EV: Why should it matter?

A

AviCKter

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In probability theory, the expected value (or expectation) of a random variable, is the long-run average value of repetitions (trails) of the experiment it represents.
The formula for expected value,
EV = x1*p1+x2*p2+x3*p3+....+xk*pk
, where x1, x2, x3, ...,xk is the possible outcomes of the experiments, each occurring with probability p1, p2, p3, ..., pk. Sounds complicated, right? Let me explain with an example.

When we're tossing a coin, the possible outcomes of the experiments are {Heads, Tails}, each with a probability of {1/2, 1/2}; i.e. the coin has a probability of coming heads 50% (1/2 of the time) and probability of tails 50% (1/2 of the time). Simple. In this case, we cannot calculate the EV directly, because we cannot multiple a outcomes with its probability (i.e. we cannot multiple Heads with 1/2 or Tails with 1/2).

Lets take another example, lets say we're rolling a dice, the possible outcomes of the experiments are {1, 2, 3, 4, 5, 6}, each outcome having a probability 1/6. Assuming the fair dice, the EV = 1*1/6+2*1/6+3*1/6+4*1/6+5*1/6+6*1/6 = 3.5.
Now this doesn't mean that in any given trail, we'll get a value of 3.5, simply not possible because 3.5 is not an outcome of the experiment. It simply means that if we were to roll the dice infinite number of times, the average value of the outcome would be close to 3.5.

Great, but how does it translate to gambling situation?
Lets take a simple example, lets say you and your friend have decided to wager $10 on the outcome of a coin toss, if a head comes your friend pays you $10, if a tail comes you pay your friend $10. As we've already seen, the possible outcome for the experiments are {Heads, Tails}, each with probability of 1/2. Remember we couldn't simply find the expectation of that experiment earlier (we were unable to multiple "Heads" or "Tails" with 1/2, a number), but in this experiment we've replaced the outcome with a monetary figure (number). So in case a head comes, you win $10 and if a tail comes, you lose $10. So the outcome of the experiment from your perspective is {+10, -10), this can be multiplied with the probability.
Assuming a fair coin, EV= +10*1/2+(-10)*1/2 = 0
So the expected value of the game is $0, i.e. in the long run (running the experiment infinite number of times), you expect to break-even (no gain, no loss). This doesn't mean that in a particular trial, you'll have a $0 profit, i.e. in each trail either you'll win $10 or lose $10.

You with me so far?

How do we take this concept into poker?
Again lets start with a simple example, lets say you're playing Heads-Up and each of the player has 10 Big Blinds. The SB posts 0.5bb, and you're sitting in the BB, post 1bb, before any card is dealt. Suppose the SB goes all-in and you look at your cards, and you see:ah4::ac4:. As you may or may not know, the best hand against AA is 65s, which has an equity of 23%. Lets say you know for sure that he has :6s4::5s4:, and have already decide to make the call (at least you should), what is your expected value from the game?
EV = 0.77*11 + 0.23*(-9) = +6.4bb , the 11bb that's actually in the pot and the 9bb that you have to invest to make the call.
So in the long run for this experiment you're expected to win +6.4bb when you make the call.

Even still, I know the concept but why should I make these many calculations?
Because poker is a multiple street game & you have a decision to make, and you cannot make a decision without carefully considering the different options (Call/Fold/Raise) in front of you. Now most beginners try to make the decision based on the number of outs they have, i.e. if I have a flush draw, I have 9 outs, which roughly translates into 36%, by the river. But the problem with this is that you're already assuming the opponent's hand, a particular hand or even a small set of hands, like on a board with :10d4::js4::4s4:, holding a :as4::5s4:, you're assuming that he has a J or a 10. Which might or might not be correct, but simplifies the problem. The more advanced players try to range the opponent, give the opponent a particular set of hands and calculate their equity against that range. Now, there's no problem with equities, but it doesn't give us a clear answer to whether we should call, fold or raise. That's where EV comes into play, EVs tell us what action to choose from the multiple options in front of us.

I do understand that its tough to do these kind of calculations on the table, you really need a good brain to be able to make these kind of calculation on the go. And as we know, most of us aren't that fortunate, so what we can do instead is make these kinds of calculation off the table and use it while playing.

The whole point of this post is to make you understand that the game has evolved and the players (opponents/playing field) have become much more knowledgeable (tougher). Now to stay in the competition, you need to be able to stand on the same plank as the competition and the only way you'll ever be able to beat them, unless off course, you're just blessed by the Luck-God/Goddess.

Hope it makes sense.

Disclaimer: All the ideas I've presented is not something new, its been discussed over and over again in multiple materials by multiple authors.
 
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freestocks

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Good, clear writing. I think you're right, opponents are tougher.
 
terryk

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I don`t think you can explain this any better,,,well done.This should help alot of people:D
 
Clasher

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Not a fan of math n calculating n stuffs but will try learning this thanks for the info..
 
terryk

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Not a fan of math n calculating n stuffs but will try learning this thanks for the info..
Me neither,,,only math i use is for counting the $ ;) (but i respect the math guys):)
 
Clasher

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Me neither,,,only math i use is for counting the $ ;) (but i respect the math guys):)
Math play may be good play but im not too interested in this kind of stuff..😀
 
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Rational Madman

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You cannot put in math the likelihood of an opponent engaging with you in a situation as it depends heavily on their cards and emotional state which will fluctuate base don past few hands.

EQ matters more than IQ and bet sizing and EV from a big or small bet matters far more based on the emotional state of the opponent. Opponents on bad-luck tilt will usually fold to you and opponents on euphoria tilt will likely call big bets but some humans are the total opposite and always expect their luck to change which you cannot ever calculate, you can only observe.
 
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rushdaman

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Hey great post with alot of beneficial knowledge thats fairly easy to follow. Cheers.
 
Clasher

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You cannot put in math the likelihood of an opponent engaging with you in a situation as it depends heavily on their cards and emotional state which will fluctuate base don past few hands.

EQ matters more than IQ and bet sizing and EV from a big or small bet matters far more based on the emotional state of the opponent. Opponents on bad-luck tilt will usually fold to you and opponents on euphoria tilt will likely call big bets but some humans are the total opposite and always expect their luck to change which you cannot ever calculate, you can only observe.
Nicely quoted Madman.....
I want to know the meaning of EQ?
 
Alucard

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Really good thread Avi. Thanks again for taking your time to post this.
I think this should be bookmarked or starred.

Question - when you assign ranges & not exact hands for the V to calculate your hand equity, how does it affect & how should you calculate EV?
 
A

AviCKter

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Really good thread Avi. Thanks again for taking your time to post this.
I think this should be bookmarked or starred.

Question - when you assign ranges & not exact hands for the V to calculate your hand equity, how does it affect & how should you calculate EV?

When we calculate the EV in poker, the probability that we're considering, is nothing, but the equity of your hand vs a particular range. Remember the EV formula, x1*p1+x2*p2...+xk*pk, here xi is the outcome and pi is the probability of that outcome. So in poker, xi is nothing but chips (that we stand to win or lose), and pi is the equity. Lets take an example: I'm simply copy-pasting someone's HH (hope its alright with OP)

SB: 286,079 (71.5 bb)
BB: 523,132 (130.8 bb)
MP: 60,168 (15 bb)
Hero (CO): 219,439 (54.9 bb)
BTN: 304,898 (76.2 bb)

Preflop: Hero is CO with A
club4.gif
Q
club4.gif

MP folds, Hero raises to 8,800, BTN raises to 29,600, 2 folds, Hero calls 20,800

Flop: (67,700) 4
heart4.gif
T
club4.gif
5
club4.gif
(2 players)
Hero checks, BTN bets 32,000, Hero raises to 189,339 and is all-in, BTN calls 157,339

As you see in this hand, the Hero starts with 55bb, and calls a 3-bet OOP. The flop comes :4h4::10c4::5c4: and the pot is 67700. And he has to decide how to proceed.
So in this case, he checks to the Villain, which is a standard line. And the Villain bets 32000 in the pot. Now the Hero has to decide what he's going to do. Should he call here or should he re-raise (if he re-raises, to how much value) or should he just fold?

I assigned the Villain a no-folding range {99+, AK}, i.e. the Villain is never folding to whatever action you take. Against that range, you have an equity of 46.59% on that flop, i.e. 46.59% time you'll win with your hand against the Villain's range and 53.41% time you'll lose.

So lets get into the EV calculations for the different actions (Fold, Call or Raise):

EV(fold) = 0
Why is that? Because you're simply folding your hand without putting any more chips in the middle. So the xi value is 0 and even if we multiply any number with 0, the calculation will simple turn out to be 0.

EV(call)=0.4659*99700-0.5341*32000 = +29359.03
Here the x1 = 99700, which is the chips in the pot when we're to take any action,
x2 = 32000, the amount of chip we need to make the call
p1 = 46.59%, that we found out by using some equity calculator, putting our hand vs his range {99+, AK}; that's the chance of our hand winning at showdown (river)
p2= 53.41%, that's his chance of winning at showdown

EV(check-raise all-in) = 0.4659*257039-0.5341*189339 = +18628.5102
Here, the pot size that we expect to win was, x1 = 67700+32000+157339 = 257039 (157339? Since we're going all-in for the rest of our chip and that's the amount he needs to make the call as he's already put in 32000 in the pot)
x2 = 189339 (that's the stack we we're left with after the flop)
p1 = 46.59%, chance of us winning against his range
p2 = 53.41%, chance of him winning against our hand

Off course this equation is simple, but there's one more consideration, we were assigning a no-fold range to the Villain, where in actuality there must be some part of his range which would c-bet/fold. So the complete range of Villain might look something like {22+, A2s+, A9o+, KTs+, KJo+, QJs, JTs} with which he made the play on the flop, of which he'll fold a certain % of hands. As you can see before any other calculation, the equity against this range with our holding AQ would be different than the previous case where we were assigning a range of {99+, AK}. So that fold-range of our Villain has to be taken into account.

In reality, the actual EV calculation in poker =
(%time Villain folds)*(chips we gain when Villain folds)+(%time Villain calls)*(Total pot * %chance of winning - Chips we stand to lose * %chance of losing)

chips we stand to lose = Chips we made the play with (either call, re-raised or gone all-in)

I just calculated three lines of action, there's one other line too. Given the stack depth, Check-Raise (to a certain bet) and that can be calculated too, for the completeness of the solution.
 
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AviCKter

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You cannot put in math the likelihood of an opponent engaging with you in a situation as it depends heavily on their cards and emotional state which will fluctuate base don past few hands.

EQ matters more than IQ and bet sizing and EV from a big or small bet matters far more based on the emotional state of the opponent. Opponents on bad-luck tilt will usually fold to you and opponents on euphoria tilt will likely call big bets but some humans are the total opposite and always expect their luck to change which you cannot ever calculate, you can only observe.

The only problem with this thought process is this: You're assuming that the opponents are going be emotional about the decisions they make, rather than making a rational choice. If you think they cry after losing a tournament or two, you've no clue.

You got to understand this, for these young prodigies, the wizards, the soul-crushers, the new age poker players, it's just a job. And you don't mix work with your personal emotions, do you?
 
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AlexTheOwl

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Avi,

You're doing a nice job of explaining this. Do you normally do the calculations this way, or do you use a software tool to calculate the EV?

Madman,

Some players tighten up after a big loss, others play like maniacs. If you have evidence of how this specific player reacts, that evidence can certainly be part of your decision.
The players who have had thousands of big losses usually just carry on playing the way they normally do.
 
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A

AviCKter

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Avi,

You're doing a nice job of explaining this. Do you normally do the calculations this way, or do you use a software tool to calculate the EV?

Madman,

Some players tighten up after a big loss, others play like maniacs. If you have evidence of how this specific player reacts, that evidence can certainly be part of your decision.
The players who have had thousands of big losses usually just carry on playing the way they normally do.

Initially, I used to do these calculations on paper, using calculator. But once I got a hang of what I was doing, I started using Microsoft Excel for doing these, all I have to do is insert the value (equity, pot size, bet size) & my predefined calculation (the equation that I put down on my 2nd post) on the sheets gives me an answer.
 
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I don't understand the point of the post.
Are you saying to play against player hand ranges and to not play against the board?
 
A

AviCKter

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I don't understand the point of the post.
Are you saying to play against player hand ranges and to not play against the board?

Oh, no. No, what I'm saying is that there are concepts that you need to be aware of, esp. while playing online, in this case that concept is, Expected Value.

Two Reasons for learning it:
1. In poker you'll have to make decisions at each point (pre-flop, flop, turn, river) and choose from one of the actions (fold, call, raise, etc), so the question is how do you choose? Shouldn't we be making a comparison between all the different options before making a decision to go with one option vs the other? Doesn't that sound logical? That's where EV comes into play.
2. The opponents 2-3 or 5 years back were not as tough as they're today and the field keeps getting tougher and tougher each year, partially because these players are learning too hard. If you want to be beat them, then you got to understand how they approach the game. And most often then not, they're approaching it from a mathematical perspective, which gives them a clear idea of what each spot looks like or what needs to be done on what kind of situation.
 
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darthdimsky

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Killer OP and follow up with example.
 
akmost

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Nice post , well written. Apparently no one can use those calculations the moment they are playing but those who can, they do it easily with a good approach on villains' range.

You should have a good grasp of the EV thing in order to mix up your game and be a tough to read opponent. ie bet strong your draws etc.

You will help a lot of players understand how to think and how to approach the game of poker more than the basic way!

:)
 
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