I'm trying to study material around the internet regarding the EV but I find many examples that I think are written bad or wrong

Example

Suppose there is a $ 0.50 / 1.00 game where the UTG rises to $ 3, the CO 3 bet to $ 10, the Hero 4 bet to $ 22 from the button, the UTG fold and the CO pushes. If CO had an initial stack of $ 75, then what is the EV to call if we expect to have 38% equity?

Remember our process: let's start by establishing the possible results of the call. In this case, we can say that either we will win or we will lose. We must not make up for the ties because this is due to our heritage. Next, we must establish how often each of the possible results will occur. We will win 38% of the time and we will lose 62% of the time. Finally, we must find the profit of each of the possible results. If we win, then we earn the $ 75 stack of the CO, the $ 1.50 in blind, the $ 3 raise from the UTG player and our original $ 22 4-bet. If we lose, we lose our $ 53 call. Here is the equation we get with this information:

EV of Calling = (possibility of winning) (profit of winning) + (possibility of losing) (profit of losing)
Calling EV = (0.38) (101.5) + (0.62) (- 53)
Calling EV = $ 38.57 - $ 32.86
Calling EV = $ 5.71

In this example for the calculation of the ev in the scenario where we win in the "profit" that we will take into account the 22 $ used for our 4bet which does not make sense otherwise we will have to consider the 53 $ we add later, it is not correct to consider $ 22 together with $ 53 in the scenario where we do not make a profit?

it would therefore be

EV of Calling = (possibility of winning) (profit of winning) + (possibility of losing) (profit of losing)
Calling EV = (0.38) (79.5) + (0.62) (- 75)
Calling EV = $ 30.21 - $ 46.5
Calling EV = $ -16.29

is correct?


Sorry for my english but i'm an italian guys =)

The short answer is the money you previously invested does not matter at the current decision point. It is similar to the idea of opportunity cost in economics.

The long answer is that the overall EV of your line from the first decision point to the last decision point IS influenced by your previous investment. However, if you tried to evaluate this you would have to add more probabilities to your EV equation, and some of those would be dependent on things like equity or conditional on other things happening (i.e. there is a bet then call, bet then raise, bet then fold, bet then cal then bet on the next street etc). It gets messy really quickly to solve analytically.

If you are familiar with trees or graphs in computer programming you can think of your initial calculation as calculating the final node at the bottom of a decisio tree that followed the action you provided. Everything in the tree graph befor this had an EV of some kind but you just want to know the EV of the node right before the end of the hand. To calculate the EV of the entire hand, you'd start at the top of the tree and add up the EV's you hit along the path to the bottom of the tree. When you follow the full path, that is when your previous investment would get counted.

sorry but I didn't understand your answer

I place you another scenario

We are heads-up and out of position on the turn in a no-limit hold’em cash game hand. We hold JsTs on a board of As4h5d9s. The pot is currently $28 with $81 in our stack, and our opponent barely has us covered. We bet $21 on the turn. If our opponent raises us on the turn, then we fold our hand. Our opponent raises us 20 percent of the time, and our opponent folds 20 percent of the time.

The other 60 percent of the time, our opponent calls. There are nine spades left in the deck with 46 cards left to come, so the chance of a spade coming is 9/46. If a spade does not come, then we always check/fold with no chance of winning the pot even if it checks through. If a spade does come, then we go all-in for our last $60 on the river. Our opponent will fold 60 percent of the time. A total of 45 percent of the time, we will be called and win, but 5 percent of the time, we will be called and lose.

The Process of Organization

This is a much more complicated scenario than anything that we have looked at so far before. However, the math part of it is particularly easy to do if you have followed through the previous three installments of this series. The hardest part is organizing your answers. The first thing we have to do, like always, is make a list of all of the possible outcomes of this scenario.

We bet on the turn, our opponent folds.
We bet on the turn, our opponent raises, we fold.
We bet on the turn, our opponent calls, a non-spade comes on the river, we check/fold.
We bet on the turn, our opponent calls, a spade comes on the river, we bet, our opponent folds.
We bet on the turn, our opponent calls, a spade comes on the river, we bet, our opponent calls, we win at showdown.
We bet on the turn, our opponent calls, a spade comes on the river, we bet, our opponent calls, we lose at showdown.


The key points are scenarios 5 and 6

Outcome 5

Again, we’re called on the turn 60 percent, a spade comes (9/46) of the time, and we’re called in a situation where we win at showdown 45 percent of the time. That makes the chance of this outcome occurring (0.60)(9/46)(0.45). Our profit in this scenario is the $28 turn pot plus the $21 turn bet that our opponent calls plus the $60 river bet that our opponent calls for a total of $109. So our EV for the fifth possible outcome is (0.60)(9/46)(0.45)($109) = $5.76.

Outcome 6

Same Bat time. Same Bat channel. Our chance of this outcome happening is (0.60)(9/46)(0.05). Our profit is found by combining the bet we lose on the turn ($21) plus the bet we lose on the river ($60) to get a profit of -$81. Our EV in the sixth outcome is (0.60)(9/46)(0.05)(-$81) = -$0.48.

In scenario 5 our $ 21 bet on the turn is not counted in the case of a win but only considers the 28 of the pot + 21 of villains + 60 bet river when calling for a total of 109.

In scenario 6 it counts in the total all the episodes that we lose in case we are called and oppo wins at the showdown

Why in the scenario that I showed you before does it differently?

It's different because you are evaluating 2 different decision points in the hand.

In the first scenario provided you are only deciding if calling is profitable AFTER you have already decided you're going to 4 bet and villain will shove.

In the second scenario you are on the turn and deciding whether to bet the turn or not so you need to account for betting the turn in the EV equations.

If you go back to your first scenario, you could choose to decide to try to perform the EV calculation at any time during the betting and raising instead of just the call at the end.

What you would end up with is a long list of different potential outcomes and then you would need to track the money that flows into the pot at each outcome from the decision point you start at to the end of the hand.

so I have two scenarios if they are correct?

In your first post you have 3 outcomes, call and win, call and lose, or fold.

Folding isn't interesting because it's 0 EV at that point.

The probability of calling and winning and calling and losing is equity in the first example.

Tell me if I misunderstood.

1 case since I just want to calculate the EV of the call to its all in and not the whole EV of the play itself if the calculation is correct in that way because my calculation is made only on the money I have to put to call the 'allin. So it is fair to consider the 4bet episode as money already present in the pot.

While the second case refers to the example you made of the tree since my ev calculation takes all the roads from the turn to the river into consideration.

Tell me if it is clear what I wrote because I do not know
how to explain to me well but I think I understand

Yes I would say you understand.

Another way to look at it is when you or your opponent are free to make a choice in the future, you need to account for the money that goes into the pot for that choice.

If you and your opponent made a decision in the past the money is locked into the pot and the EV of that last decision is already decided/set.

sorry but i do not understand