Yes, Lawdude is asking a math question.

Here was the specific situation, although what I'm really looking for is some ideas about how to calculate what to do in a range of situations like this.

8-16 at Hustler. Hero is in the BB with 9c9s. MP, a typical 8/16 fish who plays too many hands and also calls down and folds the river a lot, limps, folds to SB, a nit, who raises (this is not an isolation raise-- this is a big hand), I call (because the raise came from the nit and I am usually behind here), MP calls.

Flop is A97 with two hearts. SB bets, I raise, MP calls, SB folds (this was probably KK or QQ), we are heads up.

Turn is the 6 of clubs, I bet, MP calls.

River is the 8 of hearts.

The basic lay of the land is this. I am never folding this hand heads-up, so the only two options are x/c vs. b/c. The question is what the math problem is to determine whether I should x/c or b/c? I imagine it depends on several different things:

a. MP's range.
b. What hands MP will call a bet with.
c. What hands MP will raise with.
d. What hands MP will bet with if I check.

So supposing I can estimate these four things, how do I calculate whether to b/c or x/c?

You also need to split his raising range into hands you beat and hands you don't beat. (Since you say you're not folding this hand he must have hands you beat... like 88?)

Also, would he bluff hands he would otherwise fold if you check?

Count the bets you win subtract the bets you lose vs his whole range(JT is 16 combos), compare the results of each scenario(bet vs check).

Here are the possible end states and pathways (POT refers to pot size in BB prior to any action)

Bet - Raise - You Win -> EVa=(POT + 2BB) x p(Raise given Bet) x p(You Win given Bet/Raise)
Bet - Raise - You Lose -> EVb=(-2BB) x p(Raise given Bet) x p(You Lose given Bet/Raise)
Bet - Call - You Win -> EVc=(POT + 1BB) x p(Call given Bet) x p(You Win given Bet/Call)
Bet - Call - You Lose -> EVd=(-1BB) x p(Call given Bet) x p(You Lose given Bet/Call)
Bet - Fold -> EVe = POT x p(Fold given Bet)

To evaluate the EV for a Bet, you add EVa-e

For a Check,
Check - Bet - You Win -> EVf=(POT + 1BB) x p(Bet given Check) x p(You Win given Check/Bet)
Check - Bet - You Lose -> EVg=(-1BB) x p(Bet given Check) x p(You Lose given Check/Bet)
Check - Check - You Win -> EVh=(POT) x p(Check given Check) x p(You Win given Check/Check)
Check - Check - You Lose -> EVi= Zero

Add EVf-i for EV for a Check.

Pick whichever EV is higher

So values you need are -
POT size (remember, villain could fold, so this must factor in)
p(raise if you bet)
p(you win given bet/raise)
p(call if you bet)
p(you win given bet/call)
p(fold if you bet) - this will be 1-p(raise)-p(call)
p(bet if you check)
p(you win given check/bet)
p(check if you check) - this will be 1-p(bet)
p(you win given check/check)

The p(lose) values are simply 1-p(win) for the correponding state.


More than you hoped for, I'm sure.

Eskimo Sickness' 2:1 theorem imo.

Or maybe it was BPM's 2:1 theorem, I don't remember.

Why not? What if you know for a fact that this opponent is not capable of ever raising the river without the nuts, because he's a real mubsy guy. How small would the pot have to be for you to even consider a fold? Imagine that he's so mubsy that he would not even raise the straight here. I mean, I don't mean to say that b/f is the correct play, I just mean to say why you categorically discount the fold here EVER.

b/c

his raise could be bluffs but also 2 pairs

if you check, he'll check back worse imo, but might bet 2 pair

I'm no real live player, but from what I understand this is a trivial b/f.

Putting this in terms of the number of combinations...

Divide villains range into 5 parts based on your betting
a=number of combos he raised and you won with
b=number of combos he raised and you lost with
c=number of combos he called and you won with
d=number of combos he called and you lost with
e=number of combos he folded with
f=total range=a+b+c+d+e

Then EVbet is
(POT+2BB)(a/f)+(-2BB)(b/f)+(POT+1BB)(c/f)+(-1BB)(d/f)+POT(e/f)


Then divide villain's range into 4 parts based on your checking
g = number of combos he bet and you won with
h = number of combos he bet and you lose with
i = number of combos he checked and you won with
j = number of combos he checked and you lost with
k=g+h+i+j (should equal f above)

EVcheck is
(POT+1BB)(g/k)+(-1BB)(h/k)+POT(i/k)

Beeker, thanks for the math. I'm going to start plugging in numbers from different situations (I have about 6 hands that i transcribed over my cell phone and I'm working on b/c vs x/c as a leak.

Bella, i can't b/f here for much the reason Dragon says. There's an ace on the board and thus there may be some hands that we beat in a typical opponent's value range. Plus-- and this is definitely a leak-- I have a very had time b/f'ing this strong a hand-- I tend to b/f a range of weaker showdown value hands and some bluffs.

That's fine in this case, but think about it this way. At which pot size would b/f become an option? Why?

Well, presumably the smaller the pot size, the lower the odds we are getting on the call of the second bet, correct? So we need to be good more often to b/c rather than b/f.