For simplicities sake say I have intel on a player and I know that he double barrel cbets 99 times out of 100. Now say I have a 10-1 nut draw on this guy on the flop but I know if I miss my money card I’ll have to fold to his 2nd barrel.

So if he cbets half the pot (3-1) on the flop I know I’ll have to get 7 more bets out of his stack to justify a call vs a fold. Let’s say I call and I hit my nut card but he checks that 1 time in 100. Will I still need to extract 7 more bets from his stack OR closer to 2 more bets because after all I did get two opportunities (free card) this unique time?

It doesn’t matter if he bets the turn like he does 99% of the time or if it’s that single time he checks, assuming his check/bet decision is random.

We don’t know which card will come and what he does. Therefore we use our best expectations. If he bets $100 99% of the time and $0 1% of the time, his average bet size on the turn is $99.

Not different from us calling all-in with AA preflop even though we know that our opponent will beat us sometimes. On average, that play is still profitable.

ok so we still do need to get a fraction under 7 more bets off him then?

for simplicity reasons, I would start with looking at a spot where villain bets the turn and we have to find out how much we need to win on the river to make the call profitable. If you start looking from flop to turn, there's another street coming after so the results won't be as meaningful, especially if you're accounting for situations that are not 1 or 0.

So our example might be this: We have air right now (and won't ever bluff the river) but have a 10% shot of making the nuts on the river. There's $100 in the pot and villain bet's $50, offering us odds of 3:1 to call.

So we need to pay $50 now in order to win a final pot of $x 10% of the time. The other 90% we lose the hand.e win $50 on average. Pot on after our turn call is $100 + $50 + $50 = $200.

So how much do we have to win on the river to make the turn call profitable?

What is your decision point? As I understand it, it is calling or folding villain’s flop bet of half pot. So this is the situation where $ or chips is in pot size units for pot = amount prior to villain c-bet = 1.0:

Pot = 1.0 Villain bets ½ Pot. Pot = 1.5. Hero call amount = 0.5. Hero card equity: 1/11 (10 to 1 against)

EV fold = 0

EV call = (1/11)*(0.99 *(X+1.5) +.01*1.5) – (10/11) *0.5

Explanation You hit a winner with prob. = 1/11. Villain bets turn 99% of the time, with bet = X. So if you hit the nut card (1/11), you win X + pot = X+1.5. 1% of time villain checks so you bet, villain folds and you win the pot of 1.5. If you don’t hit (10/11) you fold and you lose the flop call of half-pot = 0.5. Setting EV to 0, X= 3.5.

I don’t know how you got 7 more Bets, but you haven’t defined the term Bets and maybe my interpretation of the situation is wrong.

I haven't done the maths for this example, but just generally speaking, you shouldn't be calling pre or floating the flop light with hands that usually need to see all 5 cards to make a strong holding if your opponent is very aggressive. With hands like 86s (random example), a lot of your EV comes from taking free cards or by having cheap bluffing opportunities (e.g. when villain is one-and-done and check-folds the turn often). Vs an aggro player, you won't get the chance to realize your equity cheaply, since he'll make big bets and blow you off the pot.
Like in the old days you could pot TPTK vs a whale and usually get 2 streets from his gutshot, and then sometimes get another bet called if binks a useless third pair on the river. In today's games, people bet a bit smaller and give you more opportunities to take the pot away. The opponent you described doesn't sound like the type that will let you do that, so you're basically playing solely to hit your hand... and you're not getting the right price. It would be much trickier/closer if you said he only bets the turn about 50% of the time.

stat to answer your question. i need 10-1 and he offers 3-1 so i need the difference when i hit. i.e 10 minus (his half pot bet) 3 = 7 more units/bets required.

no wait i misinterpreted your reply. you are suggesting that we include the assumed avg bet on the next street (even though this one time it was specifically $0) which means i will only need to extract close enough to 5 more bets/units on the river to justify the flop call vs a fold.

thanks madlex you're totally right here. I know it's difficult to grasp the idea of what i'm talking about.

And arty youre right in assuming the average joe blow will cbet the turn half the time so in this instance if we prepare a decision based on that input it would look something like this thought process:

i need 10 to 1 he is giving me 3 to 1, on the turn he will give me on average half of a half pot bet ( $50 a.k.a 1 more unit) so on the river we will need to extract 6 more units in total. Holy **** this is a light bulb moment for me. Next level stuff

but this gets confusing because do i really need 10-1 (gutshot) if i will get a free card half the time. Perhaps instead of 10-1 it's 7.5-1 because of this reason. damnit i'm confused again. i think the updated revised thought process now looks like this

i need 7.5-1 and i'm getting 3-1, i will get 1 more unit on the turn so i will need 3.5 more units on the river