Poker Math Question - Implied odds EV on the turn w/river bluffs
 

If you have a situation where you are facing a bet on the turn with money behind, and you will only put further money in on the river if you hit your nut outs, it seems like calculating EV is as follows:

1) Size of pot on river not including any additional money hero puts in if villain can be assumed to put in additional money * chance of hero hitting outs * probability that V will put in further money on river
2) Size of pot on turn including our call * chance of hero hitting outs * probability V will put in no additional money if hero hits
1 + 2 = Implied EV of calling the turn with no intention of bluffing?

But now if we include a river bluff component where we will sometimes bet the river with a missed draw, so we actually will sometimes lose the river bet, what should that math look like? I imagine we've got to do a fancy matrix including the percent chance that we will bluff, and calculate negative EVs on those points in the matrix where hero bluffs the river and villain calls?

If there's a nice thread that walks through this stuff, please let me know what to search on, or throw me a link?

Thanks

Re: Poker Math Question - Implied odds EV on the turn w/river bluffs
 

there isnt a nice easy thread once u include the bluff component (or even without it). Though there doesn't need to be as this calculation is unneeded. If the bluffs u make on the river are +EV generally, which you assume or you wouldnt make it (even if you sometimes get called and lose), you can ignore that component with your turn calculation. If you make -EV bluffs after your turn calls, dont bother calculating the EV of the total line, just work on plugging that leak in your river play.

the reason theres not a nice easy thread even without this added possibility (bluffing the river) is because implied odds are tricky to estimate, (you need to know frequencies how often you win x1, x2.... xN from him when you hit or lose x when u flush up but the board pairs and hes got a boat).

Generally you should assume implied odds are better in position than out of position as you have more information about your opponents actions then ( less chance it checks down and no money goes in on the river when you hit, esp with flush draws which people are generally weary of)... so actually implied odds are a bit better with straight draws in NL as well.

estimating this vs any one opponent any one hand is difficult (its the art aspect involved in the science or math of poker, making assumptions to apply the math), sometimes with stack sizes you assume they ll be pot committed to call an all in on the river (or u know they suck and always stack off once theyve put in any decent amount of money and get to the river) and you can count his whole stack. But usually you should estimate conservatively, if calling the turn and then getting villains whole stack on the river isnt much better than break even pot odds you should fold, if expressed odds are kinda close (3:1 2.5:1) and the guys got a PSB for the river behind or ur getting 2:1 or less but ur effective stacks are deep and your drawing to the nuts you should pretty much always call.

Re: Poker Math Question - Implied odds EV on the turn w/river bluffs
 

Thanks for the advice. Let's simplify the example.

Assume we've got a limit hold'em game, and we are HU on the turn with 4BB in the pot. We have 1.5BB behind, and villain has the same. We have 8 winning outs and plan to bet any additional scare card. We assume villain will call about 50% of the time on any scare card, whether or not we actually have a made hand.

What's the math for the expected value here?

Re: Poker Math Question - Implied odds EV on the turn w/river bluffs
 

let's say

P = pot size on turn, including villian's bet
TB = how much you have to call on turn (eg villian's turn bet size)
RB = how much you will bet on river (let's assume you bet the same size whether you're bluffing or not)

XC = probability villian will call river bet
XB = probability that we bluff on a missed draw
X = probability of our draw coming in

Then expected value of calling the turn bet is:

EV = X(P + RB*XC) + (1-X)*(XB*(XC*(-TB-RB) + (1-XC)*(P)) + (1-XB)*(-TB))

kind of a long formula but it makes sense. If we make all bets pot sized, then TB = P and RB = 3P, and the formula reduces to

EV = X*P*(1+3*XC) + (1-X)*P*(XB*(2-5*XC) - 1)

Re: Poker Math Question - Implied odds EV on the turn w/river bluffs
 

Is stuff like this covered in The mathematics of poker, Bill Chen? If not, where might I find stuff like: "how often does your opponent have to fold to make this play profitable?" etc..

Re: Poker Math Question - Implied odds EV on the turn w/river bluffs
 

[ QUOTE ]
let's say

P = pot size on turn, including villian's bet


[/ QUOTE ]

That's exactly what I was looking for thanks!