# The Mathematics of Limit Holdem - Part 2

- Fixed-Limit
- FL

(9 Votes)
6795
## Description

In the second part of the Mathematics of Limit Holdem Leader22 focuses on River HU OOP play, overcalling the river, condidtional probability, conditional EV and calling the river with a player behind.

## Comments (20)

newest first#2

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#4

http://de.pokerstrategy.com/forum/thread.php?threadid=836621

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I also think that the formulas are quite confusing.

He made the assumption that villain will check back every hand that he would fold to a bet, so the formulas are true as we win the same amount, BUT ONLY if Villain never folds a better hand, because then betting wins the pot and check/folding loses it.

This is a very important condition and if it's not the case, the formulas are definitely wrong.

so @Leader22:

I think these assumptions are simplified, first Villain mustn't ever bluff after our check and second Villain mustn't ever valuebet a hand that he would have folded to a bet.

I think that's not vey often the case, it's true that Villain will seldom valuebet a hand that he would have folded against a bet, but it's quite often the case, that Villain bluffs a hand against a check, that he would have folded against a bet. And that needn't to be many hands, as we loose the whole pot by check/folding instead of betting, so there will be huge differences in the EV.

I looked through it one more time and I don't think that your formulas are true.

Th EV of bet/fold and bet/call is wrong I think. The first term is [1-P(raise)] and ismultiplied by our Equity against Villains call. This first term considers the cases, in which Villain doesn't raise, but it doesn't consider the case if Villain folds, because if Villain folds, our EV is 0 as you set the EV to 0. But in your formula, you are not considering that, as 1-P(raise) incoludes Villains folds and that is also multiplied by the second term, but it hast to be multiplied by 0, so the formulas of EV(b/f) and EV(b/c) are wrong I think.

#8

You also sound very bored and just reading what is standing on your papers and that quite fast, I just can't listen to that, I'm sorry, I stopped watching after 8 minutes.

Maybe you can improve that in your next video.

#9

You are correct. I was hoping to simplify the situation by removing that term, but overlooked what you pointed out. I am endeavoring to fix it as we speak.

HamburgmeinePerle,

"as 1-P(raise) incoludes Villains folds"

I tried to explain in the video but it maybe did not come across clearly that P(raise) is the probability of a raise within raising and calling.

#10

EV(c/f) = P(check back) * (Equity vs check back) * pot

EV(c/c) = (Equity vs bet) * (Pot + 1) - (1 – Equity vs bet) + P(check back) * (Equity vs check back) * pot

EV(b/f) = P(call) * [Call Equity * (Pot +1) – (1 – Call Equity)] - P(raise) + P(fold) * pot

EV(b/c) = P(call) * [(Call Equity * (Pot +1) – (1 – Call Equity)] + P(raise) * [Raise Equity * (Pot+2) – 2 * (1 – Raise Equity) + P(fold) * pot

#11

EV(b/c) = P(call) * [(Call Equity * (Pot +1) – (1 – Call Equity)] + P(raise) * [Raise Equity * (Pot+2) – 2 * (1 – Raise Equity)] + P(fold) * pot

Also note that now P(raise) refers to the probability of a raise within all options (call, raise, fold).

Sorry for the inconvenience.

#12

Unfortunately, there is an error in the equations on the slide about playing the river hu oop. The correct formulas follow:

EV(c/f) = P(check back) * (Equity vs check back) * pot

EV(c/c) = (Equity vs bet) * (Pot + 1) - (1 – Equity vs bet) + P(check back) * (Equity vs check back) * pot

EV(b/f) = P(call) * [Call Equity * (Pot +1) – (1 – Call Equity)] - P(raise) + P(fold) * pot

EV(b/c) = P(call) * [(Call Equity * (Pot +1) – (1 – Call Equity)] + P(raise) * [Raise Equity * (Pot+2) – 2 * (1 – Raise Equity)] + P(fold) * pot

Hopefully I will have a chance to fix the commentary/vid shortly.

#13

EV(c/c)=P(bet)*[(Equity vs bet) * (Pot + 1) - (1 – Equity vs bet)] + P(check back) * (Equity vs check back) * pot

Although it didn't work out in the vid I like the Idea of simplification for some special situations. Assume Villain bets the same hands he raises and checks the same hands behind he calls. And assume Villain doesn't fold. If we set EV(c/f)=0 we get (after some calculations):

EV(c/c)=P(bet)*[(Eq bet)*(Pot+2)-1]

EV(b/f)=2*P(call)*(Eq Call)-1

EV(b/c)=EV(b/f)+P(raise)*[(Eq raise)*(Pot+4)-1]

These formulas are simple, but hold only under these assumptions.

#14

I dont agree with your calculation here.

0.3*0.4=12% is wrong, because it does not include that both opponents have the same hand.

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"EV(c/c) = (Equity vs bet) * (Pot + 1) - (1 – Equity vs bet) + P(check back) * (Equity vs check back) * pot" is still worng, isn't it?!

the term P(check back)*(EQ vs check back)*pot is true, these are the cases if Villain check behind, we win the potsize according to our equity.

But the other two terms ar still wrong I think. They both have to be multiplied with P(Villain bets) as Villain doesn't bet to 100%, in the cases in taht he doesn't bet, wedoesn't loose or win anything else than the pot according to our equity.

oh sorry, San Wogi said it already, didn't read it.

#18

Good job! It's very nice now with a clear range and explanations what part of his range Villain's folding, calling raiseng, checking and betting.

Well done!

Did you count the handcombos ourdo you have a program (pokerazor?) for that?

I counted it, it's alright now.

Though, there are a few small mistakes according to the equity we have against the several parts of his range:

we don't have 100% against the range he checks back and the one he calls to our bet (which is the same range), as he also has 1 combo of JJ in his range. So our equity is only 99.1% against his calling range and the range he checks back.

And I got an equity of 12.7% against the range he is betting to our check, this is the following range:

AKs, AJs-A2s, KcJc, KcTc, Kc9c, Kc8c, Kc7c, JcTc, Jc9c, Jc8c, Jc7c, Tc9c, Tc8c, Tc7c, 9c8c, 9c7c, 8c7c, AKo, AJo-A4o (118 combos).

This leads to the following results for our EV:

EV(c/f) = 0.253*0.991*6.25

= 1.56702 BB

EV(c/c) = 0.747*(0.127*(6.25+1)-(1-0.127))

+ 0.253*0.991*6.25

= 1.60269 BB

EV(b/f) = 0.253*(0.991*(6.25+1)-(1-0.991))

- 0.652

+ 0.095*6.25

= 1.75721 BB

EV(b/c) = 0.253*(0.991*(6.25+1)-(1-0.991))

+ 0.652*(0*(6.25+2)-2*(1-0))

+ 0.095*6.25

= 1.10521 BB

these should be the right results.

#19

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In any case, I want to thank HamburgmeinePerle and SanWogi for there assistance. In the future, I think I'll take a more structured approach like I did in the new version to avoid any more misadventures in simplification. Likely this will also add more clarity for some as well.

titze2,

I don't really agree. What you're saying is sort of true in the sense that the likelihood that we beat player B depends on whether we beat player A. However this is very small effect unless the ranges are extremely narrow. If you need a really accurate number though, you could obtain the ranges from pokerazor and then use poker stove to get the exact value. Perhaps I should have gone that route, but in the end I think it makes little difference.

I will try to add slide numbers in the future.

#21