Hi,
Just read Dan N's chapter in SS2 on Triple draw and just would like to know how to calculate the odds of catching draws etc (a quick method for the table and a long version so I know how it works) so I can compare pot odds to te chances of drawing out etc.

Also reading the chapter Dan doesn't really talk about pot odds when making decisions- is 2-7 not so much a pot odd game?

Thanks for any help

Well the math for 1 card draws is fairly straight forward. probability of hitting for three draws is just: 1 - (#outs/#unknown cards at first draw)(#outs/#unknown cards at 2nd draw)(#outs/#unknown cards at 3rd draw)


an expansion of the 4/2 rule for outs in hold 'em for triple draw is:

5.5/4/2 so if you have an 8 outer with all 3 draws remaining, you are ~44% to hit it by the end. With 2 draws: ~32%. With 1 draw: ~16%

Now drawing multiple cards...this gets more confusing for math, so there arent any simple rules here.

lets say you have a smooth 2-card 7 draw with two high cards in your hand (i.e. 237KJ, 247QT, 267KQ, etc...): Whats the chance you'll end up with a 7? an 8? by the 3rd draw?

Draw 1:
Cards that help you improve to a 7: 12
8s: 4
Cards that do not help you improve: 31
Cards remaining in deck: 47

Improve to a 7 on first draw:
(12/47)(8/46) = 0.04440333

No improvement:
(31/47)(30/46) = 0.43015726

Improve to a one card 7 draw:
(12/47)(34/46) + (31/47)(12/46) = 0.360777058

Improve to an 8 on the first draw:
(16/47)(12/46) - (12/47)(8/46) = 0.04440333

Improve to a one card 8 draw:
(4/47)(34/46) + (31/47)(4/46) = 0.120259019

2nd draw:
Still drawing 2:
Cards that help you improve to a 7: 12
8s: 4
Cards that do not help you improve: 29
Cards remaining in deck: 45

Improve to a 7:
(0.43015726)(12/45)(8/44) = 0.02085611

No improvement thru 2 draws:
(0.43015726)(29/45)(28/44) = 0.176407927

Improve to a one card 7 draw:
(0.43015726)(12/45)(32/44) + (0.43015726)(29/45)(12/44) = 0.15902784

Improve to an 8:
(0.43015726)(16/45)(12/44) - (0.43015726)(12/45)(8/44) = 0.02085611

Improve to a one card 8 draw:
(0.43015726)(4/45)(32/44) + (0.43015726)(29/45)(4/44) = 0.053009279


Drawing 1 to a 7:
Cards that help you improve to a 7: 8
8s: 4
Cards that do not help you improve: 33
Cards remaining in deck: 45

Improve to a 7 or better:
(0.360777058)(8/45) = 0.064138144

Improve to an 8:
(0.360777058)(4/45) = 0.0320690718

Still drawing 1 to a 7:
(0.360777058)(33/45) = 0.2645698425

Drawing 1 to an 8:
Cards that help you improve to an 8: 12
Cards that do not help you improve: 33
Cards remaining in deck: 45

Improve to an 8:
(0.120259019)(12/45) = 0.03206907

Still drawing 1 to an 8:
(0.120259019)(33/45) = 0.0881899473


3rd Draw:

Still Drawing 2 to a 7:

Cards that help you improve to a 7: 12
8s: 4
Cards that do not help you improve: 27
Cards remaining in deck: 43

Improve to a 7:
(0.176407927)(12/43)(8/42) = 0.009377166

Improve to an 8:
(0.176407927)(16/43)(12/42) - (0.176407927)(12/43)(8/42) = 0.009377166

No improvement: (3 total bricks)
(0.176407927)(27/43)(26/42) = 0.068570523

Improvement to a 4-card 8 low or better (still worthless):
(0.176407927)(16/43)(30/42) + (0.176407927)(27/43)(16/42) = 0.08908307

Drawing 1 to a 7:

A.
Drew 2 on 2nd draw:
Cards that help you improve to a 7: 8
8s: 4
Cards that do not help you improve: 31
Cards remaining in deck: 43

Improve to a 7:
(0.15902784)(8/43) = 0.029586575

Improve to an 8:
(0.15902784)(4/43) = 0.014793287

No improvement:
(0.15902784)(31/43) = 0.114647978

B. Drew 1 on 2nd draw:
Cards that help you improve to a 7: 8
8s: 4
Cards that do not help you improve: 32
Cards remaining in deck: 44

Improve to a 7:
(0.2645698425)(8/44) = 0.0481036077

Improve to an 8:
(0.2645698425)(4/44) = 0.0240518039

No improvement:
(0.2645698425)(32/44) = 0.192414431


Drawing 1 to an 8:
A.
Drew 2 on 2nd draw:
Cards that help you improve to a 8: 12
Cards that do not help you improve: 31
Cards remaining in deck: 43

Improve to an 8:
(00.053009279)(12/43) = 0.01673339

No improvement:
(0.053009279)(31/43) = 0.0432279248

B. Drew 1 on 2nd draw:
Cards that help you improve to a 8: 12
Cards that do not help you improve: 32
Cards remaining in deck: 44

Improve to an 8:
(0.0881899473)(12/44) = 0.0240518

No improvement:
(0.0881899473)(32/44) = 0.0641381435

Okay......now, the total chance of improving a 2 card 7 draw to a 7 is:
0.04440333 + 0.02085611 + 0.064138144 + 0.009377166 + 0.029586575 + 0.0481036077 = 0.2164649327

The total chance of improving a 2 card 7 draw to a 8 is:
0.04440333 + 0.02085611 + 0.0320690718 + 0.03206907 + 0.009377166 + 0.014793287 + 0.0240518039 + 0.014793287 + 0.0240518 = 0.2164649327

So with a smooth 2 card 7 draw you have a 21.6% chance to improve to a 7 and a 21.6% chance to improve to an 8.

In comparison, hitting a one card 8 draw is simply:
1 - (35/47)(34/46)(33/45) = 0.59636139

With a one card 8 draw you have a 59.6% chance to improve to an 8.

What about a one card smooth 7 draw?
1st draw:
make 7: 8/47 = 0.17021277
make 8: 4/47 = 0.08510638
still drawing: 35/47 = 0.74468085

2nd draw:
make 7: (35/47)(8/46) = 0.1295097
make 8: (35/47)(4/46) = 0.0647549
still drawing: (35/47)(34/46) = 0.55041628

3rd draw:
make 7: (35/47)(34/46)(8/45) = 0.09785178
make 8: (35/47)(34/46)(4/45) = 0.04892589
fail to improve: (35/47)(34/46)(33/45) = 0.4036386

With a smooth one card 7 draw you have a 39.8% chance of making a 7 and a 19.9% chance of making an 8.
This reaffirms how strong this draw is.

As far as pot odds go...you should pay attention to them obviously. But its more about good predraw selection and taking advantage of position. In a loose low limit game, if you raise your good hands predraw the pot will be bloated enough where quite a few one card draws are chasable. If you are drawing 1, and you are certain that the bettor is standing pat, you probably shouldnt call if you aren't getting 3 to 1 or better on the last draw. Rarely draw 2 on 3rd street (if ever)...the only way this can ever be an okay call is if you KNOW your opponent is drawing still...AND the pot is fairly large.

Thanks for that. EXCELLENT POST.

wonderful post!
thanks

great post indeed! i also wanned to know more about the math of 2-7