#mathology

In our final introductory EV post, we will summarize the relevant math for the three major types of betting decisions – call, bet and raise. We use the following notation assuming a heads-up situation:

fe = fold equity, the probability your opponent will fold to your bet or raise

eq = card equity, the probability you will win the hand if there is no fold (+1/2 ties)

Pot = the amount in the pot just prior to the betting action

Beth = the amount of Hero’s bet

Betv = the amount of Villain’s bet/raise

Call = the amount of your opponent’s bet/raise that you have to call

Raise = the amount of your raise (excluding the call amount)

Below are the EV equations for each of the major types of Hero betting decisions:

HERO CALL. The pot is 100 and Villain bets 25 which Hero has to call to stay in the hand. He has estimated his showdown equity to be 40%:

               EVcall = eq*(Pot + Betv) - (1-eq)*Call

               = 0.40*(100 + 25) – 0.60*25 = 35

HERO BET. The pot is 100 and Hero makes a ¾ pot semi-bluff bet where his equity is 20%. He estimates fold equity of 50%. Villain’s call amount is the bet amount.

               EVbet = fe*Pot + (1-fe)*[eq*(Pot+Beth) - (1-eq)*Beth]

               = 0.50*100 + 0.50*[0.20(100+75) – 0.80*75] = 37.5

HERO RAISE. The pot is 100 after villain has bet. If Hero bets he has to first call Villain’s bet of 50. He decides to make a raise-of 200. He thinks Villain will fold about 10% of the time. Hero’s equity is estimated to be 30%.

EVraise = fe*Pot + (1-fe)*[eq*(Pot+Raise) - (1-eq)*(Call+Raise)]

= 0.10*100 + 0.90*[0.30 *(100 + 200) – 0.70*(50+200)] = -66.5

We can operate on these equations in various ways to determine, for example, the card or fold equity needed to assure +EV.   The basic approach in most cases is to set the EV value to ≥ 0 and solve for the equity or dollar value that will satisfy the condition. This then will be the critical value that gives a positive Hero EV[1].

To illustrate this, let’s take the simple case of calling an all-in bet and finding the critical minimum equity. For this situation, the EV equation is the following:

EVcall = eq*(Pot + Betv) – (1-eq)*Call

On setting EV to ≥ 0, we have

eq*(Pot+Betv+Call) ≥ Call, or

eq ≥ Call/(Pot+Betv+Call)

Noting that (Pot + Bet)/Call = Pot Odds, if you divide the right-hand numerator and denominator by call, you get

eq ≥ 1/(PotOdds + 1).  This can also be expressed as

eq ≥ Call/Total Pot,

so interpreting equity as your expected share of the total pot, for + EV, the equity has to be at least equal to the fraction of the pot that you just invested.

The following table shows the equation for a number of critical factors to assure +EV calling against one opponent along with the critical value for the example application values. It assumes the call amount is equal to the bet amount.

#For eq < 0.50. All bets are +EV for eq >= 0.50

The first result tells us that for the given chip/dollar values, you need pot odds of at least 3 to 1 to call the bet if your equity is 25%. Since your pot odds are actually (100+60)/60 = 2.67 to 1, calling will be a     -EV decision.  The second result tells us that the minimum equity needed for a +EV decision is 27.3%, higher than the 25% you have, again indicating a call will be -EV.  The table also shows that Villain’s maximum bet for you to have +EV is 50 when Pot = 100, or with a bet of 60, the pot must be at least 120 for a +EV call.

We conclude with three important points. Unlike earlier posts, we did not stipulate that the betting was an all-in situation. Therefore, if not all-in, it’s likely there will be future betting so the results have to be considered as a first-cut analysis. There are ways to consider future bets such as implied odds analysis which we will deal with at some length in the future.

The second point is to note that the card equity estimate should be what we call realized equity, the equity likely to be achieved that accounts for future folds. Equity calculators provide a showdown equity, which assumes there are no folds. However, if Hero or Villain fold, then the realized equity will differ; it will be higher than showdown equity if Villain mistakenly folds more than Hero does or lower otherwise, where a mistake is folding a winning hand. This will also be a future topic. 

The final point is to note that we have assumed that equity and bet sizes have been estimated but we did not go into much detail on how that is done. Such important factors as ranges, position, stack sizes and opponent characterization are involved but to focus on the math we leave that for another time and place. So, we do suggest that in doing the math as outlined here one should then modify the result as applicable for those factors not explicitly considered.

The above text introduces some fundamental aspects of proper Hold’em poker play through EV analysis. It has only scratched the surface but seeing, understanding and implementing the fundamentals is a necessary first step to becoming a winning player.

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We’ll consider here a button steal to win the pot with a zero equity hand by having both blinds fold to a button open-raise. In that case we assume the win amount is 1.5bb, the typical sum of the  forced blind bets. Let fe be the probability both blinds fold and let Bet be the button bet amount. Then the EV equation is

EV = fe*1.5 - (1-fe)*Bet

For determining the maximum bet size to break even (EV=0) given fe is estimated, we have

Bet < 1.5*fe/(1-fe)

For determining the required fold equity (both blinds fold), we have

fe > Bet/(1.5+Bet)

Since fe is for both players folding, the simplest approach is to assume each player folds with probability fe^0.5. Obviously, other combinations of individual fold equities will give the desired combined fe.

The following table provides some results:

Note that the individual fold equities have to be fairly high for all cases even when the button is short-stacked and his bet is only 1bb.

Example: If each player will fold 80% of the time, they both will fold with probability equal to 64%. In that case hero’s breakeven bet size with pure air should be no greater than 2.67bb.

Example: If hero goes all-in with a bet of 5bb, to break even the combined blind fold equity should be at least equal to 77%.

A poster asked if his rough approximation for an EV-based decision was reasonable. He actually used the basic EV equation to see if the EV was greater than zero.  But, you don’t normally have to calculate EV. I think an easier way, or at least an equivalent way, is to see if your equity is greater than the amount you have to contribute to the total pot if you call.

“…  if you held 2 random unpaired cards and the flop of 3 cards is dealt, you will have improved to a pair about 1/3 of the time. Therefore, in a heads-up pot, 1/3 of the time you make a hand, 1/3 of the time your opponent makes a hand and 1/3 of the time no one makes a hand.”

Quoted above is a posting on c-Betting. The author goes on to suggest that one may think by c-betting he can “claim” the hand for the 1/3 of the pots he hits and for the 1/3 of the pots when no one hits.

The above might seem reasonable to some but it is misleading as it doesn’t cover the actual possibilities. It’s true that with no pair you will hit the flop about 1/3 of the time, with a hit being making a made hand . Then, the possibilities for hero and villain are as shown below along with the approximate occurrence probabilities:

Improving on the Flop?

So rather than 1/3 hit and 1/3 no one hits that were incorrectly used IMO, the actual probabilities are 2/9 for only you hit and 4/9 for no one hit. Yes, the sum of 2/3 is the same as the misleading numbers but the ratio between a very good result (you hit and villain doesn’t) and a so-so result (both miss) is significantly different, namely 1-1 vs. 1- 2. Also, the notion that you can claim the hand if neither hits is very optimistic.

In any event, we can say that 2/3 of the time your opponent will not hit on the flop so if you were the aggressor pre-flop having made the last raise then c-betting can be a profitable move but as in so many situations, it depends.

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