## 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:

Know your chances vs your opponents hanrange

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.

## 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.

## Your opponents winning hands:

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.

Read another post about playing the river: :

poker tutorial1.5 playing the river