Mathematical PLH decision

2-2 PLH, average stack 400, I have table covered


Folded to the cutoff who makes it 8 to go, recent history has been good against his folding to my raises (and he can get out of line a bit in attacking the blinds) so I make it 28 to go on the button with QJo (flame away, I'm a maniac), folded to him and he calls, pot is 60, he has 335 left in his stack (I know from the action)


flop is KT6r, he hesitates a bit and leads 25, I read this as weakness and feel he will fold to a big raise (once before he led 5 into a $12 pot and I raised pot and he folded very quickly), I raise 110 (pot)... he thinks for a while and eventually pushes all in for exactly 200 more, so I have an open ended straight draw to the nuts and need to call 200 more to win a 530 pot, what do I do?

You will get your str8 31.5% of the time. However, if he flopped a set he will fill-up 22% of the times you get your str8. You win 24.7% giving you odds of 1:3.06; the pot is giving you 1:2.65. A take home quiz is what if you think he has 2 pair?

Here's my analysis of the situation. At the time I approximated it as I'm getting 5:2 on a 7:3 shot, so I can call. Being more precise afterwards...


Backdoor flush possibilities turn out to make a difference here because the decision is so close, so I'll just make up the suits of the board relative to my hand as:


I hold QdJs, board is KhTc6d


I need my percentage equity in the pot on a call to be greater than 200/730=27.40% to make a call correct. In order to determine my equity in the pot, I computed my equity against all his plausible holdings and then multiplied by his number of outs and a weighting corresponding to the probability that he plays each hand this way (which is 0 for most hands because of preflop). I got the following


Holding - Equity - Combinations - Consistancy

set - 25.86% - 9 - 1/3 (weighting averaged over KK,TT, and 66)

KdTd - 28.59% - 1 - 4/5

other KT - 30.10% - 8 - 4/5

AA - 23.84% - 6 - 1/3

AcKc - 27.78% - 1 - 2/3

AdKd - 27.88% - 1 - 2/3

other AK - 28.69% - 10 - 2/3

KcQc/KcJc - 30.20% - 2 - 1/3

KdJd - 30.30% - 1 - 1/3

other KQ/KJ - 31.82% - 15 - 1/3

QQ - 31.82% - 3 - 1/10

JJ - 41.82% - 3 - 1/10

QcJc/QhJh - 47.73% - 2 - 1/10

other QJ - 50.00% - 7 - 1/10


Thus my equity in the pot on a call is


(9/3)25.86%+(4/5)28.59%+(32/5)30.10%+(6/3)23.84%+(2/3)27.78%+(2/3)27.88%+(20/3)28.69%+(2/3)30.20%+(1/3)30.30%+(15/3)31.82%+(3/10)31.82%+(3/10)41.82%+(2/10)47.73%+(7/10)50.0% = 825.12%


divided by


9/3+4/5+32/5+6/3+2/3+2/3+20/3+2/3+1/3+15/3+3/10+3/10+2/10+7/10=277/10


to get


825.12%/27.7 = 29.79% > 27.40%.


Thus I make 29.79%(730)-200 = $17.5 on a call, assuming my weightings for consistancy are correct. However, against most opponents it's hard for my call to be wrong. AA turns out to be the worst hand for me, with a set being second worst (kind of a surprise), and everything else the call is immediately correct against. It's just hard for me to conceive of any players whose consistancy weightings will be such that the equity calculation goes from a call to a fold.


Alex

to me it is not the percentage of times you will win, etc,etc


for this one hand it is pot odds vs drawing odds


with either 6,7, or eight outs (if he has one or two aces) you get TWO draws, thus drawing odds are about 2-1..your pot odds are substantially better thefore you should go for it


also I think no need to consider any outs he may have to improve his hand...if he could make a full it would not be with one of your straight cards...it appears that he does not need to draw out, you do.

But if he has a set, Alex could spike on the turn and will still get re-spiked about 1/4 of the time.


I think that should weigh into it a fair amount. I think that makes Alex's worst hand to be against a set since the set will also fill up about 2-1 (a little less, but rough estimates will do) and thus he'll lose when they both hit.


If he has AA, then he's get a re-draw 1/3 of the time Alex makes his hand on the turn and there again he'll lose about 1/4 of the time.


That should weigh into the analysis since 1/4 is large enough to affect the decision. However, since he could have AK or just a single pair, the re-draw possibility should be weighted the same way he weighted the hands he put the opponent on.

I called. Turn came a 6, river an 8, I chopped with QJo.