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Maths Behind 2-7 Triple Draw


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#1 medleyoz

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Posted 15 April 2007 - 06:18 PM

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

#2 Vman96

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Posted 20 April 2007 - 07:33 AM

View Postmedleyoz, on Sunday, April 15th, 2007, 6:18 PM, said:

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: 128s: 4Cards that do not help you improve: 31Cards remaining in deck: 47Improve to a 7 on first draw:(12/47)(8/46) = 0.04440333No improvement:(31/47)(30/46) = 0.43015726Improve to a one card 7 draw:(12/47)(34/46) + (31/47)(12/46) = 0.360777058Improve to an 8 on the first draw:(16/47)(12/46) - (12/47)(8/46) = 0.04440333Improve to a one card 8 draw:(4/47)(34/46) + (31/47)(4/46) = 0.1202590192nd draw:Still drawing 2:Cards that help you improve to a 7: 128s: 4Cards that do not help you improve: 29Cards remaining in deck: 45Improve to a 7:(0.43015726)(12/45)(8/44) = 0.02085611No improvement thru 2 draws:(0.43015726)(29/45)(28/44) = 0.176407927Improve to a one card 7 draw:(0.43015726)(12/45)(32/44) + (0.43015726)(29/45)(12/44) = 0.15902784Improve to an 8:(0.43015726)(16/45)(12/44) - (0.43015726)(12/45)(8/44) = 0.02085611Improve to a one card 8 draw:(0.43015726)(4/45)(32/44) + (0.43015726)(29/45)(4/44) = 0.053009279Drawing 1 to a 7:Cards that help you improve to a 7: 88s: 4Cards that do not help you improve: 33Cards remaining in deck: 45Improve to a 7 or better:(0.360777058)(8/45) = 0.064138144Improve to an 8:(0.360777058)(4/45) = 0.0320690718 Still drawing 1 to a 7:(0.360777058)(33/45) = 0.2645698425Drawing 1 to an 8:Cards that help you improve to an 8: 12Cards that do not help you improve: 33Cards remaining in deck: 45Improve to an 8:(0.120259019)(12/45) = 0.03206907Still drawing 1 to an 8:(0.120259019)(33/45) = 0.08818994733rd Draw:Still Drawing 2 to a 7:Cards that help you improve to a 7: 128s: 4Cards that do not help you improve: 27Cards remaining in deck: 43Improve to a 7:(0.176407927)(12/43)(8/42) = 0.009377166Improve to an 8:(0.176407927)(16/43)(12/42) - (0.176407927)(12/43)(8/42) = 0.009377166No improvement: (3 total bricks)(0.176407927)(27/43)(26/42) = 0.068570523Improvement to a 4-card 8 low or better (still worthless):(0.176407927)(16/43)(30/42) + (0.176407927)(27/43)(16/42) = 0.08908307Drawing 1 to a 7:A.Drew 2 on 2nd draw:Cards that help you improve to a 7: 88s: 4Cards that do not help you improve: 31Cards remaining in deck: 43Improve to a 7:(0.15902784)(8/43) = 0.029586575Improve to an 8: (0.15902784)(4/43) = 0.014793287No improvement:(0.15902784)(31/43) = 0.114647978B. Drew 1 on 2nd draw:Cards that help you improve to a 7: 88s: 4Cards that do not help you improve: 32Cards remaining in deck: 44Improve to a 7:(0.2645698425)(8/44) = 0.0481036077Improve to an 8:(0.2645698425)(4/44) = 0.0240518039No improvement:(0.2645698425)(32/44) = 0.192414431Drawing 1 to an 8:A.Drew 2 on 2nd draw:Cards that help you improve to a 8: 12Cards that do not help you improve: 31Cards remaining in deck: 43Improve to an 8:(00.053009279)(12/43) = 0.01673339No improvement:(0.053009279)(31/43) = 0.0432279248B. Drew 1 on 2nd draw:Cards that help you improve to a 8: 12Cards that do not help you improve: 32Cards remaining in deck: 44Improve to an 8:(0.0881899473)(12/44) = 0.0240518No improvement:(0.0881899473)(32/44) = 0.0641381435Okay......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 = 0.2164649327The 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 = 0.2164649327So 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) = 0.59636139With 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 = 0.17021277make 8: 4/47 = 0.08510638still drawing: 35/47 = 0.744680852nd draw:make 7: (35/47)(8/46) = 0.1295097make 8: (35/47)(4/46) = 0.0647549still drawing: (35/47)(34/46) = 0.550416283rd draw:make 7: (35/47)(34/46)(8/45) = 0.09785178make 8: (35/47)(34/46)(4/45) = 0.04892589fail to improve: (35/47)(34/46)(33/45) = 0.4036386With 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.

#3 medleyoz

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Posted 03 May 2007 - 06:39 PM

Thanks for that. EXCELLENT POST.

#4 mamaboy

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Posted 16 January 2008 - 08:42 AM

wonderful post!thanks

#5 dinho_style

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Posted 22 January 2008 - 02:56 PM

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




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