Is poker a game of skill or chance? This question has been discussed and argued in many places and is the center of the arguments for and against legalizing Texas holdem and other forms of poker in many places, including online.

The answer to this question boils down to the mathematics behind the game. If the math shows one player can win more often than another based on the mathematical and statistical truths about Texas holdem then the game is one of skill.

Let’s look at a few facts before moving on.

Fact 1

Texas holdem is played with a deck of 52 playing cards, consisting of the same four suits, and 13 ranks in every deck. You know each deck has an ace of spades, and ace of hearts, an ace of clubs, and an ace of diamonds. The same is true for kings, queens, and all of the ranks down through twos.

Fact 2

Over a long period of time each player will play from each position at the table an equal number of times. In other words, each player will play in the small blind, the big blind, under the gun, on the button, etc. an equal number of times as other players. If you take two individual players it might not be 100% the same, but it’ll be close. When you take thousands of players and average their times played in each position mathematically they each play the different positions an equal number of times.

Fact 3

The rules in each game are the same for every player at the table.

Fact 4

The player that starts the hand with a better two card starting hand wins the hand more often than the player with a worse hand. This has been proven by computer simulations that run millions of hands and consider every possible outcome.

The reason all of this is important to Texas holdem players is that you can use all of this math to help you win.

Though there are thousands of possibilities on every hand of Texas holdem,you can use the fact that everything is based on a set of 52 cards to predict outcomes and possibilities at every stage for every hand.

Here’s an Example:

If you start the hand with two aces as your hole cards, you know that the remaining 50 cards in the deck only have two aces. The remaining 48 cardsconsist of four of each rank below the aces. At the beginning of the hand youdon’t know where any of the other cards are located, but as the hand progressesyou learn where some of them are located.

Continuing with the example, if the flop has an ace and two fours, you hold afull house. You also know the only hand at this time that can beat you is fourfours. Because two fours are on the flop, the number of times a single opponenthas the other two fours is 1 in 1,326 hands. This is such a small percentage ofthe time that you always play the full house in this example as if it’s the best hand.

How do we know the number of times the opponent has the other two fours?

Because two fours are on the flop, let’s say the four of hearts and the fourof diamonds, so you know that your opponent has to have the four of clubs and the four of spades. The chances of the first card in their hand being one ofthese two cards are two out of 52. If they get one of them as the first cardthat leaves the single other card they need out of 51 unseen cards, or one outof 51.

You multiply two over 52 times one over 51 and this gives us the 1 out of 1,326 hands.

Some of the math we discuss on this page can be complicated and the truth is some players won’t be able to use it all. But that doesn’t mean they can’t be winning Texas holdem players. The math covered in this section forms the building blocks for the advanced math covered lower on the page.

Every Texas holdem player can use the basic math included in this section, and if you aren’t using it yet you need to start right away.

At the most basic level of Texas holdem everything starts with your startinghand. As we mentioned above, mathematically the player who stars the hand withthe better starting hand wins more than the player with the inferior hand.

This means the first math lesson you need to learn and start using is to playbetter starting hand on average than your opponents. While this can getcomplicated, especially in games with many multi way pots, you still need tolearn how to play better starting hands.

If you take nothing else from this page, if you simply tighten up yourstarting hand selection it’ll immediately improve your results.

It’s difficult to directly relate position to mathematics, but the main thinto know is the later your position, the better your chances to play in apositive expectation situation. We’ll discuss expectation in a later section,but it’s important to understand that having position on an opponent is a strongadvantage that equates to a mathematical advantage over the long run.

One of the most important skills Texas holdem players need to develop is theability to determine the number of outs, or cards remaining in the deck that cancomplete the hand they’re drawing to. You use this information to determine yourchances of winning the hand as well as to determine the pot odds. Pot odds arediscussed in the next section, but they show you whether or not a call isprofitable in the long run when an opponent makes a bet.

We can determine how many outs you have because we know what’s in the deckand what we need to improve our hand. If you have a king, queen, jack, and 10after the turn you know any of the four aces or four nines complete yourstraight.

This means you have eight outs. You’ve seen six cards, so the deck has 46cards remaining in it. Don’t make the mistake of thinking about the cards thathave been folded or your opponent holds. You haven’t seen these cards so anyunseen card is still considered a possible river card.

In other words, on average, if you play this situation 46 times you’re goingto complete your straight eight times and not complete it 38 times.

You should always consider how many outs you have in every situation whileplaying. B knowing your outs you have another piece of information that can help you make profitable decisions throughout the hand.

The next question many players ask after they learn how to determine their outs is how they can use this information to make more money at the table. This is where pot odds come into play.

Pot odds are simply a ratio or comparison between the money in the pot and the chances you have of completing your hand. You use this ratio to determine ifa call or fold is the best play based on the information you currently have.

If you consider the example in the last section concerning the straight draw,you know that the deck holds eight cards that complete your straight and 38cards that don’t. This creates a ratio of 38 to 8, which reduces to 4.75 to 1.You reduce by dividing 38 by 8.

The way you use this ratio is by comparing it to the amount of money in thepot and how much you have to put into the pot. If the pot odds are in your favorit’s profitable to call and if not you should fold.

Example:

If the pot has $100 in it and you have to make a $10 call the pot is offering10 to 1 odds. You determine this the same way as above, by dividing $100 by $10.

If you’re in the situation described above of drawing to a straight on theriver you can see that a call is correct because the pot is offering 10 to 1 andyou have a 4.75 to 1 chance of winning.

On the other hand of the pot has $100 in it and you have to put $40 in to seethe river the pot is only offering 2.5 to 1 odds and your chances of hittingyour straight are still 4.75 to 1 so you should fold.

Pot odds can get complicated, especially when you start considering how theywork when you’re determining the correct play with both the turn and river tocome.

Fortunately charts are available to quickly check the odds of hitting yourhand based on how many outs you have. We’ve included one next so all you have todo is determine your outs and compute the odds the pot is offering. Then comparethe two to see if it’s profitable to call or fold.

The "2 and 4" rule is a quick and simplified method to estimate your potential pot odds and help you make decisions during a hand. The rule is often used on the flop to quickly assess whether it’s profitable to continue with your hand based on the current pot size.

On the Flop:

• Count the number of outs you have. Outs are the cards that can improve your hand.
• Multiply the number of outs by 2 to get an estimate of your chances of improving your hand by the turn.

On the Turn:

• If you didn’t improve on the flop, you can use the same outs to estimate your chances of improving by the river.
• This time, multiply the number of outs by 4 to get an estimate of your chances of improving by the river.

Decision Making:

If the estimated percentage of improvement (based on the "2 and 4" rule) is higher than the percentage of the bet relative to the current pot size, it may be profitable to continue with the hand.

The formula is a rough approximation and doesn’t provide an exact probability, but it gives you a quick and easy way to make a decision based on basic pot odds. Remember that this is just a rule of thumb and may not be as accurate as more precise calculations. For more accurate pot odds calculations, you may want to use a poker calculator or more advanced techniques.

Number of Outs	Turn+ River (2 rounds left)	River Only (1 round left)
1	22.26 to 1	45 to 1
2	10.9 to 1	22 to 1
3	7 to 1	14.33 to 1
4	5.06 to 1	10.5 to 1
5	3.93 to 1	8.2 to 1
10	1.6 to 1	3.6 to 1
11	1.4 to 1	3.18 to 1
12	1.22 to 1	2.83 to 1
13	1.08 to 1	2.54 to 1
14	0.95 to 1	2.29 to 1
15	0.85 to 1	2.07 to 1
16	0.75 to 1	1.88 to 1
17	0.67 to 1	1.71 to 1
18	0.6 to 1	1.56 to 1
19	0.54 to 1	1.42 to 1
20	0.48 to 1	1.3 to 1

When you’re determining your pot odds for the turn and river you determine them on the turn and then if you don’t hit your draw you determine them again onthe river. This often happens, especially in limit Texas holdem. But if an opponent moves all in on the turn you simply use the turn and river combine dodds in your decision.

Many beginning Texas holdem players look at a discussion about expectation and instantly decide it’s too hard and ignore it. When they do this theyseverely hurt their long term chances at being a profitable player.

We’ve broken down how to look at situations while playing poker in a simplemanner that almost any player can use below. Do yourself a favor and go intothis with an open mind. Once you understand it at a simple level you can learnmore as you gain experience. You may be surprised at just how easy it gets todetermine positive and negative expectation with a little practice.

Expectation is what the average outcome will be if you play the samesituation hundreds or thousands of times. Once you determine the expectation youknow if a situation offers positive or negative results on average.

Your goal as a Texas holdem player is to play in as many positive expectationsituations as possible and avoid as many negative expectation situations aspossible.

You need to understand that expectation is something that can be applied toalmost any situation in poker, but it’s also subjective in many areas.

• If you play at a table where every opponent is better than you in the longrun you’re going to lose money. This is a negative expectation situation.
• If you play at a table where every opponent is a worseplayer than you it’s a positive expectation situation because you’re going towin in the long run.

The problem is determining whether a situation is positive or negativeexpectation when you sit down at a table with some players who are better than you and some who are worse.

You can find many situations where it’s easier to determine expectationmathematically, and we’ll teach you how to do this now. While this may seemoverly complicated at first, especially to do at the table while playing, you don’t need to know exactly how negative or positive a situation is, you onlyneed to know if it’s positive or negative.

Once you determine if a situation is positive expectation or negativeexpectation you simply remember the next time you’re in a similar situation.Once you start determining expectation you’ll find that you learn mistsituations quickly and only have to think through an occasional situation at thetable.

The best way to see how to determine expectation is by running through acouple examples.

You’re facing a bet after the turn and you have four to a flush.The pot had $400 in it and your opponent bet $100. You’re certain that if youmiss your flush draw you’ll lose and when you hit your flush draw you’ll win.

In order to see the river you have to call the $100 bet. When you lose youlose $100, and when you win you get back $600. You get your $100 back plus the$400 that was in the pot plus the $100 bet your opponent made.

Many players claim that part of the money already in the pot is theirs, butonce you put money into the pot it isn’t yours. The only way to get it back isto win the pot. So you can’t consider it in any other way when determiningexpectation.

The way to see if it’s positive or negative to call is to determine what willhappen on average if you play the same situation many times. Most players findit easiest to determine by pretending to play the hand 100 times.

In this example you’re going to hit your flush 9 out of 46 times. This means19.56% of the time you’re going to win and 80.44% of the time you’re going tolose. To make this simple we’ll round these numbers off to 20% and 80%.

If you have to put $100 in the pot 100 times your total investment is$10,000. The 80 times you lose you get nothing back. The 20 times you win youget $600. 20 times $600 is $12,000. When you take the $12,000 you win andsubtract the $10,000 you lose when you play the situation 100 times, you seethat you win $2,000 overall.

To determine how much you win on average per hand simply divide the $2,000 by100 to get a positive expectation of $20 per hand. This means that every timeyou’re in this situation you’ll win on average $20.

The truth is you may win a little more because we’re ignoring the river.Because you know you can’t win if you miss your flush, you always need to fold onthe river when you miss your draw. Every once in a while you may be able toextract a small bet from your opponent on the river when you hit your flush,increasing your average expectation. Sometimes it’s even correct for your opponent to call on the river in this situation. See the next example to seewhy.

Let’s say you’re playing the same hand as above but you have astraight and your opponent appears to be drawing to a flush. You’re on theriver, the pot has $600 in it, and the board has the third suited card hit on theriver.

If your opponent was drawing to the flush, they completed it and you’re goingto lose the hand. In this situation your opponent bets $20.

In this situation you clearly have to call.

Do you know why?

The reason you have to call is because you can’t know for certain your opponent was drawing to the flush. They may be bluffing or have two pair or anyother number of hands that aren’t as good as your straight.

Let’s look at the math behind this decision.

If you play the situation 100 times your total investment is $20 times 100,or $2,000.

When you win you get $640, consisting of the original $600 pot, your opponent’s $20 bet, and your $20 call. If you win three hands you get back$1,920 for a loss of $80, or 80 cents per hand.

If you win at least four times you’re in a positive expectation situation.Four wins nets $2,560 for an overall win of $560, or $5.60 per hand.

What this means is if your opponent is bluffing or has a weaker hand justfour times out of 100 or more, calling is a positive expectation situation. Fourtimes out of 100 is only 4%. You’ll win at least 4% of the time in thissituation.

The numbers get closer the more your opponent bets on the river, and thecloser the numbers get the more you’re going to need to use what you know aboutyour opponent to determine if a situation is positive or not.

Start looking at every decision you make at the Texas holdem tables in terms of positive and negative expectation. It’s hard at first, but the more you practice, the better you’ll get at predicting if a situation offers positive expectation.

Texas holdem math is often the only thing that separates winning and losingplayers. Take the time to learn the basics now so you can improve your game inevery way possible as you gain experience. This guide is the perfect place tostart for players of every experience level.