EV is additive

[Imthenewfish]

EDIT:ADDING SUBTITLES BECAUSE IT'S 'HARD TO READ'
GENERAL THOUGHTS:We are always trying to make the most +EV play we can whenever its our turn to act, but our goal isn't really to make the most out of this one play. Our goal in any hand is to maximize the EV of it, which is the combined EV of all of our actions. This can be applied to a lot of situations a few ways. One of the most common is setmining because lolimplied odds. Our play for that street is actually -EV, but playing the hand for set-value can be +EV.

When checked to on any street, we have the option of checking also or betting. When we bet we're trying to get worse to call/fold better/(capitalize on dead money). These are how we gauge the EV of a play, but they don't tell us much about the hand itself. A side-effect of our action is that it affects villain's ranges on future streets, which directly relates to the EV of those street.

OMNOMNOM MANIEZ- We open 99 and get to the frawp HU IP. The flop comes JJ2r and villain checks to us. kay we have this nifty read that we got from playing a lot of hands vs this guy-Villain's c/c range has no Jx(mebe JJ)/22 'cos he thinks that there are a lot of turn cards that will hurt his action.

Okay, 99 on JJ2, flop showdown value, get to showdown, profit amirite??? hmm..

TRYING TO LOOK AT EV'S OF BETTING/CHECKING
Board: Jc Jd 2h
Dead:

equity win tie pots won pots tied
Hand 0: 64.509% 64.13% 00.38% 849474 5022.00 { 99 }
Hand 1: 35.491% 35.11% 00.38% 465102 5022.00 { QQ-22, A9s+, KTs+, QTs+, JTs, T8s+, 97s+, 87s, 76s, 65s, 54s, A9o+, KTo+, QTo+, JTo, T9o }

But let's take a look at his folding range to a bet..

Board: Jc Jd 2h
Dead:
equity win tie pots won pots tied
Hand 0: 76.104% 75.90% 00.20% 622194 1644.00 { 99 }
Hand 1: 23.896% 23.70% 00.20% 194238 1644.00 { AQs+, ATs-A9s, KQs, KTs, QTs, JTs, T8s+, 97s+, 87s, 76s, 65s, 54s, AQo+, ATo-A9o, KQo, KTo, QTo, T9o }
...wait, why do we care about what he's folding? This part of his range has 23.9% equity if we check, and 0% if we bet (he has a lot of oats and theres dead money)

finally equities vs continuing range
Board: Jc Jd 2h
Dead:
equity win tie pots won pots tied
Hand 0: 44.002% 43.34% 00.66% 218820 3348.00 { 99 }
Hand 1: 55.998% 55.33% 00.66% 279384 3348.00 { QQ-22, AJs, KJs, QJs, JTs, AJo, KJo, QJo, JTo }

FUTURE STREETS
so cbetting this flop vs this range is a somewhat -EV for this play (without taking into consideration him folding equity with overcards); except betting here will change the EV of the turn and change the EV of the hand in a way differently than we can see from what we've looked at already.
On the turn his range is a lot weaker and looks mostly like QQ-33. We now have about 70% equity and barrel for value. We can't really analyze it because we have to assume that the turn can be any card, but we guesstimate that on average we have about 60-65% against his continuing range. OWAIT, THE POT CAN NEVER GET SMALLER THROUGHOUT THE HAND, the pot is larger now, and with a larger bet than on the flop it will show even more profit than what we lost from just betting the flop.

FUTURE STREETS IF WE CHECK
If we check flop we still have an option to bet turn and river; betting turn is just as -EV as betting flop though because his continuing range doesn't really have any draws. The point is betting one street has a direct relation on eevees of future streets.
I'M A TARD
IDK im bored and retarded and this probably makes no sense and theres a million things wrong with it but posting it anyway

[Imthenewfish]

note for the number of equity we have against the hands hes folding to be relevant we need to know how many combos there are and how many total combos of hands are in his range on the flop. i just threw it out there to try to show that while the amount we win from checking is pretty much the equity in the hand vs his total range, the amount we win from betting includes the dead money that we pick up so often because we're folding out hands that have equity

[spoonitnow]

Keep trying, you'll make sense eventually.

[!Luck]

I think I know what you are saying. But can't this be summarized by:

When we bet the range of our opponent changes. These change given certain flop textures is going to leave a weaker/stronger range, which we can exploit more/less on further streets.

When we get our opponent to any street with a weak range we generally profit, since I suspect fold equity profit is better than show down profit.

Something like that.

[Imthenewfish]

EV is additive
etc....

[HarleyGuy13]

Keep trying, you'll make sense eventually.

Oh snap....

Thanks for trying!

[JR9477]

So, how do you go about finding the EV of betting opposed to the EV of checking (on the assumption it's going to get checked to showdown)?

[Imthenewfish]

for betting-(equity vs calling range street 1)(pot size street 1)+(fold equity street 1)(pot size street 1)+(equity vs calling range street 2)(pot size street 2)+(fold equity street 2)(pot size street 2),etcetc. I think spoon showed an easier way to show it that i don't remember, but i think this is the same but a lot longer.
ev of checking back is just (equity vs range)(pot size) assuming it's checked to showdown

I left out a lot about how fold equity is just as important as equity edges but i don't really wanna write much more on this