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 /images/graemlins/club.gifJ /images/graemlins/club.gif in the Big Blind. Flop A /images/graemlins/diamond.gif6 /images/graemlins/club.gif3 /images/graemlins/heart.gif...checked around. Turn K /images/graemlins/club.gif...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.

[raisins]

I've been thinking about this situation some over the last couple of days and I came at it a little different. Here's what I think.

The OP's hand history had a few slight discrepancies so here is how I recreated the action:

Hero starts with 610. PF action is 125 pot. Hero has 570.

Turn Hero bets 85. Pot has 210. Villain raises to 225 (130 more). Pot has 435. 130 for Hero to call. Hero has 485.

Hero basically wins 1 out of every 4 times. That's a little simplified but it is not off by much. The only hand Hero is in bad shape against is AcTc.

Villain's hand range consists of 10% bluffs, hands that he will fold without a doubt to a reraise all in.

The formula I get for determining the EV of a raise all in on the turn is:

with x=% of Villain's non bluff hands that he will fold to a raise all in on the turn, ranging from 0-.9

EV=(.9-x)(1145(1/4)-485(1-1/4))+(.1+x)(435)

This is pretty similar to the equation given by Little Fishy with a few different numbers thrown in. The 435 instead of his 305 is because I came up with a different size for the pot as mentioned above. The more important difference is that I have 1145 for the total size of the pot that he will win 1 time in 4. Using 485 (the amount Hero has to bet with at that moment) for that portion of the equation excludes what is already in the pot.

So, like HiatusOver mentions we have to compare the EV of this equation to the EV of just calling the turn bet.

Again, ignore AcTc, and assume all of our draws win. This isn't true but if we call the turn sometimes Villain may give up on running a bluff and our nut no pair will win the pot even when we don't hit our draw. So for simplicity we'll assume that those wins cancel out the times we draw close to dead against AcTc or make a flush and lose to a boat. 12 cards win, 34 lose.

12 W (435+130) = 6780
34 L (130) = 4420
6780-4420 = 2360
EV = 2360/46 = 51.3

This assumes no further action. But if we call on the turn and we make our draw and bet out all in villain will sometimes call, especially if we make the gutshot. Here's my equation for that:

x=% of hands that Villain calls an all in on the river with. 10% of his distribution of hands are bluffs so it ranges form 0 - .9

EV=[(34*-130)+12(x)(565+355)+12(.9-x+.1)(565)]/46

If we give Villain credit for playing the river perfect he folds his hand to all scare cards then the EV for the turn call is as mentioned $51.3. Villain folding 16% or more of his legitimate hands to an all in reraise on the turn brings back higher EV.

Here's some more numbers:

Villain folds % - EV
20 - 72.75
25 - 98.625
30 - 124.5
35 - 150.375
40 - 176.25
45 - 202.125
50 - 228
60 - 279.75
70 - 331.5
80 - 383.75
90 - 435

Villain might not play the river perfectly. Here are some numbers from the second equation:

Villain calls river all in % on draw card - EV
5 - 55.87
10 - 60.43
15 - 65
20 - 69.56
25 - 74.13
30 - 78.70
35 - 83.26
40 - 87.83
45 - 92.39
50 - 96.96
60 - 106.87
70 - 115.22
80 - 124.35
90 - 133.48

So, my conclusion is that if you have a player who will bluff in this spot 10% of the time and will lay down about 1/3 of his legitimate betting hands to an all in reraise, then an all in reraise on the turn has more value than a call even against a player who will always play the river wrong. Now I know Harrington says to always calculate a player's chance of bluffing as 10%. However if you have a player that you know would never bluff, or never bluff you, then if he would fold a bit more than 40% of his betting hands (that is, all his hands) then an all in reraise would still be better than a call even if he would call your all in on the draw card with a worse hand every time.

My guess, is that from the way the hand has gone down he will fold to an all in bet on the river scare card a substantial amount of the time. Perhaps calling only 25-30%, certainly calling on the straight card and probably with AK and sets. I also think it is likely he will fold more often than 1 time in 5 to a reraise of 355 so a turn all in play would be my choice against a typical opponent.

Interesting thread.

Regards

raisins

P.S. Where it says EV that should be positive expected value, not total expected value. In other words the amount of profit on the play.

[HiatusOver]

Thanks a lot for the reply raisins, I really appreciate the effort. I am gonna read through again and see if I have any constructive thoughts.