How do I determine whether it's a good idea to fire a c-bet on the flop, using math?
Or conversely, if I am villain with, say, 78s, I would might call once here if I think the pfr gives up after c-betting once. How do I calculate the money I'm making/losing in this spot?

Put another way, how do I "see the math" behind a c-bet here, if that makes sense.



E.g.
- live 6 handed $200NL game, 100bbs deep
- folds to me in the SB, I raise to $8 with 66, BB calls
- flop is A72


Assuming:
- opponent 3bets AJ+/TT+ preflop
- calls pre with 56o+, 22+, Axo+, 79o+
- folds a 7 to two barrels
- flats with any Ace or flush draw
- raises two pair+ on flop
- folds other hands that missed like 89, JT, QJ, 44 etc

P.S where can I find further information regarding what formulas I could use in order to "do the math" behind decisions on preflop, flop, turn etc etc

Your basically asking “How do I make math based decisions”. If it’s math based, you need a metric, and since you’re presumably playing to make a profit, that makes Expected Value, EV, the clear metric choice.

For most any situation, there is an EV equation. Some exact, such as for an all-in or closing bet, some are a first cut approximation, and some rely on estimates of what future action will take place. Normally one would then adjust a math-based result with factors not directly included in the math such as having an opponent who is likely on tilt or betting to achieve a proper balance or making an opponent indifferent.

There are books and threads on 2P2 that can provide useful information..Check Mathenoobics in the Beginner’s Forum

Thanks - I will check out Mathenoobics as a start

Essentially, I want to know why betting/calling a certain amount is 'correct mathematically' or why I should be betting at all with a certain hand given my opponents range/tendencies.

All this time, I've been betting, calling etc without having a clue as to whether it is a +EV decision (versus their range/s) and thus no mathematical basis behind my decisions.

Consider one of the simpler decisions – calling an all in bet. Your choice is to call or fold. Since the EV of a fold is 0 (your stack will not change), if the EV of a call is positive, then the math says to call. Other factors may then be considered.

The applicable EV equation is

EV= eq*(Pot+2*Bet) – Bet

where

Pot = pot size before opponent bet
Bet = size of the all-in bet
eq = your winning chance

The example in Mathenoobics supposes we’re facing a call for $10 after villain has gone all in. The pot is $30. Let e = the probability that we’ll win the hand if we call.

Is your formula the equivalent to that used in the example?

EV(call) = e x $30 + (1-e) x -$10

Ignore this post. I found the answer

Okay but others may have the same question.

My simple example had villain betting $10 into a pot of $20. The pot here is the amount prior to villain’s bet. The basic EV equation is then

EV = eq*(Pot + Bet) – (1-eq)*Bet (Expected $Win –Expected $Loss)

which reduces to what I showed:

EV = eq*(Pot +2*Bet) – Bet

=0.3*(20+2*10) -10 = 2

The Mathenoobics equation defines Pot as the amount after villian bets:

EV= eq*Pot- (1-eq)*Bet

= 0.3*30 – 0.7-10 = 2

EV analyses should clearly state how pot is defined. Before or after villain bets will make no difference if the proper EV equation and values are used.

Regarding this example of villain betting $10 (going all-in) into a pot of $20 and assuming this was a turn decision..

This is basically calculating the EV of a turn call for the same amount/pot size with more money behind but with the assumption that the river goes check-check, right? Reason for asking is because IIRC the eq in:

EV = eq*(Pot +2*Bet) – Bet

is pulled from, say, PokerStove, which shows our equity vs an opponent's range in a preflop all-in situation.

Poker Stove, Equilab or any typical equity calculator does not do EV. They provide showdown equity estimates for hand or range vs hand or range for a given number of players.

The EV equation for my simple example would be exact for an all in bet or a closing call on the river. Otherwise it is a first cut approximation which basically assumes the hand is checked down after action on the current street. Estimating what may happen on future streets or using an implied odds model can address future action but the EV equation would be much more complex.

Equity estimates but with no betting on any street essentially, is this correct? Just running flop, turns, rivers


I was editing my post to ask about this but the page refreshed/logged me out.

Is there an example of this multi street calculation in our forums that comes to mind?

Download Equilab - it's free. It includes pre-flop


I recall several fairly recent examples but don't remember when or where. Use the 2p2 search function.

To give you one example, here is my EV equation for implied odds, which I won't go into but it should indicate the increase in complexity you have to deal with when considering future bets.

EV =(1-H)(-C$)+ H(C(W(P$+F$)- (1-W)(C$+F$)))+ H(1-C)P$

I don't know if this will be at all helpful, but just to follow-up on statmanhal's excellent posts in this thread (and elsewhere on 2+2), EV is no more and no less than the average number of chips you would accrue, positive or negative and typically relative to your "current" chip position, if you take a specific action (such as call, raise, fold).

The average is taken over all possible future actions by you and all of your opponents, and over all possible outcomes of any future uncertainties (such as turn and/or river cards).

Each branch of every decision point in the game tree essentially has an EV associated with it. This is why the EV calculation is simpler the fewer branches emanate from the decision option in question.

In general, the following statements seem to be true:

- River EV's are simpler to calculate than Turn EV's which are simpler than Flop EV's

- Heads-up EV's are simpler to calculate than multi-person EV's

- All-in EV's are simpler to calculate than EV's which encompass possible future betting.

There are some really excellent posts in this thread.

However, note that there is one major variable in your initial example that was not taken into account: the rake! Specifically at live small stakes NL, the rake is going to devour you in these small pots (especially from the SB where you don't even have a positional edge). Consider that at most casinos in this $16 pot example, they'd probably be taking $2-3 out just to see the flop. So once you see the flop you're actually playing for like $13 or $14 instead of $16. Plus then you have tip another $1 even if you win!

So even if we ignore future streets, your flop c-bet's immediate return would only be like $12 instead of the full pot of $16, when factoring rake and gratuity. So if you bet half pot on the flop ($8/16), then in a vacuum it would need to work 8/(12+8) =40% instead of 8/(16+8) = 33%. Imagine how that difference factors into the range of hands you should be c-betting (obviously fewer).

I would imagine if you looked at a large sample of random blind battles at live 1/2 NL that the biggest winner would be the house. Without turning this into too much of a poker strategy discussion, if we're talking about EV then there are probably more equitable paths by either, A) raising much larger preflop to where the relative rake is smaller, or B) always chopping.