## 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.

## 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

# The rake you will pay while playing very tight

In this artikel we will discuss a very tight strategy and how much money you will pay to the pokersite. The rake you pay will make a large difference in the outcome. In general you pay 5%  of the pot at showdown in rake to the house. We will show how much  rake that is by taking a tight strategy and making an excel calculation.

# A very tight preflop all in strategy

The idea is to play so tight that you can basically move all in preflop and be profitable from that strategy because of those drunken chinese players calling all ins with J5s just because they like to play suited hands. This is the all-in handselection i am talking about:

AA,KK,QQ,JJ,TT,

AKs, AQs

AKo

Which is the top 3,77% of hands.

Lets analyse the hands from a basic 1vs1  all-in point of view.

AA wins 85% of the time vs all hands

KK wins on average 82% vs all hands except AA. KK vs AA wins only 19% of the time, however the chances of facing an opponent with AA are quite low. The amount of starting hands are: (52*51)/2=1326. There are only 6 ways to deal AA ((4*3)/2), this gives a chance of 6/1326= 0,45% or 1 out of 221.

Other general examples:

KKvs TT = 81%

TTvs AKo=56%

AKo vs AQo=74%

Ako vs AJs= 70%

TT vs 98s= 81%

TT vs KQs= 54%

AQs vs QJs= 70%

AKo vs QJs= 61%

As you can see no matter what your opponent is having, you dominate him with a higher kicker (AK vs KQ, or AQ vs QJ ) or a pocket pair which usually wins against drawing hands as well as lower pairs.

## Risk of facing a higher hand

The only case you are losing is if your opponent has a higher pair. But as we allready calculated, the chance of having a certain pair is only 1 out of 221. If you play TT the chances of another playing having a higher pocket pair(AA,KK,QQ,JJ) is approximately: 4/221.  So lets say there are 9 other players on the table, in that case the chance is only 1-(1-4/221)^9 = 15,15% that somebody else is having a higher pair then you.

The majority of the time they will call you with hands they consider to be good. For instance AK, AQ, AJ, ATs, KQ, KJs QJs, 88, 99 (top8% of hands). These hands are being dealth in 106 ways, while AA,KK,QQ and JJ can only be dealth in 24 ways. As you can see the loose players will give you positive “expectancy” +-76% of the time.  I used equilab to calculate that the top 3,77% will have a winrate of 60,22% against the top 8% call.

## Running calculations in excel

So now we figured this out we can run a very opportunistic monte- carlo analysis in excel. Lets say you start with 10 buy ins (1buy in to get seated on one table with 100bb) and you get into all- in situations 10 000 times. What will happen to your account?

You went from 10 buy ins to 2187 buy ins. Wow i am going to pre-order a ferrari!!

## Please don’t fall into this trap

You have to play 265200 hands on 1cent/2cent tables to win 2187*100*2ct=4374 dollar, because higher blinds wil not give you this winrate… Besides that the costs of playing poker where not taken into account.

There are some mistakes in this calculation. One of them is the fact that you will lose blinds (approximately -1,5 BB per 10 hands). You will only get AA-TT Ak, AQs 3,77% of the time, so for 100 all-ins you will need to play 2652 hands. You will lose (2652/10)*1,5=397 bigblinds, which will be trown away waiting for those good hands, this is approximately 4 buy ins each 100 all ins.

And another problem is rake during these games. Rake is the profit of the house, after you win a pot you only get 95% of the pot or something like that, depending on the site you play on.  So instead of winning 1 buy in from your opponent, the 2 buy ins in the pot will be reduced to: 2*0,95= 1,9 buy ins, which means you win only 0,9 times your buy in, while you lose 1 buy in if you lose the hand.

Another problem is that your all-in wil not be called a lot of times, resulting in gaining only the blinds.

Let’s say your opponents call your all-in with approximately the top 8% of hands:

88+, ATs+, KJs+, QJs+ AQo+,KQo.

Yes, i know that is quite loose, but on low levels it happens quite often that players call with even far worse hands then these.

In this case at a table of 10 players your all-in will be called  1-(1-0,08)^9=0,5278, or +- 53% of the time. The other 47% of the time you will gain 1,5 bigblind (which is an assumption).

## Running the calculations inclusive the rake paid

So lets rerun our monte carlo analysis. This case we will first determine if your all in is called, or folded, inwhich case you only can win the blinds (1,5bb). In case you win your all-in you will gain 0,9 buy ins, because 5% of the pot of 2 buy ins wil go to the house in earnings. Besides all of this we asume a stable loss of 1,5bb per 10 hands. In this case for 100 all- ins we expect +-397 big blinds being thrown away, or aproximately -0,04 buy in for each all-in.

As you can see the very tight strategy is actually paying off, however the basic costs of playing the game reduced the profit by 2187 – 478= 1709 buy ins per 10 000 all ins, or 3418 dollar in rake. If you would have played these 265200 hands you would have gained 468*2=936 dollar given the fact that this system only works at the worst tables.

If you can find these conditions at the worst of tables you will be having a winrate of 18bb/ 100 hands though, which is insane if you take into account that 5bb/100 hands is considered to be the highest amount for professionals. However 18bb/ 100 is considered to be mediocre at 1cent/2cent tables. A true professional wil make 40bb/100 hands on those bad tables by playing more hands : 20-28% depending on which table size (i remember a pro showing his statistics on this).

## Why the simple tight all in strategy won’t work

I want to end this blogpost by showing what will happen with your account if you play against players that aren’t drunk and using a tight handselection to call your all in with (5%, instead of the 8% of the drunken bastards, you will have 55,5% winrate instead of the former 60,22%, however that might be not enough because of all the costs).

This shows that a winrate of a strategy can change so rapidly that you should always keep in mind that a simple strategy will lose very hard against normal players.

The next tutorial will show a very easy way to estimate your preflop chances, regardless of which hands you are holding.  Use the following link:

poker tutorial 1.3 Preflop chances

## Poker tutorial 1.3: Easy way to estimate preflop chance

In this tutorial we will teach how to estimate the preflop chance in a showdown no matter which garbage hands are being used.

88 vs AJo wins 55% of the time.

22 vs T6o wins 53% of the time

As you can see any pair wins versus any two overcards. The difference in percentage isn’t that much. It is therefore possible to generalise a lot of other preflop chances of hands  as well.

AK vs Q9 is almost the same preflop chance as T7 vs  42, because in both cases one of the two lower cards need to pair the board, while the opponent doesnt pair the board. In both cases almost the same is needed to win for the underdog, therefore the preflop chance is almost the same.

This leads to the following  preflop chance:

AA vs Q9 but also JJvs 63               Pair versus two undercards:  85-88%

QQ vs AJ but also 88 vs A7             Pair versus one overcard 70%

88vs AKo but also 22 vs T6o        Pair versus two overcards: 55%

AAvs KK  but also 33 vs 22            Higher versus lower pair: 81%

AQ vs KQ but also AJ vs AT            dominated by a higher card: 74%

AT vs KQ but also T3 vs 74            one overcard and one lower card:  60%

AQ vs KJ but also T8 vs 97            one overcard, but not one lower card: 63%

AK vs Q9 but also T7 vs 42           two overcards, vs two lower cards: 65%

+If one of both hands is suited, while the other isn’t, add 4% to the win chance of that hand

+If one of both hands is connected  while the other isn’t (with the possibility to get the straight in up and down direction, so not AK but 67) add +- 6% to the winrate

+ If one of both hands is a one-gapper, while the other has no aditional chance of making the straight: add 4% to the winrate.

+If one of both hands is a two gapper while the other has no aditional chance of making a straight add 2%

* Example AA vs 72 =88%, while AA vs 76s is having both suited and connected, which gives+ 4% and +6% chance, for a total of 22% winrate versus 12% for just two undercards.

If you know these 8 general examples and the 4 additional rules, you basically can estimate all other chances as well.

For instance estimate 88 vs A7s

This is  a 71% win -4% for flushchances is 67%for 88 to win.

Or AKs vs 76s

AKs is not a connector, because there is no up and down chance of the straight and   KA234 is not possible, however AKQJT and A2345 are possible, so let’s  assume it to be a two gapper for 2% extra chance of having the straight) So you use two overcards vs two lower cards example, which gives 65% and you subtract 4% to the lower hand (6% for the connector -2% for the two gapper for a 4% difference in favour of the connector), which gives 61% for the AKs, Both hands are suited, which will cancel eachother out, unles both hands have the same suit, inwhich AK will win if the flush shows up for a +4%.

## Poker tutorial1.4 pot odds and counting outs

In tutorial 1 to 1.3 there was plenty of information about starting hands before the flop. In this tutorial we will learn how to calculate outs and pot odds, to determine the postflop winning chances.

# Pot odds

Lets say the pot is 100 and you have to call 50. In that case you have 2 to 1 pot odds.

The concept of pot odds is important, because it makes it possible for players to stay in the hand, even if their winning chances are lower then 50%. As long as they get good pot odds they can stay in the hand. With those 2 to 1 pot odds given in point one you can stay in the hand with just 1/3, or 33% chance of winning or better. The 33% is the break even point between losing and winning money by calling these pot odds.

## Estimating odds by counting outs

Most flops miss most hands. If you have AK the chances of flopping an ace or a king, or a two pair is only 1-(44/50)*(43/49)*(42/48)=0,32, or 32% the rest of the time you will have nothing but a small chance of drawing the ace or the king on the turn or the river. There are 3 aces and 3 kings left in the deck, these make up for a total of 6 “outs“.

The 4 and 2,2% rule gives a way to quickly estimate your chances of improving your hand. When you are still waiting for the turn and river cards to come you estimate your chances of improving by multiplying  your amount of outs by 4%. For instance in the case of AK you had 6 outs, which gives you a chance of: 6*4= 24% of improving (turn and river combined).

If you have only the last card to go, you calculate the percentage by multiplying the amount of outs by 2.2%. In the case of AK that gives you 6*2,2=13,2% chance of improving to a high pair.

## Death cards

Do note that outs aren’t exactly the same as the winning cards that could come. It is always possible that an out is a death card. For instance your opponent is having a chance of getting a flush. In such case one or two of your outs may be improving your opponent to a flush while it gives you only a pair.

Lets say you have AKs and you flop two of your suit. In that case you have 13-4=9 outs that improve you to a flush. Multiplying by 4 gives you a 36% chance of getting the flush.

In the case of a straightdraw things are a bit more complicated. Lets say you have:

89TJ

In this case a 7 or a Q improves your hand to a straight. There are 4 sevens and 4 queens, for a total of 8 outs. Which gives you 8*4=32% chance to improve to a straight on the turn or the river/

There is also the possibility of having an inside straightdraw:

89 JQ

In such case you need a ten and you have 4 outs, giving you 16% chance of hitting it on either the turn or the river.

There are also more complicated situations. For instance you have 6 cards that can give you a high pair and you have a flush draw. In such case you can add the 6 and 9 together for 15 outs total.

But lets say you have both a flushdraw and an up and down straightdraw. In such case the 8 cards that give you a straight have 2 cards that make you the flush. So in that case you only have 8+9-2=15 outs.

## Outs with a set on the flop

Lets say you have the following:  88  and the flop comes:

8h3hTh

you hit your set, however your opponent has a flush, how many outs do you have now?

Well you improve to four of a kind if the last 8 hits and you improve to a full house with any 3 or any T, to pair the board, for a total of 1+3+3=7 outs on the turn and on the river you have one extra card to pair the board, which gives you 3 extra outs, for a total of 10. In such case you get: 7*4+3*2,2=28+6,6=34,6% chance of hitting the full house or a four of a kind.

In the rare situation that you have more then 16 outs, you should multiply them by 3,5% instead of 4 with 2 more cards to come.

## Hidden outs

There are also hidden outs that you may not be aware of. For instance:

You have AA.

the flop gives:

56T

your opponent has two pair now, but if a T hits, you will improve to a higher two pair: aces and tens, while your opponent will only have sixes and tens. So on the turn you have 3 extra outs and on the river even 6 outs, because the turncard could also make you a pair on the river. In this case the hidden outs give you 3*4+3*2,2=12+6,6=18,6% extra chance of winning, besides the 2 outs (or 8%) you allready had for hitting a set..

To summarize:

When you have to make a decision on the flop or turn, you count the amount of outs. Multiply it with 4 or 2,2% to get the chance and calculate the pot odds you need to call:

1:1     =>   50%+

1,5:1  =>   40%+

2: 1     =>  33%+

2,5:1   =>  28,5%

3:1      =>  25%+

4:1      =>  20%+

5:1      =>  16,6%+

## Implied odds

It isn’t always necessary to have pot odds to make the call. In such case you need enough implied odds.  You have large implied odds if you have multiple opponents with very large stacks in the hand and you have a chance of getting the nuts. If you want to know more about implied odds go to:

Implied odds and expected value

## Poker mathematics tutorial 1.5: playing the river

On the river you either have the winning hand or the losing hand. Besides that the pot is allready big because of the previous betting rounds. According to statistics the average pokerprofessional usually gets to see the showdown only 20 to 25% of the hands with an average winrate of 55% (9-handed).

## Showdown value

So what type of hands can you expect going to the showdown 20% of the time?  These are hands like: two pair, three of a kind  a straight or a flush (and higher ofcourse).

A top pair top kicker is actually not that great of a hand on showdown (when playing on a 9-10 player table), you only go to showdown with it if you can keep the pot small enough or if your opponent is a maniac playing like 35-45% of the hands he has been dealth. If your opponent payed your bets off multiple times you can expect him to have a decent hand, which usually means two pair or higher.

## Using a math problem as an example how to think about your chances on the river.

The lottery with 3 doors to choose from is a decent way to understand how pokerprofessionals think about poker. Let me explain:

You are in a gaming show and you won a chance to win a ferrari. There are 3 doors presented to you and only one of them has the ferrari behind it. You choose door number 1. The commentator walks to another door and opens it to show that there is nothing behind that door. He then asks you if you want to change your choice. Should you change your choice?

The answer is yes, you should change your choice to the other door and win the ferrari 2 out of 3 times.

Sounds crazy right? Lets explain the reasoning behind it:

You choose door 1:

1/3 chance the ferrari is behind it

2/3 chance there is nothing behind the door

Lets show this in a table. F means the ferrari is behind the door and n means nothing, there are 3 possibilities:

Now one door is being removed from the game, showing that the ferrari is not behind it. This means that you actually gained more information than you had before. In the table you see that by switching you will actually win 2 out of 3 times.

## Example of narrowing down the hands your opponent may have

The same idea is the case with poker to. You start the hand without any information, but during the hand you will have 4 betting rounds and these should be used to find out what your opponent may be having on the river.

Lets give a very quick and visual way of showing this.

Your opponent raises to 150 preflop from middle position he is a decent poker player (playing on a table with 9 players).  Everybody before him folds. This should give us the following handrange:

You are sitting on the big blind with AA and you decide to slowplay by just calling 100 for a pot of 325.

The flop is:

you decide to bet 150, your opponent raises to 450.  By doing this he is actually reducing his potential hands to:

Two pair

Three of a kind

A  pair and a flushdraw

An (over)pair and a two ended straightdraw

Straightflushdraws

Two overcards and a flushdraw

Or a crazy bluf based on the possible fact that you bluf the flop way to often.

According to calculations by pokerstrategy.com’s equilab, the aces are actually winning only 60,2% of the time in this situation. However we may know more about this player. For instance that he is loose and aggressive, but he also knows when to quit the hand.

We decide to go to the turn (pot= 1225):

You bet 600 and your opponent calls (pot=2425). You move to the river:

A blank. You decide to check and your opponent moves all in for 3000 more. What do you do?

Fold ofcourse! Allthough the river card is a blank the chances of you winning this hand are only 43% according to equilab (and the handrange he may have), which is enough to call based on pot odds (3000 in a pot of 5425 or 1,8 to 1, or 36% to break even). However based on his call after the 3rd flushcard (and 4th towards a straight) hit the board and the  all-in move on the river your opponent is actually showing great strength, which gets rid of a lot of missed hands in his range. You decide to take out most of the missed hands which gives your opponent a range of:

Inwhich case the suited hands are only hearts. The river equity calculated AFTER taking the carbage out of his handrange, gives an equity of only 25%, which means you need 3 to 1 pot odds to break even.

You decide to fold and your opponent shows:

He had a nut flushdraw and an inside straightdraw and two overcards on the flop and he decided to semibluf us with a great amount of outs to back him up. After hitting the flush on the turn he decided to trap us by just calling, hoping that we would bet the river.

## Conclusion

What this hand shows us is that by looking at the betting patterns of the player you can actually deduce that your chances are actually way smaller then you calculate, in this case you had 0% and you should have known it. It is a bit like the 3 doors math problem. The chances change much more then you expect if you get new information during each of the betting rounds.

The methodology in this guide can also be calculated based on hand combinations your opponent may be holding on the river. This advanced method can be found on:

Calculating equity before showdown

## Poker tutorial 1.6: knowing your opponent’s handrange

In this poker guide we will discuss the importance of knowing your opponent’s handrange. If you have a decent estimation of what your opponent may be holding you can estimate your chances of winning the hand.

## Preflop handrange

Preflop handranges (as you can see in the table above) of players are very important to determine if you call, raise or fold. The table above gives a quick insight in the preflop winrates of certain handranges considering one opponent. In general you estimate your opponent’s handranges based on the position he is playing and how often he played a hand in the past.

## Postflop equity based on your opponent’s handrange

Estimating your chances post flop while taking into account your opponents hand ranges is a bit harder, because you need to assume some flops.

We will work with the following hand ranges:

5% your opponent is a tight player who raised from early position with 9 players behind, or from late position he may have reraised or rereraised. The top 5% of hands are: 88+,AJs+,KQs,AKo

15% your opponent may have called a preflop raise, or may have raised himself from middle position

The top15% of hands are: 77+,A7s+,K9s+,QTs+,JTs,ATo+,KTo+,QJo

25% your opponent called from late position

The top25% of hands are: 66+,A2s+,K6s+,Q8s+,J8s+,T8s+,A7o+,K9o+,QTo+,JTo

We will assume that you go to the flop with one opponent.

## Postflop chances versus your opponents handrange

In the table below you can see the chances vs handranges on certain flops.

1.Do note that top pair is a good hand, but the kicker could make a difference between having 70% chance of winning or only 57%.

2.Do note that having an overpair doesn’t always mean that you are favorite in the hand. AA could have 84%, while TT only has 46,6% as an overpair. This is because of the assumption that your opponent is playing the top5% of hands. So if your opponent showed a lot of preflop aggression your TT overpair may not be any good. Also note that almost all overcards are potential outs for your opponent in this situation.

3.Second pair is in general not the type of hand to overplay, only 30- to 60% chance of winning the hand depending on the hight of the card, the kicker and your opponents handrange.

4.Third pair or even 2 overcards are on average even worse then second pair giving you chances of: 24- to 57%.

## Chances of your opponent having a draw on a draw-heavy board

Below is a table of chances of AA on a flop with a potential flushdraw and a potential straightdraw.

1.The chance of your opponent having the flushdraw is only 6,5%

2.The chance of your opponent having any type of straightdraw (except backdoor straightdraws) is 60%

3.The chance of an already made straight or set is 7,9% on this flop

4.Do note that there is 30 to 40% chance of loosing this hand if you are holding an overpair.

5.On other straight- heavy boards the chances  of your opponent having the straightdraw are a bit smaller: only 29,7% on a board of 598. That is a very large difference with the T97 board.

## A paired board

In case of a paired board with aces the chances are as follows while you are holding KQ ofsuit:

1. Do note that the king high is actually no good in the case your opponent plays 5% or even 25% of the hands.
2. The chances of having a set on this AA8 board (without taking into account the 2 more cards to come) are actually 28% and there is even 7% chance of having an already made full house. Do note that these chances are only the case with tight handranges of 5%. In the case of handranges of 25% the chances are smaller.
3. If you are playing heads up with your opponents handrange of 70% the chances of your opponent having three of a kind or a full house is only 12,1% on this type of flop. Your opponents handrange makes all the difference in the choice you make on a AAx board.

## To summarize:

1.It is important to know your chances on the flop and adapt your way of playing after seeing the flop. Knowing your opponent’s handranges is very important as well. Usually it is highly dependent on the amount of players at the table, your opponents position and the preflop action.

2.The chances of your opponent to win against your overpair may be larger then you expect, especially on certain boards.

3.On flops without any draws you can assume your top pair top kicker to be good. Be aware thought that the chances could be much lower if you are having a poor kicker.

4.Second pair and bottom pair are occasionally good, but should be played with small pots.

5.When you see a flop with a high pair, for instance AAx, your opponents handrange is very important in finding out your chances. If your opponent is playing any random hand (~70%) he has only 12% of having three of a kind or better. If you put your opponent on playing the top5% of hands, because of preflop action at the 9-player table, you can assume your highcard king to be worthless.