Can someone explain to me how exactly to solve each equation?
In EV Part 1, we discussed the simple case of Hero calling an all-in bet and followed that up with Part 2 where card and pot odds are used to make the call or fold decision. Here, we will consider Hero rather than Villain making an all-in bet against a single opponent. If we know Villain will call, the math is essentially the same but realistically we have to assume he sometimes folds, so the math has to include the possibility of an opponent fold. We will define fold equity, fe, to be this possibility.[1] Here is the applicable EV math


EV of Hero bet = EVfold + EVcall

EVfold = Pr(V folds)*Pot = fe*Pot

EVcall = Pr(V calls)*(eq*(Pot + Bet) – (1-eq)*Bet) = (1-fe)*(eq*(Pot + 2*Bet) - Bet)

EV = fe*Pot + (1-fe)*(eq*(Pot +2* Bet) – Bet)

where

Pot is the pot before Hero bets[2]

Bet is the all-in bet size which Villain must call to see a showdown

Example: With a pot of 100, Hero has a stack of 70 and is considering an all-in bet. He estimates his equity to be about 20% against a fairly tight Villain. Let’s first calculate EV if Villain always calls the all-in bet (fe=0): EVcall = 0.20 * (100+2*70 – 70) = 0.20*240 – 70 = - 22.

If it helps here's the link to the thread: https://holdemmathology.tumblr.com/p...d-value-part-3 im just having trouble breaking down the equation and would really appreciate it if someone can help show me how it is being solved step by step.

I like to break these down into really simple concepts. So, the equations you're using look fine, but whenever I did these calculations I'd just break it down.

So:


Total EV = EV of call + EV of fold

EV of fold is just the frequency that your opponent folds and you win whatever is in the middle.

EV of fold = (frequency fold)*(pot in middle) ---> does not include the bet you're making

EV of call gets sort of broken down into two parts, which you did in your OP. When opponent calls we lose sometimes and we win sometimes. First we need to know how often he is calling which you can write as just (1-frequency of fold) and then we need to know our equity vs. the calling range. So there is two parts and both get multiplied by their frequency, or you can combine the terms. You have how often he calls multiplied by how often you win multiplied by the pot you can win, and then you subtract how often he calls multiplied your bet (this is amount to lose) multiplied by the frequency you lose this bet (1-equity).

In your example where we have 20% equity, then when he calls we win the pot + our bet (this is the portion he is calling, we don't include our bet itself this is why you have 2*70 - 70), so that's the part where we win, then the other part would just be how often we lose. So this would be frequency of getting called * frequency of loss (1-equity) * amount to lose.

So to break it down further:

EV_tot = EV_call + EV_fold

EV_call = (freq of call)*(equity)*(pot to win) - (freq of call)*(1-equity)*(amount to lose)

EV_fold = (1-freq of call)*(pot in middle)

(freq of call) = (1 - fold frequency)
(pot to win) = all money in middle + the bet you're putting out since this is what opponent is calling
(equity) = equity

(1-equity) = basically how often you lose when called
(amount to lose) = your bet or raise amount in that node

Sorry if this is a little long winded, could be a bit more succinct, but hopefully this helps.

so if i had to plug in some numbers how would i plug them in and actually SOLVE the equation? Let's say villain calls about 60% of the time and fold 40% of the time. I try to plug the numbers in and actually solve with the equation but I keep getting numbers that dont make any sense. Would you be able to help guide me through that with hypothetical numbers being plugged in?

sure, just post a hand history or say the situation

so for this equation

EVcall = Pr(V calls)*(eq*(Pot + Bet) – (1-eq)*Bet) = (1-fe)*(eq*(Pot + 2*Bet) - Bet)

Just to my understanding Pr(V calls) = the percentage or probability we think villain calls? and (1-fe) the amount of time he folds?


So we're on the turn and i shove, with what I think ill have as being 20% equity

Pot before our shove= 100
Our shove= 70
Villain to call= 70

Pr(V calls) = probability villain calls

(1-fe) = (1 - fold equity) = Pr(V calls)

Folding frequency + calling frequency = 1 when there is no raising

They would be equivalent. FE would be fold equity or folding frequency
-----------------------------------------------------------------------

If you take the equation I posted above which is more just written out then:

EV_call = (freq of call)*(equity)*(pot to win) - (freq of call)*(1-equity)*(amount to lose)

(freq of call) = 1.00 because you have him never folding

(equity) = 0.20 or 20%

(pot to win) = pot before your bet + the amount he calls = 100 + 70

(amount to lose) = your bet = 70

EV_call = (freq of call)*(equity)*(pot to win) - (freq of call)*(1-equity)*(amount to lose)

EV_call = (1)*(0.20)*(170) - (1)*(1 - 0.20)*70

= 34 - (1)*(0.80)*70

= 34 - 56

= -22

Here is toy game example of your example:

OOP has 77 and always jams

IP has JJ, TT, 99, 88, and 55 and always calls (so played with 77 wins 20% of the time)

Board is 2 2 2 2 3 and action is played on the river

Effective stacks: 70 chips

Starting pot = 100 chips

See image 2 where is says OOP ev = -22 chips.

so yea this makes sense, i guess the challenging part is that when I do input different variables (lets say i put 40% call frequency) and i solve the problem, the EV number doesnt match up with the chart that is presented in the link





Example on the chart it says for 40% FE you will have ev of +26.8...but when solving the equation i dont get that result...am i doing something wrong or looking at it wrong?

Hi
I am the author of the Hold’em Mathology blog. I think Brokenstars did a fine explanation, especially since he didn’t find fault 😊.

Here is the equation of interest.

EV = fe*Pot + (1-fe)*(eq*(Pot +2* Bet) – Bet) :

In the example, eq = 20%, the pot is 100 and you, as hero, bet 70. You then stated with fe = 40%, you don’t get the result 26.8.

Lets do it step by step:

fe*Pot = 0.40*100 = 40

(1-fe) = 0.60

(eq*(Pot +2* Bet) – Bet) = 0.20*(100 + 2 * 70) - 70 = 0.20*240 - 70 = 48 – 70 = -22

Putting it all altogether

EV = 40 +0.60 *(-22) = 40 -13.2 = 26.8

I hope this helps.

interesting so are these numbers specific for this situation? or will EV = the same for every situation of bet-to-pot ratio? because i noticed if i were to bet 700 into a pot of 1000 then 40% fe becomes -EV? or is that wrong?

How about your write out the equation and numbers you're using and we can see where you're going wrong.

Well, lets solve it.

So given variables are:

calling frequency = 60%

equity when called = 20%

pot start = 100

bet = 70

folding frequency = 40%

-------------

EV_tot = EV_call + EV_fold

EV_call = (freq of call)*(equity)*(pot to win) - (freq of call)*(1-equity)*(amount to lose)

EV_fold = (1-freq of call)*(pot in middle)

----------------------------------------

Let's start with EV_fold since it is simple:

EV_fold = (1-freq of call)*(pot in middle)

= (1-0.60)*(100)

= 0.40*100

= 40

-------------------------

EV_call = (freq of call)*(equity)*(pot to win) - (freq of call)*(1-equity)*(amount to lose)

= (0.60)*(0.20)*(100+70) - (0.60)*(1-0.20)*(70)

= 20.4 - 33.6

= - 13.2




EV_tot = EV_fold + EV_call

EV_tot = 40 - 13.2

EV_tot = 26.8

If you're getting an answer of 51.2 you might be mixing up call and folding frequencies. In your last msg you said "(lets say I put 40% call freq)", but then you're referencing 40% folding freq at the bottom.

Also, you can see statman's equation combines some terms to make it easier to plug and chug, but I find that it's harder to visualize that way. It could be that is what is confusing you as well. I always try to break these equations down into their parts. In this situation there are 3 possibilities:

1. Villain folds and you win pot

2. Villain calls and you win pot

3. Villain calls and you lose pot
--------------

He is combining some terms in the (2.) - (3.) part

You can see in my equation (which is equivalent to his), that the EV_call has two parts. One is for when villain calls and you win, one is for when villain calls and you lose. This is just a function of your equity. Equity = how often you win, and 1-equity = how often you lose.

AHHH got it now!! I know where i went wrong, this makes much more sense thank you so much Brokenstars for taking the time out!

That is great news. I am glad I was helpful.

So last question, can this be used on either flop turn and river? or should just be used for one specific street? do numbers change if it used on flop as oppose to river? I would assume not...

also is there a way or equation used to find the FE needed in order to get +EV instead of plugging in numbers we assume villains fe amount would be? like lets say i want to bet X amount into the pot and want to know if it's +EV, can i use a formula to find the FE needed in order to BE? Or did i just have to assume villains FE each time and use the equations listed above in order to see if its +EV/BE?

The EV equation applies for any street, with reservations. As written, it is exact if the bet is all-in, or it can be assumed the hand is checked down, or it will close the action on the river. Otherwise there will likely be future bets or a fold, so its use has to be considered as a first cut look at the situation.

There are ways to include future action such as implied odds or assuming a constant betting pattern but they are much more complicated.



The critical value of any of the elements in the equation can be determined for meeting a target EV. For +EV, the target is 0, so you set the equation to = 0 and solve for the critical value of FE or whatever variable is of interest.

The Holdem Mathology blog you referenced has a number of articles where this is done.

Yes, you would solve the equation for FE, assuming you know your equity when called, when the equation equals 0.

EV_tot = EV_call + EV_fold

EV_call = (freq of call)*(equity)*(pot to win) - (freq of call)*(1-equity)*(amount to lose)

EV_fold = (1-freq of call)*(pot in middle)
--------------------------


Let (freq of call) = 1 - FE where FE = fold equity, then:

EV_tot = EV_call + EV_fold

EV_call = (1 - FE)*(equity)*(pot to win) - FE*(1-equity)*(amount to lose)

EV_fold = FE*(pot in middle)

-------------------------

solve for FE when EV_tot = 0 to get fold equity required to be +EV when taking equity into account:

0 = (1 - FE)*(equity)*(pot to win) - FE*(1-equity)*(amount to lose) + FE*(pot in middle)

-------------------------

statman's equation would be easier for this to solve. There are a few alternatives, though.


1. Assume you have 0 equity and the required fold equity will simplify to your bet/pot (I'd recommend you input equity = 0 in the equation and simplify it down so you can see this)

2. Use online calculator resource found @ https://redchippoker.com/fold-equity-calculator/

3. Use wolfram alpha and just solve for the FE variable making it be x.


To illustrate (3.) given the conditions of:

equity = 20%
pot = 100
bet = 70,

then:

0 = (1 - FE)*(equity)*(pot to win) - (1-FE)*(1-equity)*(amount to lose) + FE*(pot in middle)

0 = (1 - FE)*(0.20)*(170) - (1-FE)*(1-0.20)*(70) + FE*(100)

FE = 11/61 =~ 0.18 which matches the value in the table --> you can see his table that he created for when equity = 20% FE vs. EV, at 0 the value is ~0.18

As statman states here, if you're taking into account future streets or implied actions, then it gets very complicated very quickly and the usefulness more or less dissipates with an increase in complexity. Really, the only thing that is practically useful is just understanding that when equity = 0, required fold equity for +EV on that street is just bet/pot.

Last question for statman hal.... for this scenario....is here raising 200 MORE or 200 TOTAL?

HERO RAISE. The pot is 100 after villain has bet. If Hero bets he has to first call Villain’s bet of 50. He decides to make a raise-of 200. He thinks Villain will fold about 10% of the time. Hero’s equity is estimated to be 30%.

EVraise = fe*Pot + (1-fe)*[eq*(Pot+Raise) - (1-eq)*(Call+Raise)]

= 0.10*100 + 0.90*[0.30 *(100 + 200) – 0.70*(50+200)] = -66.5

The pot = 50. Villain bets 50; pot is now 100. Hero decides to make a raise-Of 200 ; thus his investment involves calling the 50 that villain bet and then investing an additional 200, so his raise-To is 250.

It is not unusual to read poker discussions where the author doesn’t make clear if it is a raise-of or raise-to.

Note that in the call part of the equation, Pot + Raise-of is the amount hero collects if he wins while Call + Raise-of is hero’s investment that he loses if he doesn’t win the hand, assuming no future betting.

ahhh okay that makes sense thank you statman. I am loving the articles!