Basic Poker Statistics

The main underpinning of poker is math – it is essential. For every decision you make, while factors such as psychology have a part to play, math is the key element.

  1. Basic Poker Statistics Games
  2. Basic Poker Statistics Definition

In this lesson we’re going to give an overview of probability and how it relates to poker. This will include the probability of being dealt certain hands and how often they’re likely to win. We’ll also cover how to calculating your odds and outs, in addition to introducing you to the concept of pot odds. And finally we’ll take a look at how an understanding of the math will help you to remain emotional stable at the poker table and why you should focus on decisions, not results.

What is Probability?

Probability is the branch of mathematics that deals with the likelihood that one outcome or another will occur. For instance, a coin flip has two possible outcomes: heads or tails. The probability that a flipped coin will land heads is 50% (one outcome out of the two); the same goes for tails.

Ben Affleck is such a good poker player, he once won the California State Poker Championship, which had a prize of $356,000 and a qualification for the World Poker Tour Championship. The Yakuza got their name from the worst hand dealt in poker. Ya 8 + Ku 9 + Za 3. Learning poker statistics is also as crucial as understanding the hands in a poker game. You have to be familiar with the most widely used stats to understand how you should play. Likewise, knowing your stats gives you the chance to improve areas of your strategy. Important note: Poker can be as complex depending on the game and players.

Probability and Cards

When dealing with a deck of cards the number of possible outcomes is clearly much greater than the coin example. Each poker deck has fifty-two cards, each designated by one of four suits (clubs, diamonds, hearts and spades) and one of thirteen ranks (the numbers two through ten, Jack, Queen, King, and Ace). Therefore, the odds of getting any Ace as your first card are 1 in 13 (7.7%), while the odds of getting any spade as your first card are 1 in 4 (25%).

Unlike coins, cards are said to have “memory”: every card dealt changes the makeup of the deck. For example, if you receive an Ace as your first card, only three other Aces are left among the remaining fifty-one cards. Therefore, the odds of receiving another Ace are 3 in 51 (5.9%), much less than the odds were before you received the first Ace.

Want to see how poker math intertwines with psychology and strategy to give you a MASSIVE EDGE at the tables? Check out CORE and learn poker in the quickest and most systematic way:

Pre-flop Probabilities: Pocket Pairs

In order to find the odds of getting dealt a pair of Aces, we multiply the probabilities of receiving each card:

(4/52) x (3/51) = (12/2652) = (1/221) ≈ 0.45%.

To put this in perspective, if you’re playing poker at your local casino and are dealt 30 hands per hour, you can expect to receive pocket Aces an average of once every 7.5 hours.

The odds of receiving any of the thirteen possible pocket pairs (twos up to Aces) is:

(13/221) = (1/17) ≈ 5.9%.

In contrast, you can expect to receive any pocket pair once every 35 minutes on average.

Pre-Flop Probabilities: Hand vs. Hand

Players don’t play poker in a vacuum; each player’s hand must measure up against his opponent’s, especially if a player goes all-in before the flop.

Here are some sample probabilities for most pre-flop situations:

Post-Flop Probabilities: Improving Your Hand

Now let’s look at the chances of certain events occurring when playing certain starting hands. The following table lists some interesting and valuable hold’em math:

Basic poker statistics cheat

Many beginners to poker overvalue certain starting hands, such as suited cards. As you can see, suited cards don’t make flushes very often. Likewise, pairs only make a set on the flop 12% of the time, which is why small pairs are not always profitable.

PDF Chart

We have created a poker math and probability PDF chart (link opens in a new window) which lists a variety of probabilities and odds for many of the common events in Texas hold ‘em. This chart includes the two tables above in addition to various starting hand probabilities and common pre-flop match-ups. You’ll need to have Adobe Acrobat installed to be able to view the chart, but this is freely installed on most computers by default. We recommend you print the chart and use it as a source of reference.

Odds and Outs

If you do see a flop, you will also need to know what the odds are of either you or your opponent improving a hand. In poker terminology, an “out” is any card that will improve a player’s hand after the flop.

One common occurrence is when a player holds two suited cards and two cards of the same suit appear on the flop. The player has four cards to a flush and needs one of the remaining nine cards of that suit to complete the hand. In the case of a “four-flush”, the player has nine “outs” to make his flush.

A useful shortcut to calculating the odds of completing a hand from a number of outs is the “rule of four and two”. The player counts the number of cards that will improve his hand, and then multiplies that number by four to calculate his probability of catching that card on either the turn or the river. If the player misses his draw on the turn, he multiplies his outs by two to find his probability of filling his hand on the river.

In the example of the four-flush, the player’s probability of filling the flush is approximately 36% after the flop (9 outs x 4) and 18% after the turn (9 outs x 2).

Pot Odds

Another important concept in calculating odds and probabilities is pot odds. Pot odds are the proportion of the next bet in relation to the size of the pot.

For instance, if the pot is $90 and the player must call a $10 bet to continue playing the hand, he is getting 9 to 1 (90 to 10) pot odds. If he calls, the new pot is now $100 and his $10 call makes up 10% of the new pot.

Experienced players compare the pot odds to the odds of improving their hand. If the pot odds are higher than the odds of improving the hand, the expert player will call the bet; if not, the player will fold. This calculation ties into the concept of expected value, which we will explore in a later lesson.

Bad Beats

A “bad beat” happens when a player completes a hand that started out with a very low probability of success. Experts in probability understand the idea that, just because an event is highly unlikely, the low likelihood does not make it completely impossible.

A measure of a player’s experience and maturity is how he handles bad beats. In fact, many experienced poker players subscribe to the idea that bad beats are the reason that many inferior players stay in the game. Bad poker players often mistake their good fortune for skill and continue to make the same mistakes, which the more capable players use against them.

Decisions, Not Results

One of the most important reasons that novice players should understand how probability functions at the poker table is so that they can make the best decisions during a hand. While fluctuations in probability (luck) will happen from hand to hand, the best poker players understand that skill, discipline and patience are the keys to success at the tables.

A big part of strong decision making is understanding how often you should be betting, raising, and applying pressure.
The good news is that there is a simple system, with powerful shortcuts & rules, that you can begin using this week. Rooted in GTO, but simplified so that you can implement it at the tables, The One Percent gives you the ultimate gameplan.

This 7+ hour course gives you applicable rules for continuation betting, barreling, raising, and easy ratios so that you ALWAYS have the right number of bluffing combos. Take the guesswork out of your strategy, and begin playing like the top-1%.

Conclusion

A strong knowledge of poker math and probabilities will help you adjust your strategies and tactics during the game, as well as giving you reasonable expectations of potential outcomes and the emotional stability to keep playing intelligent, aggressive poker.

Remember that the foundation upon which to build an imposing knowledge of hold’em starts and ends with the math. I’ll end this lesson by simply saying…. the math is essential.

Related Lessons

By Gerald Hanks

Gerald Hanks is from Houston Texas, and has been playing poker since 2002. He has played cash games and no-limit hold’em tournaments at live venues all over the United States.

Related Lessons

Related Lessons

Share:

Introduction to the Mathematics of Poker

Let me get you a little bit into the math of poker. First of all, let’s consider its importance on a hand-to-hand basis. First of all, when you’re dealt a hand in poker, and it’s your turn to act, depending on the action that you got from the other players who acted before you, you have to make the decision either to raise, call or fold.
First of all, you consider your hand. Is it a strong hand that I could get all my money in pre-flop with and be profitable? Or do I have a drawing hand, like a suited connector? And what amount of big blinds can I call at most in this spot to be able to play my hand profitably on the river? Can I re-raise as a bluff and will this bring me profit in the long run?
If all the answers here are no, no and no then you’re going to fold. But are you sure you calculated it correctly? We’ll get into depth on these matters in this article.

Also, let’s say you hold a strong top pair and you’ve got to the river with it, and your opponent over-bet shoves. Do we have the right odds to call here? What is he representing here? And can we put him on a range? And after that can we call profitably? These questions will be answered also.

Pot Odds

I always go for percentages everywhere in poker, because I find it a lot easier to understand and also apply. The first thing I learned when I was starting out with poker, was pot odds, and I think they should be the foundation to every poker player’s knowledge.
To explain this as simple as do re mi, for you to be profitable, the breakeven equity that you need to make a call when you get bet into, is the amount that you have to put into the pot divided by the total pot size (including your bet – so the amount that you’re winning when you have the stronger holding).

Let’s first take an example:

Basic Poker Statistics Games

You are playing poker against Phil Ivey, and he bets into you on the river. You’re currently holding AJ, and the board is A2742. Some guy told you that he could either have AT, AJ, AQ or AK, and you’re facing a 2/3 pot bet.
Let’s face it, you’re behind. But what is the amount of hands that beats you relative to the size of the bet you have to call? Good question.
So you beat AT and get beat by AQ and AK, and split with AJ, which we shall discount because it’s 0 EV.
Obviously, the same number of combinations exist for AT, AQ and AK, so we win 1 time out of three!
Our estimated equity is 33%, but are the pot odds low enough? If they are, it’s a sure call!

This is how you calculate:

You need to call 2/3 into a pot that will contain your 2/3 bet, the pot size which is 3/3 and the opponent’s bet which is also 2/3. That means that you have to call 2/3 to win a pot of 7/3, so your breakeven equity will be:
(2/3)/(7/3)*100(to display in percentage) = (2/7)*100 = 28%.
We know from the logic above that we have 33% equity, so we have greater equity than the pot odds, so even though we’re behind, we can still call here profitably!

Basic Poker Statistics Definition

You have 1 dollar and you hate bluffing, but someone offers you the chance to crack his pocket AA’s with 72o and win 100 dollars if you do it. Assuming this is a legit deal, you only need to put in 1 dollar to win 100 so this is
(1/100)*100=1% breakeven frequency.
Last time I checked Equilab, 72o has around 12% chance to crack aces, so it’s a sure call. I would take this deal every time I get the occasion. Here are some default pot odds that you should know by heart, because they will prove very useful when thinking about calling:
  • 1/3 Pot – 20%
  • 1/2 Pot – 25%
  • 2/3 Pot – 28%
  • 3/4 Pot – 30%
  • 4/5 Pot – 31%
  • Full Pot – 33%
  • 1.5x Pot – 37%
  • 2x Pot – 40%

Fold equity

If you’re the one who’s betting, there’s always a combination of your hand’s equity and your total fold equity involved. Let’s say you’re bluffing the river this time, and you’re wondering how to determine the amount of times he has to fold to make your bet profitable, look no further!
Let’s say that we’re betting with a hand that, if we get called, we can never win, like a busted flush draw on the river. If we think that the opponent will fold enough times, we can make this bet.
The formula is: Breakeven Fold Equity = (your bet) divided by (the sum of your bet and the pot size).
Thinking about this, it becomes quite logical that if you bet full pot, you need him to fold 50% of the time, because 1/(1+1) = 1/2 = 50%. Some other frequencies that are good to remember are:
  • 1/3 Pot = 25%
  • 1/2 Pot = 33%
  • 2/3 Pot = 40%
  • 3/4 Pot = 42%
  • 4/5 Pot = 44%
  • 1.5x Pot = 60%
  • 2x Pot = 66%
  • 3x Pot = 75%.
It’s rumored that a certain player named Isildur1 has used the latter numbers to his complete advantage! What if you thought about that first? You’d be probably playing the higher stakes. So now, having showed you how cool this math stuff is, let me show you how to apply it.

Pre-flop actions

First of all, let’s get into notice some flop hitting probabilities:
  • A pair – 29%
  • Two Pairs – 2%
  • A Set (when holding a pocket pair) – 12%
  • Trips – 1.35%
  • A Full house – 0.09%
  • Four of a Kind – 0.01%
  • A pair or better - 32%
  • A flush holding 2 suited cards – 0.84%
  • A flush draw holding 2 suited cards – 11%
  • A straight with suited non-gapped connectors – 1.31%
  • An open-ended straight draw with non-gapped – 10.5%
  • A straight with one-gapers – 0.98%
  • An OESD with one-gapers – 8.08%
  • A straight with two-gapers – 0.65%
  • An OESD with two-gapers – 5.2%
  • A gutshot (suited connectors) – 16.6%
  • Any unsuited hand flopping 2p+ - 3.45%
  • Any suited hand flopping 2p+/flush – 4.29%
  • Suited connectors flopping 2p+/straight/flush – 5.59%
Having known all these, we are now inclined to play a more calculated game pre-flop. Although the reason for calling a raise pre-flop should not be pure mathematical, sometimes it does help to know how often you’ll flop an OESD + FD + pair or better with suited connectors, which is close to 50%.
I don’t like playing small suited connectors in multi-way pots, because usually you get dominated by higher flushes and you get the dummy end of a straight. They also aren’t that great to 3-bet against a raiser because they don’t have blockers, so what are we supposed to do with them when we’re not stealing blinds?
Well, let’s say that the EP raiser is opening a wide array of hands, like 14% and there’s also a caller, so we’re thinking with 56s in the BTN. This is a perfect spot to do the squeeze play, because 1) you’re getting a massive amount of folds, and 2) When you get called you will flop a lot of strong draws that you can represent along with your pairs and 2p+.