A Mathematical Reflection on Draws

Introduction

In this article

• strong draws and weak draws
  
• busted draws
  
• refining your sense of the game
  

If you've read the articles in the Gold Section attentively, you already know how to play strong and weak draws, as well as how to act when holding a busted draw on the river. Semi-bluffs, bluff bets, balancing and free cards aren't new concepts, either.

This article will provide you with an abstract reflection on playing drawing hands. A purely mathematical reflection. It will help you up your NL game another step and refine your general sense for the game.

Definition

Before we begin, you need to know the exact definition of a draw. You can find it to the right.

From a mathematical standpoint, you have a drawing hand any time your pot equity is greater than 0%.

There is no maximal pot equity, which, once reached, no longer allows us to speak of a draw.

You may intuitively think you can no longer speak of a draw when a hand's pot equity is greater than the average, but this is not correct.

Take a look at the following examples:

Player 1: 7, 7
Player 2: 7, 5
Player 3: A, K

Flop: 2, 3, 4

Equity distribution:
Player 1 (7, 7): 27.461%
Player 2 (7, 5): 22.595%
Player 3 (A, K): 49.943%

As you can see, Player 1's equity is less than average (33%), even though he would win the showdown at the moment. His hand does not meet the definition of a draw from above, but would be considered so looking solely at the pot equities.

Now look at Player 3. He would not win the showdown at the moment, even though his equity is above average. And he clearly has a draw on the flop.

This example may seem a bit contrived, but a similar situation can arise in a heads up confrontation:

Player 1 (K, Q):Equity: 72.322%
Player 2 (2, 2): Equity: 27.678%

Flop: J, T, 5

Both examples clearly show that there is no amount of pot equity, which, once reached, no longer allows a hand to be defined as a draw.

The importance of draws in No Limit Hold'em

The fact that NL Hold'em is a drawing game is what makes poker so exciting. In fact, most of the various forms of poker are drawing games, in which you draw for a hand. The only popular form that does not revolve around drawing is Chinese poker.

The nature of the game as a drawing game gives rise to the concept of pot equity, which takes the entire game in a whole new direction. It can be mathematically correct to pay x with a weaker hand, when the amount of dead money in the pot is y.

You can (re)act based on future events that can shift the balance in hand strength among the players. The concepts of implied odds and reverse implied odds are based solely on this.

You can clearly see that draws are the most important part of the game, as they can change the balance in hand strengths through the course of the hand and allow you to adjust your play according to the new variables that arise. This is where the skill of the player becomes of importance.

Understanding the concepts to come means understanding the game of poker.

The redraw

Equity tells you the probability (in %) of winning the showdown with hand range x against hand range y. The ranges x and y must be comprised of at least one hand.

The Equilator program can simulate a situation and give you a concrete (and nearly exact) equity value within seconds.

You can call a bet even as an underdog if you are getting the right pot odds because of your hand's equity. You learned about these concepts as a Bronze member and know how to calculate the equity of various hands based on outs.

This question has not been addressed in any article so far and will now be examined from a mathematical standpoint.

Player 1: A, K
Player 2: Q, J

Board: K, 7, 2

Equity distribution:
Player 1 (A, K): 61.4%
Player 2 (Q, J): 38.6%

Player 2 can win the hand with a flush, two pair, trips or a straight.

Player 1: K, K
Player 2: Q, J

Board: K, 7, 2

Equity distribution:
Player 1 (K, K): 73.5%
Player 2 (Q, J): 26.5%

In this example, Player 1 has a redraw for quads and a full house. This is seen by the large shift in equity compared to Example 3. This jump in equity cannot, however, be traced back solely to Player 1's new redraw; Player 2 now has a completely different draw, as well.

In Example 3, aside from this flush draw, Player 2 could win with runner runner two pair or trips. This is not possible in Example 4. This is why Player 1's jump in equity cannot be traced back solely to his new redraws.

The Equilator program allows you to enter dead cards, which are then not factored into the equity calculations. These cards are basically removed from the deck. This allows you to compare equity in two constellations, which shows you the impact a redraw can have.

In Example 3 the following cards are dead:
Q, Q, Q, J, J, J, K, A, A, A

In Example 4 the following cards are dead:
Q, Q, Q, J, J, J, A, A, A, A

EXAMPLE 3:

Board: K, 7, 2

Player 1 (A, K): 57.98%
Player 2 (Q, J): 42.02%

EXAMPLE 4:

Board: K, 7, 2

Player 1 (K, K): 73.78%
Player 2 (Q, J): 26.22%

There are now exactly 35 cards in the deck. Player 2 has the same outs in both constellations; Player 1, however, can only redraw to a full house/quads in Example 4.

15.8 points (an increase of 27.25%) has moved to Player 1.

But how exactly does this redraw help Player 1?

His outs on the turn are sevens, twos and the last king, a total of 7. If he doesn't hit on the turn, he will have 10 outs to hit on the river.

The probability that Player 1 will hit his redraw is:

1 - ((28/35) * 24/34)) = 1 - (48/85) = 43.53%

The probability that Player 2 will hit his flush is:

1 - ((27/35) * (26/34)) = 1- (351/595) = 41.01%

This means Player 2 will complete his flush and still lose to a full house/quads 17.85% of the time.

As you can see, a redraw always causes a shift in equity. The stronger your opponent's draw, the more easily you can see the impact of a small redraw. This shift in equity depends, of course, on the strength of the redraw. A redraw on the flop for a full house/quads is the strongest redraw you can have; its impact on equity is naturally the greatest of all redraws.

The blocking card

There is another way you can weaken an opponent's draw - the so-called blocking card. Any time you hold one of your opponent's outs, he is less likely to complete his draw.

This leads to the question:

Take a look at the following examples to see how blocking cards affect equity.

Board: K, 7, 2

Player 1 (7, 4): 45.25%
Player 2 (Q, J): 54.75%

Board: K, 7, 2

Player 1 (7, 4): 51.31%
Player 2 (Q, J): 48.69%

You can see the shift in equity (6.06 points, or a shift of 13.39%) due to the fact that Player 1 holds two blocking cards.

And of course, the greater the percentage of his outs you block, the better off you are. If, for example, your opponent has a gutshot + an overcard and you have 1 or even 2 blocking cards, you will notice a greater shift in equity than if your opponent had a flush draw + overcards and you only hold one of his flush draw outs.

Commitment with a draw

Some players never know whether or not they should commit themselves with their draw. Some are too careless, others aren't even aware of the concept. This section will focus on committing raises and playing with draws.

When we talk about committing, we mean being in a situation in which the EV(Call) > EV(Fold) after your opponent's push due to the pot odds.

An often cited example is the pre-flop 4-bet, in which a player commits himself to the hand and will have to call if his opponent pushes.

1$/2$ No-Limit Hold'em (6 handed)

Hero $200
MP $130

Pre-flop: Hero is UTG with A, K
Hero raises to $7, MP reraises to $25, 4 folds, Hero reraises to $70, MP reraises to $130 (All-In)

If you put your opponent on a tight hand range (JJ+, AK), your AKs gives you 42.14% equity.

Since your opponent is clearly in a raise-or-fold situation, you can save yourself the complicated post-flop equity calculations and assume your opponent will never just call your 4-bet.

You can also look at the EV of an all-in call and a 4-bet alone:

EV(All-In Call) = 0.4214 * $203 - (1-0.4214) * $60 = $85.5442 - $34.716 = $50.8282
EV(4-Bet) = f * $35 + (1-f) * (-$63)

You must then add an additional term to calculate the EV of both actions together:

EV(4-Bet + All-In Call) = f * $35 + (1-f) * (EV(Call after commitment) - $63)
EV(4-Bet + All-In Call) = f * $35 + (1-f) * ((0.4214 * $203 - (1-0.4214) * $60) - $63)
EV(4-Bet + All-In Call) = f * $35 + (1-f) * ($50.8282 - $63)

You can only determine whether or not the entire chain of action will be + or - EV when you include the fold equity you generate.

Let f = 0.3:

EV(4-Bet) = 0.3 * $35 + (1-0.3) * (-$63) = $10.5 - $44.1 = -$33.6

EV(4-Bet + All-In Call) = 0.3 * $35 + (1-0.3) * ($50.8282 - $63)
EV(4-Bet + All-In Call) = $10.5 - $8.52 = $1.97974

The 4-bet itself is -EV, whereas the all-in call after committing with a 4-bet is +EV. You cannot, however, simply add both values together to determine the EV of the entire chain of action.

You see that you created a +EV situation ($1.97974) by 4-betting and committing yourself to calling a push, assuming your opponent folds 30% of the time and pushes 70% of the time.

But this example doesn't really show the purpose of a committing raise. Take a look at a somewhat more complicated example that will show you the real strength of a committing raise.

1$/2$ No-Limit Hold'em (6 handed)

Hero $200
CO $129

Pre-flop: Hero is BB with T, 9
2 folds, CO raises to $7, 2 folds, Hero calls.

Flop: ($15) K, Q, 2
Hero checks, CO bets $12, Hero raises to $32, CO reraises to $122 (All-In)

You have a flush draw + gutshot, which gives you 12 undiscounted outs. Your flush draw isn't very high and your gutshot has you on the idiot's end, so you should discount 2 outs. With 10 outs, your equity is app. 40%. You therefore need 1:1.5 pot odds to make a +EV call.

You have to pay $90 to stay in a $169 pot. Your pot odds are 1:1.87. Calling would be +EV. You can therefore assume you would be committed to the hand if you check/raise and your opponent pushes all-in.

The question now is: Would check/raising and committing yourself with your draw be good?

In this case, good means creating a +EV chain of action with your check/raise on the flop. To determine this, you can take a similar approach as with our pre-flop 4-bet example.

Assumption:

On the flop you have 40% equity against your opponent's pushing range.
f = fold equity.
c = probability that your opponent calls your check/raise.

EV(All-In Call) = 0.4 * $169 - 0.6 * $90 = +$13.6
EV(opponent calls c/r) = EV(Post-flop)

Your EV(post-flop) is > 0, since you can always c/f the turn and play a break even line if your opponent calls. Since, however, you are a good player, you can assume that you can usually beat this line depending on opponent and board texture. Your EV(post-flop) will therefore be greater than 0.

EV(c/r flop) = f * $27 + (1-f) * -$32

EV(chain of action) = f * $27 + (1-f) * (c * (EV(Post-flop)-$32) + (1-c) * (EV(All-In Call) -$32))
EV(chain of action) > f * $27 + (1-f) * (c * (-$32) + (1-c) * ($13.6 -$32)

Possible assumptions for c and f:

c = 0.6, f = 0.2

EV(chain of action) > 0.2 * $27 + (1-0.2) * (0.6 *(-$32) + (1-0.6) * ($13.6 -$32))
EV(chain of action) > $5.4 + 0.8 * ((-$19.2) - $7.36)
EV(chain of action) > $5.4 - $21.248 = -$15.848

We will use the following variables to make the term more general:

• Fold equity : f
• Pot size : p
• Bet size : b
• Probability that your opponent calls your c/r : c
• EV(All-In Call) : y
• EV(Post-flop) : x

Which gives us the following term for calculating the EV:

EV(chain of action) > f * p + (1-f) * (c * (x-b) + (1-c) * (y-b)

If you get a negative value for the calculation above, you can use the following to determine how much EV your post-flop line must have (in case you get called) for the chain of action to be +EV:

x > (f * ((1-c) * y - p - b) - (1-c) * y + b)/(c(1-f))

In this example, you get a value for x of 'x > $33.016'. This means that you must, against this calling and pushing frequency, create this much EV on the turn to make your check/raise profitable.

Now for a look at this example with somewhat more realistic calling and pushing frequencies:

c = 0.25, f = 0.55

EV(chain of action) > 0.55 * $27 + (1-0.55) * (0.25 *(-$32) + (1-0.25) * ($13.6 -$32))
EV(chain of action) > $14.85 + 0.45 * ((-$8) -13.8)
EV(chain of action) > $14.85 - $9.81 = +$5.04

As you can see, with these values you have a +EV check/raise when you check/fold the turn in case your opponent calls (this is, of course, unrealistic, since you would not give up on the turn if you complete your flush).

Whether or not check/raising and committing yourself is really the best line has not yet been answered, since we would first have to calculate the EV of all other possible lines before we can determine which is best.

So was it right to raise and commit in this example? Why did we bother doing all that math?

You've seen that a committing raise can be +EV depending on your opponent's calling and pushing frequencies. This example should have, however, shown that a direct push would probably have been the better line. Without having to calculate the EV of pushing, we can assume that you would quite often lead your opponent to push with a better draw if you check/raise. And your equity against a better draw is, unfortunately, worse than against a made hand.

If you were to therefore overbet the flop and push, you can assume your opponent would be more likely to fold a better flush draw, which is good considering your equity vs. his calling range. If, on the other hand, you were holding a nut flush draw + gutshot, your equity would increase when your opponent is willing to go all-in with a draw, since you have his draw dominated.

The strength of a committing raise lies in influencing your opponent's hand range, so that your equity is higher than it would be if you were to push directly. This assumes that you generate the same amount of fold equity with both lines, but will end up playing against two different hand ranges.

Note:

We've ignored the problem of transparency to focus on the mathematical aspect. If your play is too obvious, you will be giving your opponent information:

• Direct push = weak draw
• Committing raise = Nut draw + monster

With the right read, your opponent could adjust his calling and pushing frequencies to create situations in which you need a very high EV(post-flop) for a committing raise to be +EV (see the T, 9 example hand).

How important is balancing your own hand range?

When it comes to playing with draws the question arises: How important is balancing your own hand range? Is it a problem if your opponent knows your hand range is comprised 100% of draws, or is there little he can do with this information?

We will use the following example to address this question from a theoretical standpoint:

You are heads up and in position on the turn. Your opponent bets into you.

• Hand Range A) Your range is completely made up of draws and you have 25% equity against your opponent's betting range.
• Hand Range B) Your range contains some nut hands and a few weak draws and you have 25% equity against your opponent's hand range.

Which hand range is better and what is the real difference? You have the same EV with both hand ranges when it comes to showdown value. If you were to then push with your entire range and your opponent call with his entire betting range, you would win 25% of the showdowns in both cases. There doesn't seem to be much of a difference at first glance.

When you then think about how to play against your opponent's hand range, you will see that your options are limited with Hand Range A. The best way to play your hand would be to call the turn bet on the basis of odds and outs and then bluff/raise on the river. If you complete, you bet/raise for value. If you miss, you still have the right bluffing frequency to keep your opponent calling your value raises when you do hit. The main disadvantage for you is that you have to wait for the river to confront your opponent with a difficult decision that will force him to make mistakes.

If, as with Hand Range B, your range is made up of a few nut hands and weak draws, you can find the right line on the turn and raise your legitimate hands for value and semi-bluff with your draws. You also have the chance to call a turn bet and then raise for value on the river with your nut hands and balance your hand range by bluffing with your draws.

You will have more options and an overall easier time playing your hand and balancing your hand ranges in specific situations when you have a mixed range, since you can represent a number of different types of hands.

Any time your opponent can put you on a clear type of hand, he will have an easier time playing against your range and you will have a harder time balancing your lines. You will always have a higher EV with a mixed range than with a polarized one.

Semi-bluffing and bluff frequencies

You can find a number of articles dedicated to semi-bluffs in the Gold Section. For now we will stick a purely mathematical approach to semi-bluffs.

We will start by looking at all-in semi-bluffs.

In order to determine the EV of a semi-bluff, you need the following information:

• Fold equity : f
• Equity vs. opponent's calling range: E
• Pot size: p
• Your bet size : b

EV(All-In Semi-bluff) =  f * p + (1-f) * (E * (p+b) - (1-E) * b)

With this term you determine your profit after a fold and then add your profit/loss after a call. You obviously make a profit any time your opponent folds, which means a semi-bluff can only be -EV when your opponent calls. And it just so happens that this part of the term shows that your equity against your opponent's hand range plays a large role when it comes to making a profit or suffering a loss.

The key term is therefore:

EV(Opponent calls) = E * p - (1-E) * b

You make a profit when this term is greater than 0. Rewrite the term with b and p to the left to see the role equity plays.

0 < E * p - (1-E) * b
(1-E) * b < E * p
b/p < E / (1-E)

Here is a chart with bet size/pot size ratios and the minimal amount of equity needed for +EV:

You immediately see: The greater your equity, the greater the b/p ratio may be for your semi-bluff to be profitable.

If you can estimate your opponent's calling range accurately enough to also estimate your equity against his range, you can use this chart to make perfect all-in semi-bluff raises, as you can create situations, in which you can only profit. And remember: Both terms don't have to have a positive value. What counts is the final EV after adding both terms.

The second and more complicated case is non-all-in semi-bluff raises.

A mathematical reflection on the situation becomes very difficult, since there are so many possible assumptions to be made about future actions. If you have a strong nut draw, you can refer to the reflection on committing raises to examine your play from a mathematical standpoint.

This isn't so easy when it comes to weak draws, since you have even more options after the flop. You could ...

• Either still get away from your hand, or
• see a card you can bluff on and represent a different made hand or a different draw.

A simple mathematical analysis is, unfortunately, not possible.

One problem that can arise with non-committing semi-bluffs is that your opponent might raise and force you to fold. This leaves a particularly foul aftertaste when you would have gotten a free card had you not semi-bluffed and only end up having to fold and give your entire pot equity.

Here are a few mathematical thoughts that should shed light on the situation.

1$/2$ No-Limit Hold'em (6 handed)

Pre-flop: Hero is Button with 5, 4
3 folds, Hero raises to $7, SB folds, BB calls.

Flop: ($15) K, T, 2
BB checks, Hero bets $12, BB calls.

Turn: ($39) A
BB checks, Hero ?

Should you semi-bluff with your gutshot or check behind and hope to hit? And will you only continue to play if you hit the straight after checking behind?

This is not an easy question to answer without any information on the stack sizes. You and your opponent both began the hand very deep:

Hero $409
BB $500

Your two options on the turn are either a pot size semi-bluff bet or a check behind.

For our mathematical analysis we will assume your opponent will check/raise all-in 1/3 of the time and fold 2/3 of the time. You will, of course, fold to a check/raise.

The EV of a bluff is easy to determine:

EV(Bluff) = 2/3 * $39 - 1/3 * $39= +$13

A semi-bluff would be +EV. But what about your EV after checking behind? If you complete the straight, you can assume that you will be able to stack your opponent x% of the time. The odds of completing a gutshot are 4:44 -> 1:11.

EV(check) = 10/11 * 0 + 1/11 * (x * $429 + (1-x) * $39)

If you manage to stack your opponent 33% of the time you hit, your EV for checking behind would be:

EV(check) = 10/11 * 0 + 1/11 * (1/3 * $429 + (1-1/3) * $39)
EV(check) = 0 + 1/11 * 1/3 * $429 + 1/11 * 2/3 * $39
EV(check) = 0 + $13 + $2.36 = +$15.36

Semi-bluffing is therefore +EV. But what about a check behind? If you complete the straight, you will be able to stack your opponent x% of the time.

The odds of completing are 4/44
EV(check) = 10/11 * 0 + 1/11 (x * 1100 + (1-x) * 100)

If you can stack your opponent 33% of the time (like the 33% c/r all-in bet), your EV will be:

EV(check) = 10/11 * 0 + 1/11 (1/3 * 1100 + 2/3 * 100)
EV(check)= 0 + 1/11 * 1/3 * 1100 +1/11 * 2/3 * 100
= 0 + 33.33 + 6.06 = 39.39

Checking has a higher EV than semi-bluffing, even though a semi-bluff would also be +EV. You can move the math around to see exactly how often you have to stack your opponent for the EV(check) to be greater than the EV(Semi-bluff).

13 < 1/11 *429x + 1/11 * 42.9 *(1-x)
13 < 39x + 39/11 - 39/(11x)
13 < 39x + 3.54 - 3.54x
9.45 < 35.46x
x> 9.45/35.46
x > 0.2667 -> 27%

Under our assumptions, checking behind is more profitable than semi-bluffing when you can stack your opponent 27% of the time you complete.

This example took a relatively negative look at checking behind, as we neglected to consider the possibility of hitting a pair on the river and winning the pot (after checking down). We also neglected to mention any +EV bluffing spots for the river.

But we have yet to answer the question: When is it best to risk facing a check/raise and having to fold? We'll start by listing the factors that play a role in answering this question:

• The probability of getting your opponent to fold.
• The probability being check/raised off your hand.
• The probability of completing your draw.
• Your implied odds in case you complete.

These factors are all closely related and can't be analyzed on an individual basis. You can, however, reach a few general conclusions:

At this point we must define the value of a draw and which factors can influence the value of a draw.

• Your draw has a lot of outs.
• The effective stacks are large.
• Your opponent is a calling station.
• You are drawing for the nuts.

The last three points are a matter of implied odds, which are, of course, only important if you see the river card and can get more action.

This may contradict your intuition; you might think that you should bet for value and to build up the pot, since your hand does have some value. You could then make an even larger value bet if you improve on the river.

But that is not entirely correct. Remember, when calculating the EV(semi-bluff), the value of your hand is of no importance. Your profit comes when your opponent folds. The value of your draw does matter when you check behind, since you can make a higher profit in the river if your value increases, which means the overall EV(check) increases.

Compare the board textures in this example hand:

Board 1: Q, 9, 9, 5
Board 2: Q, 9, 5, 2

You have J, T on both boards.

On which board would you check behind, and on which would you think about semi-bluffing?

Remember the arguments found above. You have lower implied odds on Board 1, which means your draw has less value. You are only drawing for a straight, whereas opponents could be drawing for a flush or full house. This means you could complete and still lose an expensive showdown.

This decreases the EV of checking on this board. You don't have this problem on Board 2. Here you are drawing for the nuts and don't have to worry about losing to a better hand if you complete. This is why Board 1 is more suitable for a semi-bluff with an OESD than Board 2.

The weaker your draw, the more suitable it is for a semi-bluff. If you get check/raised, you can simply fold (and give up less equity by folding).

Another concept that you should keep in mind when holding a draw and not semi-bluffing all-in is the "pressure of further bets."

You can use a semi-bluff to put the pressure on weak players and possibly generate even more fold equity, since your opponent will often be sitting on a marginal hand and facing reverse implied odds when you have a large stack to fire from.

1$/2$ No-Limit Hold'em (6 handed)

You are playing live. You are on the turn and have forgotten what happened in the previous betting rounds. You've even forgotten your own hand, but don't want to take a second look, since this might give your opponent a read. What you do know is that you can't bet for value.

Turn: J, T, 7, 6

Your opponent checks to you and you consider making a pot size bet. Let's look at this situation in two different scenarios:

• You don't have any chips left in your stack and are all-in.
• You still have 4 times the size of the pot in your stack after making a pot size bet on the turn.

In which scenario do you put an opponent with a hand like KJ, KT (or an even weaker made hand) under more pressure?

Your opponent will have a harder time making the correct decision when you have a large amount of chips still left in your stack, even though he has the same hand on the same board. He has to think twice about calling this turn bet and about how he will react to another bet on the river. If he doesn't have two pair/trips, just about any card on the river is going to be ugly.

In the first scenario, he would be getting 1:2 odds after you push all-in and would probably be able to call with top pair. But he has a problem in the second scenario. He not only has to call this bet, but risks having to pay even more to get to the showdown - and all that while sitting in a reverse implied odds situation.

He knows you are a good player and that you make the right amount of river bluffs with busted draws, which is why he won't know if he should call you this time or not.

As you can see, your own hand is secondary (you could have QJ, 88, 85 etc.) when a lot of weak and strong draws are possible. The key difference is the decision your opponent has to make, even though he has the same TP in both scenarios. The remaining stacks make your opponent think twice about calling a turn bet and paying again on the river, since he knows the river card is going to be ugly and he can't improve his hand by much.

In the first scenario he is getting 1:2 odds and can make a comfortable call, since he doesn't have to worry about incurring any further costs. His reverse implied odds are:

(Pot size bet turn + amount x) : (Pot size bet turn + amount x + Pot size turn)

If you check the river, they would be the same 1:2 as after pushing. If, however, you go all-in on the river, they will be 5:6. This is a huge difference and a very uncomfortable situation for your opponent to be in.

The thing you should take from all this is that you should tend to bet the turn again when you have a draw and a remaining stack and think that your opponent will fold to a x size bet on the turn due to the pressure of further bets, when he would otherwise call a x size bet on the river after a check behind on the turn, since he would no longer have to worry about reverse implied odds.

Bluff frequencies

We've mentioned bluff frequencies a few times in the course of this article, which is why we will conclude by saying what exactly bluff frequencies are.

The reason why you throw in an occasional bluff with a busted draw even though you have no equity is to put your opponent is situations in which he doesn't know if he should call your bet, because he is afraid of paying off your value bets too often. If you can find the correct ratio between bluff bets and value bets, your opponents will not be able to exploit your strategy.

You need two parameters to calculate such a bluff frequency:

• The pot size : p
• The size of your bet: b

The bluff frequency is then defined by these parameters:

B = b / (b + p)

Your opponent will never know if he should call your river bet or not when you play with this bluff frequency. For your opponent, EV(Call) = EV(Fold).

Aside from wanting to unsettle your opponent, you also want to make a direct profit when you bluff. To calculate this profit, you need to estimate the probability that your opponent will fold to your bluff. In other words, fold equity.

EV(Bluff) = f * p - (1-f) * b
0 > f * p - (1-f) * b
f > b/(b+p)

If your opponent folds more often than b / (b + p), you make a profit with your bluff. If he folds less often, you will make a profit with your value bets. Your opponent will not be able to adapt to the perfect bluffing frequency.

Bluffs are easy to describe from a mathematical standpoint, the art lies in putting the theory to practice. You simply won't be able to know exactly how often you must bluff to perfectly balance your value bets. You also won't be able to estimate your fold equity 100% accurately, since this varies from one opponent to the next and does not automatically increase with the bet size.

Conclusion

Be sure to read the other articles in the Gold Section to help you develop a concrete understanding of playing with draws. This article only focused on the mathematical aspect and aimed to improve your understanding of No Limit Hold'em.

You hardly have to time to calculate the EV of all your options when playing; luckily, this isn't necessary. The important thing is taking the time to study the theory off the felt, as this will help you make the right decisions at the tables.