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This is a tutorial for calculating poker probabilities from scratch. It is what I learned in my degree and even beforehand when I created and 5-card draw hand strength calculator. The intention of this post is to make a cleaner community without similar questions cluttering our Q&A site, with more people understanding the importance of.

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.

The chapter Operating with and Weighting the Odds of the book Texas Hold’em Poker Odds for Your Strategy, with Probability-Based Hand Analyses deals with the rules of estimating and evaluating the probabilities of complex gaming events in order to use them in a strategy, by operating with the odds of the simple events. How Important Are Your Texas Holdem Poker Hand Pot Odds? Your pot odds inform many of your decisions in Texas Hold’em poker. Of course, but there are other factors as well. It is crucial to study your opponents and observe their tendencies to better appreciate the quality of their bets. If you want to learn about an important concept that’s very related to pot odds, read this article on using implied odds. Till’ next time, good luck, grinders! NOTE: Want to become a poker boss in under an hour? Start making profitable plays with confidence when you get the $7 Postflop Game Plan.

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.

Why Is This Important?

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 cards
consist of four of each rank below the aces. At the beginning of the hand you
don’t know where any of the other cards are located, but as the hand progresses
you learn where some of them are located.

Continuing with the example, if the flop has an ace and two fours, you hold a
full house. You also know the only hand at this time that can beat you is four
fours. Because two fours are on the flop, the number of times a single opponent
has the other two fours is 1 in 1,326 hands. This is such a small percentage of
the 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 four
of 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 of
these two cards are two out of 52. If they get one of them as the first card
that leaves the single other card they need out of 51 unseen cards, or one out
of 51.

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

Basic Texas Holdem Math

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.

Starting Hands

At the most basic level of Texas holdem everything starts with your starting
hand. As we mentioned above, mathematically the player who stars the hand with
the 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 play
better starting hand on average than your opponents. While this can get
complicated, especially in games with many multi way pots, you still need to
learn how to play better starting hands.

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

Position

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

Outs

One of the most important skills Texas holdem players need to develop is the
ability to determine the number of outs, or cards remaining in the deck that can
complete the hand they’re drawing to. You use this information to determine your
chances of winning the hand as well as to determine the pot odds. Pot odds are
discussed in the next section, but they show you whether or not a call is
profitable 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 deck
and what we need to improve our hand. If you have a king, queen, jack, and 10
after the turn you know any of the four aces or four nines complete your
straight.

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

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

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

Pot Odds

The next question many players ask after they learn how to determine their
out sis 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 deter

Important Poker Probabilities