hands where you draw to 742:

75432: 3.8% chance of a tie
76532: 7.7% chance of winning
76542: 10% chance of winning, 3.8% chance of a tie
85432: 13.8% chance of winning
86543: 12% chance of winning

87543: 14.8% chance of winning
87643: 16.6% chance of winning
87642: 17.6% chance of winning

So, a rough estimate is that by drawing two again you'll win 4-14% of the time against hands that beat any 87 (60% of his range) and 14-18% against hands that an 87 can beat (40% of his range) which gives us something like 5.4 + 6.4 = 10.8% chance of winning.

In order to be better than keeping the 8, we'd need to have about 12-13% equity against the 86's and better. It's close but there are too many dead 5's and 6's in that range, too many ties, and too many second-best 7's you can make.

This suggests that you shouldn't break your 8. Even though you're often drawing dead, breaking is not all that likely to beat the very strong hands (#1-#9) and significantly less likely to beat the weaker hands (87432 through 87653).

If there is any chance he has a 9 here (a 96, for example, would almost certainly pat) then throwing away the 8 gets worse.

DealPete · #4 07-27-2007, 07:29 PM

Thank you very much for the posts. I didn't expect to see such an in-depth analysis so quickly after I posted.
When I posted I was mostly wondering whether a typical player would be likely to have 9 in his range, since from what I've read it appears to be a mathematical error to stand pat on a 9 here against someone drawing two cards. It didn't occur to me that discarding the 8s might also be a poor move against someone who would only stand pat on an 8 or worse.
To clarify, Phat Mack, I raised with 842 in the cutoff because Negreanu in Super System II suggests raising your good 8s in late position. I wanted to discard to 8s on round 2 not because 8s are poor but because 87 can be a poor combination to draw to.

Phat Mack · #5 07-28-2007, 01:28 AM

To clarify, Phat Mack, I raised with 842 in the cutoff because Negreanu in Super System II suggests raising your good 8s in late position.

Are you in the cutoff or big blind?

since from what I've read it appears to be a mathematical error to stand pat on a 9 here against someone drawing two cards.

I'm having difficulty accepting this.

MarkGritter · #6 07-28-2007, 02:08 AM

A two-card draw is not much of a dog against a rough 9, but only if he manages to draw correctly. The numbers here:

http://wiki.lowballgurus.com/article/Drawing_to_732xx

don't count dead cards, but you can see that a 96 is a favorite (if you draw correctly to an 8 or better, middle chart on the page.)

A 98 shows up as a small dog to win if the 732 draw keeps a 9 (bottom table), again ignoring dead cards.

I reran the calculation with 98764 dead, and it appears that the 732xx is a small favorite to win, 52.5% to win or tie by keeping a 9. 732xx is a dog against 98764 by drawing to an 8; he will make a 98 or better only 43.3% of the time.

Edited to add:

However, remember the important thing is not whether a 9 would or wouldn't be the favorite here, but whether patting is the 9's best option! A 97432 might be better off drawing even if he's already a favorite to win, while 98764 might be a dog either way but has a better chance of winning by staying pat.

Phat Mack · #7 07-28-2007, 01:52 PM

I reran the calculation with 98764 dead, and it appears that the 732xx is a small favorite to win, 52.5% to win or tie by keeping a 9.

Could you run it with 9732 dead?

I wonder what percentage of the TD population would know to draw to a 9 here? I sometimes find myself hoping that no one reads these threads...

MarkGritter · #8 07-28-2007, 02:02 PM

732KQ, drawing to an 8, with 97432 dead:

Code:<hr /><pre>
7 5 4 3 2 0.0656253 0.0656253
7 6 4 3 2 0.0656253 0.131251
7 6 5 3 2 0.088349 0.2196
8 7 4 3 2 0.0656253 0.285225
8 7 5 3 2 0.088349 0.373574
8 7 6 3 2 0.088349 0.461923
9 7 4 3 2 0.00946009 0.471383
</pre><hr />

732KQ, drawing to a 9, with 97432 dead:

Code:<hr /><pre>
7 5 4 3 2 0.0556272 0.0556272
7 6 4 3 2 0.0556272 0.111254
7 6 5 3 2 0.0749411 0.186196
8 7 4 3 2 0.0556272 0.241823
8 7 5 3 2 0.0749411 0.316764
8 7 6 3 2 0.0749411 0.391705
9 7 4 3 2 0.0412865 0.432992
</pre><hr />

Your opponent, of course, doesn't actually know whether he should keep the 9 or not (and in this case he shouldn't)--- I think on balance you have to draw to the 8, but it is probably worth keeping a 9 on the last draw rather than drawing two.

Phat Mack · #9 07-28-2007, 03:01 PM

Can you generate these same numbers for two draws?

MarkGritter · #10 07-28-2007, 03:35 PM

Sure thing:

732KQ, drawing to an 8, two draws, 97432 dead:

Code:<hr /><pre>
7 5 4 3 2 0.0398955 0.0398955
7 6 4 3 2 0.0398955 0.0797909
7 6 5 3 2 0.0534262 0.133217
8 7 4 3 2 0.0398955 0.173113
8 7 5 3 2 0.0534262 0.226539
8 7 6 3 2 0.0534262 0.279965
9 7 4 3 2 0.0114111 0.291376
</pre><hr />

732KQ, drawing to a 9, two draws, 97432 dead:

Code:<hr /><pre>
7 5 4 3 2 0.0364648 0.0364648
7 6 4 3 2 0.0364648 0.0729295
7 6 5 3 2 0.048852 0.121781
8 7 4 3 2 0.0364648 0.158246
8 7 5 3 2 0.048852 0.207098
8 7 6 3 2 0.048852 0.25595
9 7 4 3 2 0.0272179 0.283168
</pre><hr />

(Of course, with two draws you ought to include another couple bricks to represent whatver you started with. That improves the draw's chances a tiny bit, up to 0.315 in the first case and 0.305 in the second.