In this video I analyze other more technical aspects of Poker such as range, position or EV calculation.

1.Range
2.Poker Factors
3.Position
4.Poker EV
4.1.Call EV
4.1.1.Call EV calculator
4.2.All-in EV
5.Other Factors
6.Heads up Strategy
7.Nash Equilibrium

In the last poker math video we saw a summary of the rules of the game, and what the probability of linking each hand is.

We then looked at the outs and odds, which are the probabilities used in poker, and learned how to calculate them and refer to our table of odds and outs.

We also saw how we can approximate these values mentally without the need for tables, and finally we learned how to use a poker calculator, and saw how the odds we calculate for our projects relate to our real probabilities.

In any case, I didn’t answer how to calculate our odds when we don’t know the cards of our opponents, and neither because a hand that has little chance of being tied can be profitable. That’s why if you were left with the doubt or are simply curious to answer these questions we go with the second part of the poker math.

1.Range

As we saw in the previous video, Poker is a game of incomplete information, since we do not know the cards of our rivals. Consequently what we can calculate comes from an estimation of what hands they might have, based on the projects they can complete with the community cards. But also knowing which hands they usually play frequently or which ones they prefer to discard in certain phases of the game.

Some players tend to play a very small number of hands, because they feel safer playing like this. Others, more recreational, prefer to use a larger number. Have you ever wondered what kind of player you are?

The reality is that we all have a filter to choose which hands we play and which we don’t, which is more or less big depending on our style of play.

Well, the group of cards that make up that estimate is known as range and is represented at the software level with a matrix where you can select which cards make it up.

If you want to calculate your hands probability taking it into account, what you should do is calculate the probabilities of your cards against that range you estimate, eliminating all those hands that are impossible or very unlikely that the opponent is playing, given his profile or the game situation. This way you will get a calculation as close as possible to the real one with the information you have. That’s why the better the estimation you make, the more accurate will be the calculation of your probability, and that’s why it’s important to acquire a technical knowledge about what ranks our opponents may have. But it is also important to know our own rank.

On a practical level, I recommend you a Tracker Software to do this, because besides calculating probabilities with ranks it is able to analyze the game of your opponents and offer you personalized statistics. It does this by saving the data of the games you are playing and in this way you can know what number of hands you play voluntarily, the percentage of times you have increased in the preflop, or any variable you consider important. A player’s statistics can always give us a long term advantage in making decisions. Although keep in mind that this is only for Online Poker.

DOWNLOAD TRACKER SOFTWARE

The truth is that this software is very, very cool. And now that you’ve seen how to calculate ranks, what else do you think you should take into account to make a decision?

2.Poker Factors

Deciding when to match a bet, raise it with a certain amount or throw away our cards is what determines how good players we are in the long run, but choosing one or the other depends on many factors, so I have made a list for you to look at:

Some rely solely on our information, and others rely on information provided to us by our competitors. But then there are others that depend solely on the game such as the number of players.  The more players there are at a table, the more likely it is that one of them will have a stronger hand than ours, as we face more players on average per round. That is why we must adapt our game with a more exclusive range, and also reduce the range we estimate of our opponents, because it is less likely to win with the same hand when there are more possible better hands.

3.Position

As we see there are a lot of factors, but not all of them have the same importance.

One of the factors considered most important in poker is the position, as it can be used to get extra information from your opponents, or also to lie. If you play in an early position, you have no information about what your opponents are going to do, and since you can’t discard any player you are forced to play a shorter range of hands. On the other hand, if you play in a late position, you get information about the bets that are made or the players that withdraw, so you have certain information that can give you an advantage and more chances to win. That makes each player adopt a different game and range depending on their position and that logically offers different results. But how much of different? Well, 9 players in over 1000 hands have been analyzed to see which position has the most advantage, and this is what has been observed:

As we had already deduced the last positions are usually the ones that offer us more profit, but curiously and against what I had said, the first position also has a quite high percentage of victories. This is explained by the fact that many players use early positions to bet or raise as if they had a strong hand, whether they have it or not.

If we make a summary, we can see that a late position is the one that benefits you the most, but the early position can also serve to bluff you.

Playing good poker doesn’t seem to be easy, you have to be good enough to overcome the losses of the game and even losses on commissions. And with all these factors our decisions depend on many things. So, is there any mathematical formula to decide what action we take?

4.Poker EV

Although position is a very important factor, its influence decreases as fewer players are left, since by reducing the number of rivals we can consequently steal fewer chips. So when there are only 2 players left at the table, which is usually the case, the position is no longer a factor to take into account and we can establish a mathematical method to know what decision we should take, based only on the size of the pot, the size of the bets and our probabilities.

For that a new concept is defined called EV that comes from (Expected value) and is based on the concept of long term gain. To know if a play suits you or not, the most intelligent thing is to calculate what average gain we would obtain if we repeated many times that same play. And that is precisely what calculates the EV, subtracting the probability of winning by the profit less the probability of losing by the loss. In this way you can know that profitability has for example to equal a bet.

4.1.Call EV

When the EV is negative, equalizing or making a call is not profitable, but if it is positive, it is a priori profitable in the long term as it is higher than the EV of folding, which is 0. Note that it is also possible to have a positive EV even though you are more likely to lose, if the benefit outweighs the risk. Since it can be more profitable to win few times with much money, than many times with little money. This answers the question of why it can be convenient to bet with a small probability. I have included this EV call calculator in case you are interested in calculating the expected value of your possible call.

CALL EV CALCULATOR

To calculate the EV mentally you can use the known pot odds, which are no more than the proportion between what we win against the bet we have to make, and which are compared with the odds (which are the times we lose for each time we win). As both are calculated with the same proportion, you only have to compare odds and pot odds to know if it is profitable to go to a raise or not, so this is a practical way to calculate the EV for the live game.

4.2.All-In EV

To know instead if doing an all-in is profitable, the thing changes, since it is important to take into account the response of our rival, which can be to go or to fold based on the amount that we bet, so the EV in this case also includes the estimated probability that one action or the other happens, and can be calculated in this way.

Which is basically the Fold EV plus the Call EV

There are many formulas to determine our actions, such as the minimum bet you have to make to remove the odds necessary to the rival to pursue their project and many others that you can search on the Internet:

AM = (Outs x P)/(50 – 2 x Outs)

5.Other factors

The truth is that although these formulas are useful, there are other factors that can also influence when making a decision:

-One of them is the confrontation cost. If you have a profitable move but few chips and you don’t have the possibility of re-buying or you simply play in the short term, it may not be smart to match or to do all-in if doing so may have a confrontation cost of losing all or a large amount of chips, compared to not confronting (especially when our probability of winning is low), so in modalities such as tournament, it is also important to evaluate our decisions based on the chips we have and those of our rivals. A mathematical model that takes this into account is the ICM model, which is used especially for the bubble phase of tournaments.

-Another factor considered the most important to change those decisions is the knowledge of your rivals, which can serve to adapt and exploit their meta-game as much as possible, increasing or decreasing the bet size, the amount we have to pay to certain rivals, or simply to avoid or encourage certain clashes, which can give us an advantage.

It doesn’t seem easy to decide with so many things, so the experience also plays a fundamental role to be a good player, although are we really able to learn all the situations of Poker?

6.Heads up Strategy

Only in the Texas Holdem for 2 players there are 10^161 different decisions possible, which is more than all the atoms that form the universe. This makes it impossible for a human to learn which is the most advantageous decision for all that immensity of situations. Although, as you have surely thought, there are programs where you can consult how to act in most of the possible cases to improve your strategy. These process millions of hands based on exploiting themselves, and offer us the most profitable actions at the statistical level, which you can use to adapt to your game and exploit your opponents, so currently to be a good player is also important to know how to interpret statistical data correctly.

DOWNLOAD STATISTICAL CONSULTATION PROGRAM

One of the most important decisions that we must take in Poker is how to play our hand in the heads up, since that conditions us a lot during the following betting rounds, that’s why it’s important to have a previous well defined strategy that you can consult. That’s why I have included a series of tables based on the study of these programs made by a professional player (Zeros) and that could be helpful if you have them at hand during the game. In them you have the action plan based on the optimal return for any particular situation in the heads up, which can be very useful for online poker, so if you are interested, you have a link in the description, along with the link to these programs that I have commented in case you want to consult another situation. (And it’s not that I want you to play Poker but if you do, do it well).

DOWNLOAD HEADS UP TABLE

Now that you have seen that Poker is a very mathematical game, the most important question of all arises, is it possible to play Poker in a perfect way? Let’s see it:

7.Nash Equilibrium

In 1951 a certain John Forbes Nash better known as John Nash elaborated a doctoral thesis demonstrating that any game with a finite number of mixed strategies has at least a Nash equilibrium. This balance is given when all the players have chosen in the best possible way and none of them can improve their strategy by much change it, as long as the others maintain theirs. This balance ensures that in a zero-sum game like poker, we will never lose in the long run.

How is this possible?

The search for this balance is not simple, because we must find an optimal combination that satisfies all players among many possible, which added to the amount of decisions that can be made makes it a computer challenge. Finding a balance in the heads up is less difficult but in the River it is much more complicated. (To understand this, try to imagine all the situations that can occur in this phase of the game)

As there are so many possible casuistry, it’s too expensive to compute a strategy for each hand or bet size, so scientists have chosen to abstract the full game to a more simplified version in terms of values and bet sizes, which allows the search for balance more efficiently, and that approximates in a very precise way to the full game through a set of algorithms. Using intelligently the calculation of this balance, they have built an artificial intelligence that plays poker for two players in a practically perfect way.

In 2017 the Libratus software managed to beat the best world professional poker players for the first time with a significant statistical advantage, which is a historical fact for games of incomplete information. And all this thanks to mathematics.

Luckily or unfortunately Nash’s equilibrium is only possible for 2 players, since it is possible to lose in long term expectation in Poker of 3 players or more, especially if one group of them gets together to eliminate another. Even if a balance is calculated for all three players, that combination does not have to be a Nash balance in itself. Consequently, if you play online poker with more than two people, you don’t have to worry for the moment, because their future is assured, although, who knows until when?

The post The Mathematics of Poker – Range, Position and EV first appeared on Math4all.

Do you think math is important in poker? In this article I will analyze poker on a mathematical level so that you know how to calculate your probabilities, analyze your actions, or how mathematics can make you a better player.

1.Texas Holdem Poker Rules
2.Texas Holdem Poker Hands
3.Texas Holdem Poker Actions
4.Texas Holdem Poker Probabilities
5.Poker as a game of incomplete and imperfect information
6.Odds and Outs
7.Texas Holdem Poker Calculator

1.Texas Holdem Poker Rules

Texas hold’em unlimited poker is a French card game with up to 23 players simultaneously, where the objective is to eliminate all players from the table, to win all their chips, or simply to win as much money as possible in cash mode. For this, 2 non-visible cards are dealt to each player, and they will decide in turns and based on their cards, if they want to bet, check, fold, call or raise.

Once all players have made a decision, the dealer draws 3 visible flop cards, which are used by each player to bind their cards in order to get the best possible hand, and proceeds to a new round of betting.

As the dealer uncovers a new card, the same procedure is repeated until finally 5 cards are uncovered. Finally, the players still at the table show their cards and the winner of the pot is the player with the best hand. The players must use their knowledge and their strategy to make the best decisions that will allow them to win.

2.Texas Holdem Poker Hands

• Pair: First of all, we have the pair, which is the most basic hand in poker formed by 2 cards of different suits that have the same value. If both players have a pair, the one with the highest value wins, and if that value is the same, the player with the highest card wins. The pair is the least valuable hand that can be linked.
• Double Pair: The double pair is composed of 2 different pairs together with any other card. If both players have a double pair, the hand with the highest pair wins, and in case of a tie, the unmatched card with the highest value wins
• Three of a kind: As its name indicates, the trio is 3 cards of different suits with the same value, together with 2 cards of any kind. In case both players have a trio, the one with the highest value wins. Note that in this modality of Poker there cannot be 2 trios of the same value, since there are only 4 possible suits and with one there are already 3.
• Straight: The Straight is made up of 5 consecutive cards of the same suit or different suits, so all the values can be ordered one by one from lower to higher. Depending on the number of cards you start with, you will have a higher or lower straight, so in case of a tie, the one with higher values will always win.
• Flush: There are 5 cards of the same suit regardless of their value. In case of a draw, the flush with the highest card wins.
  Full/Full house: Gather 3 cards of the same value and 2 cards of another, or what is the same, a trio and a pair simultaneously. When two players have a full house, the value of the highest trio wins.
• Four of kind: It’s composed of a quartet of 4 cards of different suits that have the same value, together with any card. In case of a draw the value of that card decides the winner. Although this is the most popular hand, it is not the most valuable.
• Straight Flush: It’s a straight flush made up of all the lower cards of the same suit, which makes it a rather unlikely move. When two players have a straight flush, the straight with the highest values wins.
• Five of a kind (Royal Flush): This is the best poker move and is composed of all the high cards of the same suit which are 10,J,Q,K and As. In case there are 2 simultaneous hands like this one, none of the players would win, although if you think about it, it is almost impossible to have 2 simultaneous royal flushes.

• All the hands you have seen are ordered by ranking from least to most, so you also know who would win when two different hands meet. In the event that no player manages to form either of these hands, the one with the highest card wins, and in the hypothetical case of having the same, the one with the next highest card wins.

3.Texas Holdem Poker Actions

Now that you’ve seen all the hands and their priority, let’s review what actions you can take within the betting round, so don’t miss out!

• Betting: Bets are placed in turns clockwise, and players who bet later must match the amount (call) bet at least to stay in the hand. But you can also raise if you want to force the other players to stay in the hand, and if you do so they must match the amount again until the raise is matched. (It’s like a tug-of-war).
• All-in betting: When a player bets all his chips in a hand, it is said that he is all-in, and (if another player bets a larger amount), a separate pot is created in which the player is not. If more than one player goes all in in the same hand, different pots will be accumulated. To go all in you need to have a good feeling because you risk losing all your chips.
• Betting Blinds: In the hold’em mode there are two players obliged to bet a minimum amount at the beginning of each round, which are the blinds. These are the blinds. The blinds change players in each hand, with the objective that all have equal losses/the same losses at the end. The Big Blind is equivalent to the minimum bet for that round and the Small Blind is usually half the Big Blind, but this can change. These blinds are posted consecutively to the left of the dealer, first the Small and then the Big, and are intended to force players who want to enter the hand with a minimum amount. In the tournaments the blinds are increased in quantity to accelerate the speed of the games.
• Check: When no player has made any bets in that round or the bet amount has already been matched, you can choose stay or “check” if you consider that it is not necessary to open or raise the bet with your cards. This allows you to remain in the hand, without making any extra bets.
• Checkout: The player always has the option to check out if he feels he has no options in that hand, and will lose his cards and the amount he had bet up to that point, with no possibility of re-entering.

You’ve seen what actions you can take, and what hands there are, but what chances do you think you have of linking any? Do you think it’s easy to get a straight or a poker?

4.Texas Holdem Poker Probabilities

Let’s calculate the probabilities of a Four of a Kind. To tie this hand you need 4 cards of the same value out of 5 cards that are dealt. That is: The ways you have to get 4 equal cards (with 1 possible value of 13 and 4 possible suits of 4) by the ways to get another card that is not of that value, (with 1 possible value of the remaining 12 remaining, and logically 1 suit of 4). What it actually calculates for both, are the ways to select the values by the ways to select the suits.

This gives us exactly 624 total possible ways to make Four of Kind from all the combinations you can make, but how many are there?

If we take into account that a French deck has 52 cards and you must select 5 cards, the total number of possibilities are all the possible combinations of 52 elements taken in 5 since if you think about it carefully (and not like in this video) the order does not matter and they cannot be repeated. (but this is better thought out by you)

This calculation indicates that the probability of linking a Four of a Kind before uncovering any cards is approximately 0.02%, which means that you have very little chance a priori of linking it as it’s a very unlikely hand. But what do you think are the probabilities of the other hands?

As you see the important hands like Four of a Kind or Full House are the most difficult to get, and vice versa, the less important hands are much easier and more likely, which makes sense. Note also that the only hand you can often aspire to is a pair, as hands such as three of a kind, flush or even double pair are quite complicated to achieve.

The most important thing is that none of the hands reaches 50%, and not even the sum of all of them gets 50%, which means that most of the time you are not going to get anything, although this gives you some clues on how to approach the game. Was this what you imagined?

You have already seen that to get a good hand is not easy, then what you will have normally will be cards that can conform a hand with the cards that are left to uncover, or what is the same a hand draw, this will be able to be completed or not with certain probability. But how do you calculate?

5.Poker as a game of incomplete and imperfect information

If you have ever watched a poker tournament on TV, you will surely remember a percentage that is shown next to each player’s cards. This is nothing more than the probability each player has of winning and is based on both his cards and those of the other players, including the community cards. If we think about it, this calculation is only possible because the television knows the cards of each and every one of them, so only we can see what real probabilities they have and who is playing better.

If the players could know the hand of their rivals, they wouldn’t have any difficulty to play as they could calculate these probabilities in an exact way, but unfortunately this is not possible, because we don’t know what cards they have, that’s why poker is considered a game of incomplete and imperfect information. Consequently the players have to try to collect all the possible information, to calculate something that approaches as much as possible to those probabilities.

6.Odds and Outs

For this we must introduce two mathematical concepts that are used in the world of Poker that are the odds and outs, and that serve to calculate the probabilities of completing a draw or draws.

Imagine that you have a flush draw in your turn and you want to calculate the probabilities of being able to complete it in the last betting round.

To complete you draw with the following card you need one that is of the same suit. Since a suit has 13 cards and 4 of them have already come out, there are only 9 cards left in the deck that would serve to complete your flush, and this gives a total of 9 outs.

If we want to calculate the probabilities for those 9 outs, what we will do is calculate the odds, which is the probability used in poker and unlike the usual one, where favorable cases are calculated among the possible ones, here the unfavorable ones are calculated among the favorable ones. And why like that? Because normally we have more probabilities of not linking our draw than of linking it, but this you will see later.

The favorable cases are precisely the outs and the unfavorable are all possible cases except the favorable (ie all except the outs). Finally, the possible cases are the amount of cards that remain to be shown, which we calculate by subtracting from the 52 total cards of the deck, those that have already been uncovered including also those of the rivals. If you do the calculation for your 9 outs, you will get an odds of 4.11, or in other words, for every time you win you will lose 4.11 times, which is equivalent to a 19.57% probability of linking the flush with the last card to be drawn. What do you think?

Calculating all this is not very practical, so I have created a chart, where you can check what odds you have with the outs of your draw or draws.

If you take a look you can see the table is basically composed of 3 columns, which are the probabilities that can be calculated in the stages of flop and turn ordered from left to right, where we are mainly interested in the first and last.

If we want to consult our flush draw with 9 outs in the turn we see the same 4.11 that we had calculated on our own.

DOWNLOAD POKER ODDS CHART

With these tables you can see interesting things, like that a flush draw has more probabilities than any straight draw, or that a straight draw has more probabilities if it can be completed by two points instead of one.

Notice also that any draw that includes 2 simultaneous hands (like a straight and flush draw at the same time) is always going to be more likely than either of those two separate draws. Since it can be completed in either way, it accepts a greater number of outs. Curious, isn’t it?

Another important detail is that you will almost never have more than a 50% chance, which indicates as I said before that most of the time you will not link your draw. (Even if you don’t throw in the towel yet)

On a practical level, you can calculate all this, without the need for tables, with the rule of 4 and the rule of 2 (multiplying by 4 in the flop a turn and river and by 2 in the turn). These rules provide a fairly close value and can be useful when we do not have anything at hand. If you use them for your draw, your 9 outs in the turn are (9 x 2 = 18%) which is very close to the 19.57% shown in the table. This can be enough for many players, although there are other similar rules to approximate these values. There are also different odds variants if you are interested in the subject.

After seeing all this you could ask yourself the following questions:

• If we normally have a small chance, what’s the point of betting, shouldn’t we do it only when we have a majority of possibilities?
• Even if we are calculating the odds for our draw, can’t the rival beat us with a higher draw? Or in other words, how much difference is there between calculating our odds and the real odds?

7.Texas Holdem Poker Calculator

To solve your doubts I have created a Poker Calculator where we are going to experiment a little bit with all these concepts.

With it you can see that a simple pair of 77’s has more probabilities in front of an ace, k, or that a straight and flush draw has more probabilities than a brand new pair of kings on the flop.

You can also see that a pair with low values like 66 is not as good a hand either, as it can be easily outdone by any higher pair. You can consult any hand you need.

GO TO THE POKER CALCULATOR

Now let’s create with our Poker Calculator your flush draw. And let’s suppose that your opponent aspires to an inferior draw with a pair of J’s. We press the button to calculate the result and we see that your chances of winning against their cards are 20% which is quite similar to what we had calculated.

It may seem that the odds are enough to know for sure our probabilities. But imagine that the cards would allow your opponent to have a superior draw like Poker or Full House. If we recalculate your probabilities for this possible situation, we see that the calculation no longer fits the odds. This happens because before, your probability of winning depended only on whether or not you picked up your draw, but now you also depend on whether or not your opponent gets a higher draw, which decreases your options as you can be beaten with a better hand than the flush.

Things are getting interesting…

Note that the most important detail is that although the calculation is different, it’s not too big a difference, because it is still more difficult to link a Full or Four of a kind than a flush. This indicates that the stronger your hand draw is, the more reliable your odds will be on average, since it will be more difficult for another draw to modify them. Depending more on ourselves gives us more complete information, and we must take advantage of it.

But in practice it is easy that your opponents can have better cards than you, especially when there are many players at the table or big bets are made, that’s why you should also always have in mind the possible draws or hands that your opponents can have with the community cards, especially when a big amount of money is bet.

When your tied hand is superior to all those possible ones, your victory will be assured, but when you have only one draw and there can’t be higher hands with the cards that are left to reveal, your real probabilities will be at least your odds (calculated probability).

The post The Mathematics of Poker – Odds and Outs first appeared on Math4all.