Hey! I played this hand and was unsure what my worst calling hand should be here, I thought I could probably overfold quite a bit cause he should have more Qx then me, but I want to be able to calculate it more precisely.

GTECH G2 (Boss) - (5 max) - Holdem - 4 players

BB: 208.62 BB
CO: 133.78 BB
BTN: 338.85 BB
Hero (SB): 225.84 BB

Hero posts SB 0.5 BB, BB posts BB 1 BB

Pre Flop: (pot: 1.5 BB) Hero has X X

fold, fold, Hero raises to 3 BB, BB calls 2 BB

Flop: (6 BB, 2 players) T 9 K
Hero bets 4.5 BB, BB calls 4.5 BB

Turn: (15 BB, 2 players) 6
Hero bets 11.25 BB, BB calls 11.25 BB

River: (37.5 BB, 2 players) J
Hero checks, BB bets 23 BB, Hero calls 23 BB

Let's say villain has this river betting range:
57 combos total: 41 Value / 16 Bluff


This is my river range after checking:
89 combos total


I guess my question is how many combos I need to call to make it the most EV+?

You are getting 60,5:23 on a call so you need to win 27,5% for it to be better than a fold.

Every individual hand in your range needs to have 27,5% chance or more to be a call, there isn't an optimum in this case because you are polarized between calling and folding. The EV of those are fixed for each hand.

Assuming no rake you should be calling with all of your hands that beat a bluff in this scenario because given the bet size by villain you need 27.5 % equity to call and all of your hands that beat villain's bluffs have 28% equity vs villain's range.

With rake you're going to want to fold all of your hands that only beat a bluff, but you can call with any hands that can beat some of villain's value range and his bluffs.

The formula for equity you need for calling a bet B into a pot of size P is:

B/(2B+P)

If you want to adjust for rake, R, just subtract it from the denominator like so:

B/(2B+P-R)

So in your example let's say rake is 5BB then the equity required for us to call would be:

23/(2*23+37.5-5) = 23/(78.5) = ~29.3% equity.

Now we can calculate how many value combos, V, we'll need to beat for a break even call:

(16+V)/57=.293
16+V=.293*57
V = .293*57-16 = .701

So you'll probably need your hand to beat at least 1 value combo to compensate for any rake. If all of your hands beat at least 1 value combo for villain then you can call with all of them.

Assuming the numbers are so close because of the way you set up the scenario and you couldn't get villain's range exactly at 27.5% bluffs using combinations alone.

Thank you both for your answers, I've reread them a few times but not completely sure I understand yet.

With no rake I should basically call my entire range? I thought I should never defend more than 62% of my range here given his betsize, but pott odds and his v-bet/bluff ratio goes before and makes it so I can call anything? This is just because he was off the perfect ratio with 0.5% more bluffs?

With rake I should only call Qx? Or all straights?

What is the 0.701 number? It should not be .293*(57-16) = 12 combos?

So with no rake, any hand you call needs 27.5% equity to call (23/(23*2+37.5) = .275 × 100 = 27.5%)

However, if you absolutely know villain's range is 41 value hands and 16 bluffs (and all of your hands beat his bluffs) then any hand that beats a bluff has 16/57 =.28 × 100 = 28% equity vs villain's range so you should always be calling with hands that beat a bluff. You can double check with a standard EV calc.

EV call with hands that beat a bluff:

.28*(46+37.5) - 23 = 23.38 - 23 = +.38 BB.

Yes that's right. If villain's range gave you exactly 27.5% equity you would need to defend with your bluff catchers at an exact frequency to prevent him from exploiting you from folding too much or paying off valuebets too much. Because villain is over bluffing here you can call with all of your hands that beat a bluff. If he was under bluffing then you would fold them all. This is all without considering rake, bc that is money that leaves the pot.

When rake is taken into consideration you need to beat more hands than bluffs to compensate for the rake being taken out.

With the rake you need some of your hands to not only beat villain's bluffs, but also some of his valuebets.


.701 is the number of value hands we need to bet in addition to bluffs to compensate for losing money to the rake.

Intuitively the above results should seem off to you. The equity required to call only increased by 1.8% so we shouldn't have to compensate for that much money leaving the pot, meaning we shouldn't have to beat an additional 12 combos to break even.

We need to beat 29.3% of villain's entire range to break even (.293*57 = 16.701) but we already beat 16 combos so we only need to beat .701 (i.e. 1) more combo to break even with rake.

Thanks for taking the time, understand some things a bit better now at least.

In this scenario I have no hands that beat his v-bets, only splits. How does that affect things? Have to be better calling those than folding. We have 6 combos to his 41 that splits the pot.

The stake I play on has a cap at 1.5BB.

23/(2*23+37.5-1.5) = 23/(82) = ~28.05% equity

.2805*57= 15.99 combos

So there I could actually call my entire range as long as we beat all his bluffs?

Yes. If you split 6/41 value hands that will benefit you, but it looks like villain is bluffing just enough even considering rake to make this a call with all your hands that beat a bluff but it's only 1/100th of a BB and I'm sure there is error in there.

So considering the potential for error we'll assume villain's betting with exactly the frequency required to force you to defend properly to make villain's bluffs indifferent and prevent villain from over bluffing you.

I am ignoring ties for this but keep in mind ties always mean you have 50% equity in the pot and if there is dead money in the pot there is no betsize that you would be incorrect in calling with 50% equity, so having ties is going to benefit you when defending.

So for villain's bluffs to be profitable (i.e. EV > 0) villain needs you to fold with frequency B/B+P where B is the betsize he chooses and P is the money in the pot when he chooses to bet.

So in our scenario, B = 23 and P = 37.5 so

23/(60.5) = ~.38 × 100 = 38%

So if your range is only bluff catchers and if you fold more than 38% of the time villain's bluffs have a positive expectation from a bet. If you fold less, you will be paying too much to villain's value hands to prevent him from bluffing. If you fold at exactly the correct frequency then these villain's bluffs make no more additional EV for villain and you don't pay too much to villain's value hands to prevent him from bluffing, and villain's expectation is fixed at P.