No-Limit Hold Em Math...How often does he have to fold?

[HiatusOver]

5-10 Blinds No-Limit Game. My stack (610), opponents in this hand cover me. Pre-flop : UTG raises to 40, 1-caller, I call with Q [img]/images/graemlins/club.gif[/img]J [img]/images/graemlins/club.gif[/img] in the Big Blind. Flop A [img]/images/graemlins/diamond.gif[/img]6 [img]/images/graemlins/club.gif[/img]3 [img]/images/graemlins/heart.gif[/img]...checked around. Turn K [img]/images/graemlins/club.gif[/img]...I bet 85 (Pot 125)...Pre-Flop raiser raises to 225, other guy folds its one me. Putting this guy on a normal range of hands (Dry Ace, set, 2-pair, bluff 10 percent of time), how often does he have to fold to make pushing all in for 350 more correct? He would have to call 350 and the pot would be 575 after matching his 225. By the way I am sure that just calling is +EV play over time so this is a raise or call situation, I would never fold...does that matter in the math?

Thanks for all replies in advance...I will continue to contribute to this thread I am doing the math too, and think I have a close to correct answer but am not sure I am assuming all the variables

[HiatusOver]

I realized that a range of hands is probably pretty important here...not sure how to break it down. The main hands I put him on were A-10 thru AK and maybe KK, although I think often he just bets KK once on the flop and shuts down most likely if called. The player behind him is aggressive so I think checking an Ace or even a set is pretty likely here. Also from what Dan Harrington says in the book...a 10-percent bluffing rule should always be included.

[Little Fishy]

gong all in is incorrect. if you assume villain holds AT-AK or KK 90% of the time and is bluffing %10 of the time, and that villain will always call with a legit hand than: when viallain has a legit hand he will hold (out of 39 possible holdings):
AcK 4 times and you'll win 25%
AcQ 4 25%
AcJ 4 25%
AcT 6 22.73%
AcTc 1 6.82%
AK 4 25%
AQ 4 27.27%
AJ 4 27.27%
AT 5 25%
KK 3 22.73%

meaning given a call you have a 1/39(4(.25)+4(.25)+4(.25)+6(.2273)+(.0682)+(4(.25)+4 (.2727)+4(.2727)+5(.25)+3(.2273))= 24.475% you will win

lets say that if you call you will still be ahead of a bluff 95% of the time on the end so if you only call than you will beat the legit hands 24.475% and the bluffs 95% so simply callling you will win .9(.24475)+.1(.95)= 29.73%
by calling you're investing $140 additional $ to win a total of $565 (140/565)= 23.9%, the percentage that you will end up winning is greater than your investment meaning calling is +EV

now to look at pushing all in, lets say Villain will fold 100% of his bluffs and there are the same chances of other holdings. therefore you have a 10% chance to win immediatly and a 24.475% chance to win given a call. Your risk however is $485 additional $ to win a pot of 1135 if called and $790 if not called. 90% of the time you are risking 485 to win 650 with a 24.475% chance of winning and 10% of the time you are risking 485 to win 305 100% of the time. your EV for this is
.9(650(.24475)-485(1-.24475))+.1(305)= -155.98
while your EV for just calling is:
.9(425(.24475)-140(.75524))+.1(425(.95)-140(.05))= +38.1336

clearly calling is the right move while pushing all in is very incorrect.

It's a bit late so i'm sorry if I messed up on any math, but i think i did it fine.

-little fishy

[HiatusOver]

Thanks the response and the math looks fine but u didnt really answer the question...I am not asking what the correct play is...I am asking what percentage of the time my opponent would have to fold to make the play correct. I really think he would fold a dry ace here sometimes by the way

[Little Fishy]

sorry my mistake,

to solve this simply plot an equation for EV with the probablility that he will fold a legit hand as your variable:

(.9-x)(485(.24475)-485(1-.24475))+(.1+x)(305)= EV

x is the probability that he will fold a legitimate hand and must be between 0 and .9 inclusive. when you plot this you will find that pushing all in will become a +EV move when x is .34805.

so, to at last answer your question it is a +EV move to push all in here if Villain will fold legitimate hands more than 34.8% of the time.

-little fishy

ps. hope that answers your question.

[HiatusOver]

That is very close to the number I came up with...thanks. There seems to be one more variable though. Calling is almost definitely a plus EV play here. So if calling is a +EV play, then moving all in would have to technically be more of a +EV play to make it correct. Just wanted to add that in. Assuming that calling is a $50+ EV play then I think the percentage that he has to fold when u push is moved up to close to 40 percent

[Little Fishy]

yep right on, you need a fold about 41.7% of the time to make raising more correct than calling

-little fishy

ps i liked this thread, wish we had more like it.