### Being unexploitable to poker bluffs

This poker guide will teach you how to be unexploitable to bluffs by using a mathemattical approach based on pot odds.

## Value bets and bluffs

Usually the main idea behind betting is to do a valuebet or to gain folding equity over your opponent. Regardless of what your opponent may be holding there is a minimum amount for you to call your opponent’s bet. It all depends on betsizing , how the board looks like,  and the handranges your opponent may be holding.

## Being unexploitable to bluffs

Let’s start with the general use of pot odds. If you have 2 against 1 pot odds you need to win 1/3=33% of the time to be able to call your opponent’s bet. Usually the question is if you have enough of a winning chance to call your opponent. But there is also another question to ask yourself and that is:  “how often should i call to make bluffs unprofitable for my opponent?”

Your opponent gave you 2 against 1 pot odds, but it also means that he got 1 against 1 odds himself. So if you fold more often then 50% of the time he will actually gain chips by placing this bet regardless of what he is holding.

Other odds to know about are:

A standard half size pot  continuation bet makes you call 66,7% of the time in order to make yourself unexploitable to bluffs.

General knowledge about each flop teaches us:

• Most flops miss most hands. You only hit a pair or better 32% of the time.
• In case you are holding a suited hand you have a 4 way flush on the flop approximately 11% of the time (backdoor flushdraw 41,7% of the time).
• In case you are holding two connected cards your chances of getting an 8 out straightdraw is: 10,45% and the chance of hitting a straight on the flop is 1,3%

As you can see it will be hard to call with 50% of the hands since you aren’t always having a decent pair or a draw to back you up since you won’t always play suited connectors. This means that you actually have to call  on the flop at least some of the time with two  overcards (for instance AK). The question is in which situation will you do that?

You need to know your opponent’s handranges first. Has he been loose or tight and in which position was he preflop? If he was playing from the button, cuttoff or middle position he may be playing much more hands than from under the gun. If your opponent is a loose player from a late position you may assume that you can call his bets with any two overcards that are decent enough. For instance hands like AK, AQ.

The second thing to consider is the nature of the board and the likelyhood of your opponent actually hitting that board. For instance a flop like: 332 wil not have hit your opponent the majority of the time. So if he bets he must be holding any overpair for it to be a valuebet, allthough he may be doing the same with any decent highcard hand like AK. The rest of the time he will be bluffing. On such board you may make the call with hands like AK or AQ, simply because it is unlikely that your opponent has anything better then a highcard on this flop.

When your opponent keeps betting heavily it may be to much to keep calling on the turn, because he may  be betting the river as well and maybe he will be going all in on the river. In such case you need to fold because of reverse implied odds. The fact that you may have a very large amount to pay before you can see the showdown. If things aren’t cheap enough you should fold. In other cases you may win the hand on showdown and your opponent failed his bluf attempt on the flop.

If your opponent isn’t capable of making that call on the flop, you should definately place plenty of blufs yourself. It is a well known leak of players to fold way to often after the flop, even though they  may  be getting 3 to 1 odds.

If you want to know more about playing an unexploitable game of poker have a look at the following link:

How to be unexploitable

### Poker math and statistics to help play winning poker

In general there are three types of poker math you may be familiar with. These types are all being used in different ways. For instance the most complex math is being used to study the game of poker, while simplified poker math is being used during the game. The last type of math is statistics. Poker statistics can for instance be used as a blueprint to know what winning players are doing.

## Complex poker math

This type of poker math is there to calculate the very fundamentals of poker for instance the amount of starting hands a player could have:

52*51/2= 1326

Which could give 3*2*1/1326=0,45% chance of AA for instance (because there are 6 ways AA can be dealth with 4 aces)

The amount of flops can also be calculated in a similar fashion. In this way you can calculate for instance the chances of getting a  double ended straigthdraw on the flop or a flushdraw. Or an easier example the chance of hitting a pair on the flop:

1-(44/50)*(43/49)*(42/48)=32,4%

These type of chances are important to know, but you don’t have to do this type of poker math during any game. You should just look at a table to know these important types of information.

In order to solve a lot of poker math problems you should use pokersoftware like:

Pokerstove

Equilab

My personal favorite is Equilab, because it has a lot of extra information and charts to look at. In general these types of pokersoftware are being used to study pokerhands. You estimate your opponents preflop handrange, you put in your own hand and the flop and after pushing a button you know exactly how wel your hand is doing on the flop against your opponents handrange. These type of things are important to know for every poker player and i allready did a similar type of study under the following link:

In the same way you can study your equity on the turn and the river. It is also possible to see the chances of your opponent having or getting certian hands. You can find out how big the chance is of your opponent holding a straightdraw on a certain type of flop, or a flushdraw etc. It turns out to be the most important type of study to improve your knowledge about poker.

## Simplified poker math

The poker math during the hand is a bit easier to use. Most of the time this type of math has been simplified for public consumption. For instance the usage of counting outs (cards that improve your hand)  and multiplying the amount of outs by 4% or 2,2% to estimate the equity on the turn and the river.  This type of poker math is simplified to make it easy for use. In reality the calculation should be done as follows:

For instance you are having a flushdraw and 9 outs and you want to know your chance on the turn and the river.

1-(38/47)*(37/46)=34,96%

With the 4% rule you would estimate to have 9*4=36%, which is quite decent given the fact that it is very easy to calculate.

Other types of poker math being used are quick calculations of pot odds or more complex equity calculations on the river.

In the case of an equity calculation on the river hand combinatorics are being used. An example of these type of calculations with the use of hand combinatorics is given under the following link:

Calculating equity on the river

And this is the way to study the river by the use of equilab (or pokerstove):

Playing the river

An other example of the usage of handcombinatorics is the example of a poker professional folding his pocket kings.

He was holding KK in a very early stage of the tournament while player very deapstacked. After reraising a player who he knows well he decides to fold his hand. He knows that player as a very risk averse player who would never commit his entire stack at the beginning of the tournament with anything less then AA or KK. But because the pro allready has KK himself, he quickly calculates that the chances are as follows:

Ways to deal AA: 6

Ways to deal the second KK: 1

The chance of his opponent holding AA are 6 times as large as him holding KK.

The pokerpro therefore folds his hand to find out later that his opponent was holding KK as well, something that only happens once every 1225 hands ((2/50)*(1/49)).

## Poker statistics

Poker statistics are being used to estimate how well you are playing compared to the statistics of winning players. Check also the following link:

statistics of winning players

Other poker statistics can be used to make certain decisions. For instance which tables do you want to play on? There is a chart that shows that it is easier to achieve higher winrates in 1vs1 then it is on 6 handed or 9 handed tables. Besides that the winrates on a 6 handed table are also much better then on a 9- handed table. There is also a difference in winrate if there is an ante. But ofcourse the most important information is ofcourse how often the players are playing a hand. If more then 35% of the hands are being played you know that there is plenty of room to gain an edge.

Other statistics of poker are statistics about the average profits of all the hands. For instance AA has an expected value of 2,32 BB (based on billions of hands played on the internet). Ofcourse this may change based on your position and how you play the hand as well as how others around you play. This can be done for all hands and the winning hands per position can be used to determine which hands you should play in every position. This leads to the standard loose aggressive playing style described under the following link:

Standard loose handselection

In the end the most important statistics in poker are ofcourse your green, blue and red lines:

This graph was made in a poker program called pokertracker. The green line is the profit curve (and the other green line may be the EV- curve). The red line gives your earnings without showdown and the blue line is your profit after the showdown. As you can see on the graph above this player is quite aggressive because he earns most of his money after his opponents fold. Consequently when he goes to the showdown his opponent will usually have a really good hand, which can be seen on the break even nature of the blue line.

Ofcourse pokertracker has much more important information to study. For instance the money you earn per position will show if you are leaking to much on the blinds or if you aren’t stealing enough of the blinds on the button. The entire leaktracker will be plotted to show how your stats are compared to winning players and so on you can even track the amount of rake you paid to the pokersite.

## To summarize

In general you should use the complex poker math to learn general ideas. For instance what the value is of holding a second pair on a certain type of board against your opponents handrange. These type of things can not be calculated on the spot. Simplified poker math is useable while playing poker. For instance the counting of outs or the counting of the amount of hands based on hand combinatorics.  In the end most of the information will be visible in the statistics. Which hands to play in certain position is based on statistics, but also the leaktracker that shows you how you are doing compared to winning players.

### The Nash Equilibrium for the two player push and fold game

When one of two players are playing short stacked. They can play the Nash Equilibrium push or fold game, inwhich you push all in with a certain stacksize and a certain hand with the goal to get the blinds and exploit folding equity. In the table below you see all the hands that can be pushed by 20+ bigblind stacks in green. 15+BB stacks in yellow, 10+ BB in orange and 5+BB in red.

In short you will be able to push all in against one opponent while having the top 40,27% hands and 20+ big blinds. With 45,8% of the hands you can push with 15bb, With 57,8% of the hands  you can push all in while having 10bb+. With the top 72% of hands you can push all in with 5 big blinds. In general you can push almost always under certain conditions.

## The heads up calling range of the Nash equilibrium:

Your heads up calling range is approximately 21,7% while having 20+bb, 28,7% with 15+bb, 37,1% with 10+bb and 62,3% with 5+bb.

How do you memorize all of this? In general you don’t, you just estimate your hand based on the table above.

## Having more then one opponent

In theory a similar chart could be made while having two random players sitting behind you, or maybe even 3, 4, or 5. However we will give a shortcut to these all in ranges. In general every Ax combination, pocketpair or suited connector ( 65s+) Has a decent chance even in a multy-way pot. Those type of hands should be pushed with multiple opponents behind you also like calculated before in:

http://pokermath.nl/folding-equity/

In general the tighter your opponents are the more folding equity you will get. Sometimes you can even push all in from under the gun position, because players will  more likely assume you to have one of the top hands based on your position.

## An estimation of a 3 player Nash equilibrium

The idea of the Nash equilibrium is based on the idea that you push with your opponent holding any random hand. The 40% range gives you 59% chance of winning against any random hand. Lets assume this to be a rule for 20+bb stacks.

In the case of having two opponents behind you the average best hand will become the top 50% percentile of hands. Using this handrange to find a new equilibrium you will get a preflop pushrange while two players are behind you of:

18,6% 20+BB

20,4% 15+BB

26,5% 10+BB

32,6% 5+BB

If this table is correct you need to play much tighter when you have 2 players behind you.  However this did not take into account the folding equity you have.

If you take into account the folding equity you will get much more close to 40% as the open range for 20+ BB, even if both opponents may call you with 30% of the hands:

Or the top 20%:

Or the top 10%

Even if you opponents call wit a tight range of 5% you have a positive expectancy:

So the Nash equilibrium may apply as well when having two players behind you. The expectancy is still positive. The worst expectancy is around half your range, which is 20% in this case.

I did the same with 3 players behind you which call 20% of the time:

With 3 opponents the Nash equilibrium gives a negative expectancy. Instead you should tighten up a bit to 25% of the hands according to my calculations.:

Which are the following hands:

This handrange of 25%  applies to going all in with 20bb while having 3 opponents behind you who call you with approximately 12.5% of the hands.

Going all in with more then 3 players behind you isn’t advisable. The handranges may get closer to 15-20% with 4 players and so on. The folding equity will diminish fast with more opponents behind you.

All of the above may not apply in a tournament situation where you are on the bubble (almost in the money). In such case you should try to get in the money first while shortstacked. If you have a bigger stack however, you can exploit the folding equity even more so, because no opponent wants to go out before the bubble.

### Folding equity while playing shortstacked

Folding equity is a very important concept in tournament poker. It is important to the way you should play while shortstacked.
In general the more shortstacked you are, the more reason to push all in, even tough you may not have a real hand. The definition of folding equity is as follows:

Folding equity is the concept of adding a higher winning chance (and therefore equity) by playing aggressive enough to make your opponent fold.

# Folding equity examples

You are playing a 9-handed tournament and you are sitting on the button. You are the first player to act. You are holding:

You have 10 big blinds left and there are 2 players sitting behind you.  You are not close to the bubble yet. What should you do?

Go all-in ofcourse! You have some folding equity here. Let’s say the three players behing you would call you with the top 15% of hands (hands like AT, TJs, A7s or 77 and better). How big is the chance that all three of them would fold?

The chance of all three players behind you folding to your all in is:  0,85^2= 0,72 , or 72% folding equity.

As you can see you gained a huge edge simply because of folding equity.

Now we will calculate you chances if you get called. According to Equilab you have approximately 35% chance with your 87s against your opponents handrange of 15%.

In this case you have:

72% of your opponents folding 1,5 big blinds plus approximately 1 bb in ante, which gives an expected value of:

0,72*12,5bb= 9bb

You have 25,5% chance of having one caller (0,85*0,15*2).

Inwhich case you will have 35% of doubling up your stack to 21,5-22bb (on average 21,75bb).

0,255 *0,35% *21,75bb= 1,94 BB

in 2,25% of the cases you will be called by two opponents and your winchance is 27,5%.

You will have 27,5% to triple up to 31 BB

=0,0225* 0,275*31bb= 0,19BB

So your expected value will be:

9+1,94+0,19=11,13BB

As you can see Your expected value of 11,13BB is higher then your current stack of 10BB it is therefore correct to make the move.

## What happens if your opponents are playing tighter then your assumption?

In case your opponents fold more then 85% of the time, your expectancy gets even better. For instance your opponents fold 90% of their hands against an all in.

0,9^2=0,81

0,81*12,5bb=10,12BB

As you can see you don’t have to make the rest of the calculation to see that you are allready getting a higher expectancy.

## Something else happens when you have two loose callers behind you.

Let’s say your two opponents behind you are having a big stack and a loose calling style against small stacks. They call 30% of your all ins with hands like: A5o, A2s, TJo, 55, 98s, K5s etc . You have 37,6% chance against those handranges while holding 87s.

Both fold: 0,7^2=0,49

0,49*12,5bb= 6,12BB

One player calls: 0,7*0,3*2=0,42

0,42*0,376*21,75BB= 3,43BB

Both players call (inwhich case you have 27,9% chance against both players with their 30% handrange according to equilab):

0,3^2=0,09

0,09*0,279*31bb=0,778BB

The expected value is:

6,12+3,43+0,78= 10,33BB

As you can see regardless of the handranges your opponents call with, you will be having a positive expectancy under these circumstances (87s with two players behind you and a 10bb stack), because of the folding equity.

## When is your folding equity not enough to justify going all in?

It depends on:

the amount of players behind you,

on the handranges your opponents call you with.

And with that information you can calculate your expected value.

## The Nash equilibrium

There is a heads up chart of which hands you can push all in with, it depends on the quality of your hand and the amount of bigblinds you are holding. It is called the Nash equilibrium chart.

Nash equilibrium

### “Poker brat” Phil Hellmuth does the unthinkable.

Phil Hellmuth a.k.a. the “poker brat”has been on a large losing streak lately in the WSOP 2017 tournaments he played in. He probably lost because of bad luck. You know how it goes, you go all in on the flop and your opponent hits one of his two outs to knock you out of the tournament. How does a poker pro cope with this?

When it comes to bad choices the Phill Hellmuth choose to blame it all on his new bought shoes. After all he started losing badly after buying these shoes and that can not be a coincidence. He therefore chose to do a cleansing ritual by burning the shoes in front of some of his friends. After all he called his guru, his mother and some other people as well, who all seem to agree.

Ofcourse Hellmuth needs to focus on his poker instead, but he and his friends are all poker millionaires, so why not burn some 300 dollar shoes instead? After all  it takes some sacrifice to please the poker gods, doesn’t it?

Ofcourse mathematics and psychology are the main things to focus upon while playing poker. It is well known that Phil Hellmuth uses CIA methods to determine if somebody is lying or not. It is largely based on facial expression and tics.

Supersticion is as old as human kind and still takes place when facing randomness. In the past it was a random event like an earthquake, a storm or a vulcano. Nowadays all it takes for people to become supersticious is a cardgame with millions of dollars on the line. Ofcourse they call it a joke, but what if Phil Helmuth starts a sick winning streak?

### Statistical blueprint of a poker pro. How to be unexploitable.

In the picture above you see the blueprint of winning players at a standard 9 player table. The green zones are showing the zones inwhich you are basically unexploitable by others. The white zones show the break even players. First we have a look at the values and after that we will generalize them to know the biggest leaks of players.

Preflop:

voluntary put money in pot (vpip) 15-29%

Preflop raise(PFR) : 11-19%

PFR/VPIP=55-90%

3Bet: 3-8%

Flop:

Continuation bet: 60-82%

Aggression factor AF = (Raise% + Bet%) / Call%  : 2-5

Aggression frequency: 44-61%

Fold to continuation bet:  38-61%

Fold to flop bet:  51-65%

Turn:

Continuation bet: 36-63%

Turn AF: 1-3

Turn AFq: 41-58%

Fold to turn bet: 43-66%

River:

River AF: 1-2

River AFq:  37-55%

Fold to river bet: 41-65%

Blinds:

Attempt to steal: 24-39%

Fold blind to steal:  70-86%

3bet to steal attempt: 3-11%

## Well known leaks based on statistics

1.Playing more then 29% of the hands (to much garbage) or less then 15% of the hands (to predictable).

2. When playing the correct handranges, you should raise preflop about 55- to 90% of the time when first in, or reraise when you have decent hands. The mistake you can make here is raise less then 50% of the time, which doesn’t set you up for a potential bluf later on if you miss the flop. You are giving the initiative to your opponents.

3. You should reraise within the following handranges:

3%:    99+, AQs+

8%:    88+, ATs+, KTs+, QJs, AJo+

So the mistake you can make is reraise more often then  8%.  You can also reraise against the wrong player in an early position. Another mistake is not to raise with the top hands like AA,KK,QQ,JJ and AK. When you go to the flop with to many other players these hands get way lower winrates. If you go to the flop with 8 other players your AA wins only like 30% of the time.

4. If you raised preflop (which you should do ~70% when first in), you should bet the flop 60-82% of the time. This is a lot, given the fact that most flops miss most hands. In general you hit only like 32% of the flops (with hands like AK).  Given this fact you should bet much more often then the amount of times you hit the flop. Playing to passively on the flop (betting not often enough when you missed the flop) is one of the biggest leaks. You should continue the story you told preflop: “you have the best hand”.

5. If your opponent places a continuation bet, you should fold 38- to 61% of the time. The amount of time you call or raise is larger then the amount of time you hit the flop (32%), however you will call with decent draws when you get enough pot odds or implied odds. The biggest leak is to fold way to often, or call without any draws.

6.  In general you want to call or raise 35 to 50% on the flop. A well known leak is to overplay low percentage hands like small pocket pairs when you missed the flop, inside straightdraws (only 4 outs) and other low percentage hands.

## Postflop play

7. You want to get less aggressive during the turn and the river then you where on the flop. The reason is that your opponent called a bet on the flop, which usually means a hand or a decent draw. You want to fold 40-60% of the time against turn and river bets if your opponent has shown strenght. A leak would be to keep calling your opponent with mediocre hands (like second pair).

8. After betting the flop you want to bet the turn as well like 36-63% of the time. If you check on the turn you are giving your opponent a chance to make a bluff and steal the hand. A leak is to be to passive on the turn, or way to aggressive.

9. Your optimum aggression frequency on the turn and the river is about 50%. Given the fact that you hit the flop like 32% of the time you are semibluffing with draws 18% of the time after the turn and the river your hand may  improve and you bluf less and less. Never bluffing on the turn and river is a mistake though.

## Stealing and folding the blinds

10. Blinds are a very big leak even for some of the best players, because they think in terms of pot odds and call way to often. In general you should fold the blinds 70 to 86% of the time against a raise even if you believe your opponent is trying to steal the blinds.

11. You should try to steal the blinds with about 24 to 39% of the hands. This means you can raise quite a lot of hands from both the dealer button and the cut off, simply because you are in position the rest of the hand. You can raise any ace, almost any suited king and some mediocre hands like J6 suited,  J9 to K9 ofsuit or 43s, 22+,64s,  or 84suited even. Not trying to steal the blinds is a mistake as well as trying to steal to often. Do note that you should protect your weak hands by continuation betting often enough on the flop.

12. If you are on the blinds you want to defend your blinds against a wide range of hands the button might have. You 3-bet any steal attempt if you have like the top 11% of hands ( or better). This includes hands like:

77+, A9s+, KTs+, QTs+, JTs, ATo+, KJo+

3-Betting to often to protect your blinds  is a leak.

## Showdown

13. The best players go to showdown 20- to 25% of the time with a showdown winrate of 55%. This  percentage means that they actualy fold top pair top kicker some of the time. The top 20% of hands contains hands like two pair and higher. Lower hands should usually not go to a showdown, it is better to place a bluff somewhere during the hand when you sence weakness, or just fold when your opponent is to aggressive. The hands you call with are also dependent on the nature of the board and your opponent.

The biggest leak on the river is to be much to aggressive. You should assume your opponent to have something and you only bluf in situations when you believe that there is a decent chance that your opponent might fold.

If you are playing within the ranges shown, you are unexploitable.

Players who are playing way out of these ranges are exploitable.  However there is a danger in exploiting your opponent’s mistakes, because you may become exploitable yourself. For instance your opponents are folding 80% of the time on the flop. You start betting the flop 100% of the time. Now you are exploitable by your opponent, because we know statistically you only have hit the flop 32% of the time, which means you are currently bluffing on the flop 68% of the time, which gives reason to raise or checkraise you more often inwhich case you fold with a loss much more often then you should.

The information given was for 9 player tables, however there are also comparable stats for 6 handed or heads up poker. The biggest difference are the starting hands you can play and raise with.

### Implied odds and expected value

In this poker guide we will study pot odds under extreme conditions. If there is a sincere possibility that your opponent(s) may go all in at the river you may have implied odds to call them, even though you may not have the pot odds. We will go to use an example to show this.

## Preflop implied odds by the potential of hitting three of a kind

You are holding 88 a player from under the gun raises 3 times the big blind and another player reraises. You are sitting on the big blind, do you call 8 times the blind with only a draw towards a set? You are currently holding a stack of 100bb.

Chance of flopping a set or a full house: 12,5% or 7 to 1

Pot odds: 13,5 to 8.

You can’t call based on pot odds, but let’s estimate the expected value.

## Estimating expected value based on assumptions

Both opponents have stacks of over 100bb.

There is a significant chance that the first player will rereraise all in, probably three out of ten times, inwhich case your 88 should probably be folded. The other 70% you will see the flop with two players and a pot of 27,5bb.

Lets say 50% of the time when you hit your set, one of your opponents wil call your all in on the river  (for the rest of his stack of 91 BB for a total of 27,5+91+91bb=207,5bb) and you will be winning the hand 80% of the time in such case. The other 50% of the time you will win a pot of 55 blinds ( two times the potsize on the flop, because it is likely for both players to at least try a last stab at the pot)

30% of the time you loose 9 big blinds.

70% of the time you will see the flop, and 12,5% of the time you will hit your set inwhich case you will win 27,5+91+91(the rest of the stack of one of your opponents plus the rest of your own stack)=207,5  bigblinds 80% of the time.

63% of the time you will loose 9 big blinds after seeing the flop.

7% (0,7*0,125*0,8) of the time you win

93% you will lose 9bb, which gives 0,93*9=-8,37 expected value

3,5% of the time you will win a pot of 55bb   =>expected value: 0,035*55=+1,89bb

3,5% of the time you will get one of your opponents all in while having a set for a total of 207 bigblinds. => expected value: 0,035*207,5=+7,26BB

The total expected value when playing this hand is: 1,89+7,245-8,37= +0,817BB

As you can see playing this hand because of implied odds is a very marginal play with a lot of assumptions, however the expectancy is positive.

As a general rule the stacksizes of you and your opponents should be at least 14 times the amount of chips invested preflop, because that is enough to earn to justify calling with 7% equity  (100bb/7%=14,28bb). In this case you only had 12 times the preflop investment, but you had two opponents to make up for it and you where sitting on the bigblind, which led to  an investment of 8bb.

Another rule is that you want to have two other players in the pot, because your goal is to maximise the chance of getting payed off. Even when your opponents don’t have a hand, the chance of one of them bluffing is big enough to at least get a slightly bigger pot.

In case you play with suited connectors the implied odds are less. There is a problem with the fact that the flop is often not enough to hit the nuts, which gives a problem of paying your opponents on the flop and the turn in order to see the river card. The chance of getting a flushdraw is: 10.94% and the chance of getting an 8-out straightdraw on the flop is: 10,45%. Besides that you could hit two pair on the flop 5% of the time. The problem is that you will need to pay a lot more then you need to pay to have the chance of hitting your set on the flop. This means you want to see the flop as cheap as possible with as many players as possible to pay you off in case you hit the nuts. Calling preflop because of implied odds may be a mistake, unless your investment is like 1/28th of the stacksizes.

This estimation is based on the following:

You hit two pair on the flop 5% of the time, inwhich case you have 50-70% chance to win the hand vs 2 players depending on how the board looks like. Usually you will win 3,5% of the time with the two pair.

You get a flushdraw 10,94% of the time, which means in that case you have 36% chance of getting the flush on either the turn or the river. For a total chance of  0,1094*0,36=0,0393, or 3,93%.

You get an 8-out straightdraw 10,45% of the time, which means in that case you have like 32% chance of hitting the straight on either the turn or the river. The total chance of this happening is 3,34%.

In case of backdoor draws or inside straightdraws, they usually won’t be improving often enough to justify calling any bets on the flop.

When talking about preflop implied odds, the 3,5% winrate with two pair can be used, which gives two times smaller implied odds then the former calculations of hitting the set, or 2*14,28=28,56. So you need stacksizes to be 28,56 times the preflop investment.

Incase you hit the flush and or straightdraw, you will recalculate your implied odds. For instance, if you have 36% chance of the nuts, you will be fine by paying of expensive turn and rivers incase your opponent has a large stack compared to the investments. Otherwise you will just only look at the pot odds 2 to 1 on the flop and 4 to 1 on the turn.

So lets say the potsize on the flop is 500 and you hit your draw. How to estimate your implied odds?

Lets assume your opponent is going to bet the full pot on both the turn and river to make your draw expensive.

He will bet 500, you call 500 on the flop and then he bets 1500 on the turn and you call 1500 on the turn. The pot will be 4500 to see both cards. If you see both cards you win 36% of the time. So in this case you want to get your opponent all in for 2  times as much as you have payed since the flop, or at least 4000 more. But on the flop your opponent needs to have at least 6000 chips behind to have enough implied odds to keep calling him on both flop and turn, which is 12 times the potsize on the flop.

### Poker mathematics calculating river equity

In this guide we will discuss a way to estimate your river equity.
Let’s say thay you are capable of quickly counting the number of hands your opponent might be winning with versus the amount of hands your opponent might be losing with.

In such case you could find your  estimated iver equity quickly by calculating:

Estimated river equity=

Number of losing hands+ splitpot hands / (losing hands + winning hands+ splitpot hands)   (in which case the number of losing hands is the amount of hands your opponent may be holding that aren’t good enough to win)

## Hand combinatorics for calculating river equity

For instance there are 6  ways to deal any pair with 4 cards:

(4*3)/2=6    (dividing by 2 happens because the order of the two cards don’t matter)

But how many pairs can you deal if there is already one of those cards on the board?

(3*2)/2=3

So on any flop where you have 3 decent cards you could expect there to be:

3*3=9 hands that could give your opponent a set on the flop.

Now let’s count the number of AK hands:

(8*4)/2=16   (First there are 4 A’s and 4K’s, once one of them have dropped there are only the other 4 cards left: so 8*4/2)

There are 16 ways to deal AK and 4 of those are suited hands.

But what happens if there is already any A on the flop? How many hands will be left?

The following table could make it clear:

As you can see 4 hands have been removed with one suited hand. This leaves a total of 3*4=12 hands.

What if there drops an ace and a king on the flop, how many hands are left for your opponent to have?

In such case 3*3=9 potential AK hands he may still hold to make the two pair.

So how could all of this be of use?

## Calculating river equity

You are holding:

And the the board is:

You quickly calculate that your opponent may be winning with the following hands:

AT,KT,QT,TJs,T9s,T8s

AQ

QQ, TT, KK or AA

T6 and T5 are being neglected because of the preflop action.

66 and 55 are being neglected because of the preflop and postflop action.

Two tens and four aces in the deck: 4*2=8

Two tens and 3 kings in the deck: 2*3=6

Two tens and two queens: 2*2=4

Two tens and 2 jacks (because you assume only suited TJ to be played): 2*1=2

Two tens and 4 nines (suited combinations only): 2*1=2

Two tens and 4 eights (suited combinations only):  2*1=2

Four aces and 2 queens: 4*2=8

Two queens left =1 hand

Two tens left= 1 hand

Three kings : (3*2)/2=3

Four aces: (4*3)/2=6

So there are: 8+6+4+2+2+2+8+1+1+3+6=43 hands that will win your opponent the game.

## So lets say your opponent may be holding the following losing hands  :

JJ, 99,88   (6+6+6 ways of being dealth)

QJs, Q9s, (2+2 ways of being dealth)

KJ (3 kings 4 jacks =>12 ways of being dealth)

Or total air

This counts up to: 6+6+6+2+2+6+12=34 losing hands your opponent may be holding.

KQ  for a splitpot: 2Q’s and 3 K’s left=3*2=6

6 out of 83 times you will split the pot

So your river equity in this hand is: (34+6)/(34+43+6)=40/83= 48% this is exclusive his potential bluffing hands like AK.

This leads to necessary pot odds of ~ 1:1+ to call, which almost always will be the case. However it is important to note that some of the losing hands may not be played anymore on the river (the same for some winning hands usually but in this case the turn and the river are considered blanks).

## The tighter your opponent the tighter the hands you calculate

Do note that 88 and 99 aren’t real hands to be counted if your opponent is a tight player, because he should fold those on postflop action. The JJ is a potential hand that could be held, however it would only give room for a small pot. In case there is an all in the JJ will be folded as well. If these 3 hands are assumed to be folded you subtract 6+6+6=18 hands from the potential losing hands your opponent may be holding.

This leads to 34-18=16 losing hands your opponent may be holding that lose him the hand.

In such case (16+6)/(16+6+43)=22/65= 0,338 river equity

This leads to 2 to 1 pot odds to be able to call.

Do note that there was no flushdraw on the flop, so I didn’t calculate the potential hands for that. If there is a flush chance there usually are about 15 suited hands that could make the flush (in case your opponent is playing a handrange of 15%). In general, the tighter your opponent is, the less hands that could make his flush. If he plays only 5% after the aggressive preflop action there are only 4 potential flushhands. In case your opponent may be holding a wide handrange of 25% he could have 28 hands that could give him the flush chance.

## Practice makes perfect

As you can see this method is a great way to know your chances on the river, however it will take a lot of time to learn to quickly calculate this. In the long run you will be better of knowing this methodology, because it gives you a feeling for the value of your hand, based on similar boards you calculated in the past.

poker tutorial1.5 playing the river

### Poker tutorial 1.0: Rules Texas Hold’em

In this poker tutorial we will introduce the handrankings, the betting structure and some of the crucial information to get started.

Poker is a cardgame inwhich players try to get the highest combination of 5 cards.  Either the strongest hand wins or one player manages to get all the other players out of  play during the betting rounds.

## The card combinations of poker are from high to low:

The combinations are always made by 5 cards. For instance if both players have high card “A” the second card the players are holding could make the difference. This second card is called the “kicker“.

Having 3 pairs is not possible. In case somebody has 3 pairs, only the highest two pair wil count together with the highest loose card to make up the 5 card combination.

There are many versions of poker, but texas holdem is by far the most popular. This tutorial will therefore focus on texas holdem.

## Texas holdem poker rules

1. In texas holdem each player get’s 2 cards in his hands followed by the first betting round.

2. The person directly to the left from the dealer button places a small blind also called “SB” (for instance 1 dollar) the player left from the small blind places the big blind also called “BB” (which usually is 2x the small blind, or 2 dollar, allthough there are variations in some casino’s.)

3. The player to te left of the big blind starts te betting round. His position is called “under the gun” or in short “UTG”, because he sits ajacent to the big blind. Incase there are only 2 players, the smal blind is the first person to call, fold or raise. He can “fold” throw away his hand for free, “call” the big blind (put 2 dollars into the pot), or “raise“, inwhich case the minimum raise is double the big blind, or in this case 4 dollars.

4. The positions on the poker table are called: dealer button, smallblind, bigblind, UTG, UTG+1, UTG+2, lojack (middle position), hijack (mp+1), cut off. The button is always the last one to act after the flop. During the first betting round the BB is last to act, UTG (under the gun) is first to act during the first betting round. After the flop, the smalblind wil be first. If the player is allready out of the hand the first player after him will be first to act.

## There are 3 general betting rules for texas holdem poker:

5.1 Limit: inwhich case the minimum raise is the maximum raise (+1 BB preflop and 2 BB’s starting from the turn.

5.2 Pot limit: inwhich the current size of the pot is the maximum bet a player me do.

5.3 No limit: inwhich case all betsizes are allowed starting from the minimum raise all the way to all in. This is by far the most used betting system.

6. Other players decide to fold, call or raise and the first betting round ends when everybody has called or folded. Do note that after a player raised, other players can still bet even higher, which means that the other players need to decide again wether to call raise or fold.

## The flop

7. After the first betting round ends the “flop” is shown. These are the first 3 open cards everybody shares.

8. After the flop the first person to act is the player who started the game as the small blind. He can “check” (pass his turn without putting money in the pot), or bet. Incase he bets, another player can raise even higher, call or fold.

## The turn

9. After this betting round on the flop the next card wil be showed, which is called the “Turn

10. After the turn another betting round happens.

## The river

11. If there are still players in the hand after the betting round, the last mutual card is being shown, which is called the “river“, or the fifth card.

12. A last betting round happens. If none of both players fold there will be a “showdown” which means both players show there hand to decide who wins the pot.

## Showdown

13. After the last betting round is finished both players show their hand (assuming they didn’t give up the hand during the betting round). The person last to bet or raise is forced to show his hand first. If the other players loses, he can decide not to show his hand and fold. In the other case, the player with the highest hand wins. If both players, or maybe even more players have exactly the same 5 card combination, the pot will be split among the players, this is called “splitpot“.

## General variations in poker

14.Tables both online or in a casino can hold 2 players all the way up to 10 players.

15. Besides playing poker for cash, or just pokerchips, there is also a setting called a “sit & go“, inwhich case the first players to run out of chips will leave the table and the top 3 gets payed their share of the buy in pot (or just the top 1 or 2 get money). Besides that there is a a multi table tournament format “MTT“. There are multiple types of tournaments. Slow, normal, turbo or hyper turbo. These terms basically show how fast the blinds are being raised to make players dropp out of the tournament quicker.

16. In a lot of tournaments and even some cashgames there isn’t only the big blind and the small blind to force some action, but each player also adds a small amount of chips to the pot after receiving their cards. This is called “Ante“.

## Learning your first poker strategy

The fastest way to increase your level of play in poker is to learn a decent preflop strategy based on handselection. Read the following poker tutorial to learn a decent loose handselection strategy:

poker mathematics tutorial 1.1 basic loose handselection strategy

### Poker mathematics tutorial 1.1: loose aggressive strategy

There are some guidelines for new players to significantly improve their game. The most important of them is knowing which starting hands to play and which to fold. In this first section we talk about the loose aggressive strategy.

The loose style is played in combination with plenty of agression to make people fold their hand. Besides that you play hands that have the potential to give decent value at showdown.

## Your position on the poker table

The starting hands to play are dependent on your position at the table. If there are 8 players behind you, you will need a very good hand to play, while only one player means you can play a lot more.

At a 9 player table the first 3 players after the big blind are in early position. Players 4,5 and 6 are in middle position and the dealer button, small blind and big blind are in late position.

## Playing from early position

There is a standard hand notation to write down hands easily. For instance AKo, means ace-king ofsuit, which means that both cards have different suits. AKs means that both the ace and the king are of spades, harts, clubs or daimonds.

The starting hands to play from early position when you are first to act are:

Pairs: AA, KK, QQ,JJ, TT, 99, 88,77

AKs, AQs, AJs, ATs, KQs, KJs, KTs, QJs, QTs, JTs, J9s, T9s, 98s

AKo AQo

Or to make it more visual:

These hands are all played for different reasons. In general the high pocket pairs(AA-JJ) are pre-flop monsters, which means you want others to pay you off.

The high drawing hands (like AK, AQ , AJs, KQs etc)  should be played to improve the hand to become the highest pair after the flop. If those hands don’t improve after the flop you can either bluf or get out of the hand if your opponents shows strength.

The medium pairs (TT-77) should be played with caution, these hands are not likely to be the highest possible pair on the flop, but you can get three of a kind from it (approximately 12% of the time), which is a killer if your opponent is holding  a high pair.  In other cases the small pair may hold up against the drawing hands and be the highest hand. A medium or low pocket pair wins 55% of the time against AKo. Ofcourse the nature of the board changes the way you play the hand.

The lower high potential hands in this selection are called suited connectors (QJs, JTs,T9s, 98S) , or suited one-gappers (KJs, QTs, J9s). These hands are played to get a medium pair or maybe even a straight (8%) or a flush (6%). It is important to note however that these hands can easily lose against the same pair with higher kickers. For instance QJ will lose vs AJ if a jack is flopped. These hands are cool to play, but not against a lot of agression.

## Playing from middle position

From middle position it is fine to add some weaker hands to your selection all the way down to hands like:

44+

65s, 97s+

A2s+

K9s+ Q9s

ATo+

QJo+

## Late position

From the dealer button you can play every hand with an ace in it or hands all the way  down to:

22+

43s+

85s+

K3s+

Q5s+

T6s+

T9o+, K9o+

Be aware that you are playing a lot of garbage hands, which you shouldn’t overplay if others are raising from early position. But as the dealer you will have position over others after the flop, which means you can act after the others, which is a big advantage in poker.

From the smallblind (SB) or bigblind (BB) you can call a lot of hands because of pot odds. This principle will be explained later on, however these positions are dangerous to play because you will be out of position during the entire hand starting from the flop, which means others are going to play after your turn. In general it is advised to play almost the same hands as shown above.

## The loose aggressive strategy

When playing the loose aggressicve strategy be prepared to raise 70% of the time when you are first to act and others have folded before you. Even if you miss the flop, you can bluf most players out of the pot by placing the “continuation bet” (betting the flop after you raised before the flop as well). It is advised to bet 60 to 80 percent of the time after the flop, after you raised preflop.

If you have the highest pair after the flop, or better you are in a very good position to bet, or let your opponent do the betting for you. In general you want to bet.

If you have a mediocre hand after the flop it is often better to be less aggressive. This is the case because you want to keep the pot small.

If you have nothing, or just some type of draw towards a straight or a flush you can easily bluff on the flop. One of the biggest leaks of beginning players is that they fold always after missing the flop, which means you can win 70% of the time after placing a bet on the flop. The loose aggressive strategy is effective, because of those leaks.

## Betsizing

When it comes to betsizing it is normal to bet one half of the pot, giving your opponent 3 to 1 for his money, which is very cheap. That way it is likely that your opponent wil keep paying you off. If you are affraid of your opponent improving his hand it is better to bet larger sizes of the pot all the way up to twice the pot. When playing the loose aggressive strategy you want to bet slightly larger then what is normal. After all your main goal is to make people fold.

## Playing much tighter

In the next poker tutorial we wil introduce a very tight strategy and show how big the costs of playing poker are. It also shows why you can’t only play the best hands in poker.  Please use the following link:

Poker tutorial 1.2 A very tight strategy and the costs of playing poker