Drawing three in TD 2-7

[MarkGritter]

Here are some early numbers from some TD software I've been working on.

Suppose you have 72AAA on the first draw. Then your chance of making an 875 or better are 25%. This suggests that defending your blind getting 3:1 is right on the line vs. a random playable hand (but probably OK to go all-in on in a tournament setting.)

Here's the hand distribution after the third draw, assuming you keep 8's in the first two draws, but not 9's. (hand, probability, cumulative probability.)

<font class="small">Code:</font><hr /><pre>
7 5 4 3 2 0.035757 0.035757
7 6 4 3 2 0.035757 0.071514
7 6 5 3 2 0.035757 0.107271
7 6 5 4 2 0.035757 0.143028
8 7 4 3 2 0.035757 0.178785
8 7 5 3 2 0.035757 0.214542
8 7 5 4 2 0.035757 0.250299
8 7 6 3 2 0.035757 0.286056
8 7 6 4 2 0.035757 0.321813
8 7 6 5 2 0.035757 0.35757
9 7 4 3 2 0.00634917 0.363919
9 7 5 3 2 0.00634917 0.370268
9 7 5 4 2 0.00634917 0.376618
976xx 0.0190475 0.395665
98xxx 0.0253967 0.421062
T-high 0.0677599 0.488822
J-high 0.0721034 0.560925
Q-high 0.0765221 0.637447
K-high 0.0810161 0.718463
A-high 0.0213963 0.739859
pair of deuces 0.0650598 0.804919
pair of threes 0.0221935 0.827113
pair of fours 0.0221935 0.849306
pair of fives 0.0221935 0.8715
pair of sixes 0.0221935 0.893693
pair of sevens 0.0650598 0.958753
pair of eights 0.0221935 0.980947
big pair 0.00860259 0.989549
trips 0.0104272 0.999976
very bad 2.35179e-05 1
</pre><hr />

What about 23? The chances of an 875 are somewhat higher, while the chance of a 7 is lower.

<font class="small">Code:</font><hr /><pre>
7 5 4 3 2 0.035757 0.035757
7 6 4 3 2 0.0377202 0.0734772
7 6 5 3 2 0.035757 0.109234
8 5 4 3 2 0.035757 0.144991
8 6 4 3 2 0.0377202 0.182711
8 6 5 3 2 0.035757 0.218468
8 7 4 3 2 0.035757 0.254225
8 7 5 3 2 0.035757 0.289982
8 7 6 3 2 0.035757 0.325739
9 5 4 3 2 0.00634917 0.332089
9 6 4 3 2 0.00787791 0.339966
9 6 5 3 2 0.00634917 0.346316
9 7 4 3 2 0.00634917 0.352665
9 7 5 3 2 0.00634917 0.359014
976xx 0.00634917 0.365363
98xxx 0.0253967 0.39076
T-high 0.0692886 0.460048
J-high 0.0736321 0.533681
Q-high 0.0780508 0.611731
K-high 0.0825448 0.694276
A-high 0.0217785 0.716055
pair of deuces 0.0662064 0.782261
pair of threes 0.0662064 0.848467
pair of fours 0.0233801 0.871847
pair of fives 0.0221935 0.894041
pair of sixes 0.0233801 0.917421
pair of sevens 0.0221935 0.939615
pair of eights 0.0221935 0.961808
big pair 0.00860259 0.970411
trips 0.0104272 0.980838
straight 0.0191387 0.999976
very bad 2.35179e-05 1
</pre><hr />

(Both hands have about a 40% chance of being pat with a J or better after two draws, though, so position will help significantly.)

Finally, what does a legitimate hand like 732AA look like after three draws?

<font class="small">Code:</font><hr /><pre>
7 5 4 3 2 0.072155 0.072155
7 6 4 3 2 0.072155 0.14431
7 6 5 3 2 0.072155 0.216465
8 7 4 3 2 0.072155 0.28862
8 7 5 3 2 0.072155 0.360775
8 7 6 3 2 0.072155 0.43293
9 7 4 3 2 0.0147192 0.447649
9 7 5 3 2 0.0147192 0.462368
976xx 0.0147192 0.477088
98xxx 0.0147192 0.491807
T-high 0.0612363 0.553043
J-high 0.0635957 0.616639
Q-high 0.0659552 0.682594
K-high 0.0683146 0.750909
A-high 0.035337 0.786246
pair of deuces 0.0538903 0.840136
pair of threes 0.0538903 0.894026
pair of fours 0.0105509 0.904577
pair of fives 0.0105509 0.915128
pair of sixes 0.0105509 0.925679
pair of sevens 0.0538903 0.979569
pair of eights 0.0105509 0.99012
big pair 0.00457137 0.994691
trips 0.00530869 1
</pre><hr />

This is just part of the story, as this is not a full simulation taking into account standing pat early, implied odds, drawing to a 9 on the last draw or discarding an 8 early, etc.

[canis582]

#1 you are awesome

#2 now do A-5. It should be easier.

[TakeMeToTheRiver]

Great info! You da man.

Just a couple of questions for clarification -- I understand that you are discarding 9s on the first two draws -- what is the standard on the third draw? Are you always keeping 6s in the simulation?

[MarkGritter]

[ QUOTE ]
Great info! You da man.

Just a couple of questions for clarification -- I understand that you are discarding 9s on the first two draws -- what is the standard on the third draw? Are you always keeping 6s in the simulation?

[/ QUOTE ]

For these numbers, I keep an 8 and discard a 9 on all three draws. I always keep a 6 or 7, except:

6-high straight: discard the six.
7-high straight: discard the six.
8-high straight: discard the seven.

It is not too hard to fiddle with these rules, although when I finished the code last night I only implemented switches for 'draw to 7', 'draw to 8', or 'draw to 9'.

The times when it is right to draw to a 9 are situation-dependent, I think, and rather than try to encapsulate that I'd rather work on a full hot+cold perfect-information simulation, where you and your opponent(s) take the optimal draws. (Which is still a long way from covering the complexity of the game, but should be interesting anyway.)