Probability Poker Texas Holdem

It’s not uncommon for people to hear of cheating when they hear the term “card counting”, but the technique doesn’t actually have anything to do with cheating at all. What’s more is that you don’t need to be a math wiz to be able to learn how to do it.

Nevertheless, it’s not uncommon for people to want to learn how counting cards works, and in poker specifically, it can be an effective strategy that can give you the edge over your opponents. When you first start learning how to count cards, you only need to get a hang of three simple things, namely Texas Hold’em odds, counting your outs and pot equity. This guide will show you the basics of counting cards so that you can improve your Texas Hold’em game:

How is a Texas Hold’em poker odds calculator useful? A poker odds calculator shows you the exact odds of your hand winning in any scenario. For example, you can give yourself pocket Aces, opponent 1 pocket Kings, and opponent 2 pocket Queens. The poker odds software will. Probabilities in Two-Player Texas Hold 'Em Introduction. This page examines the probabilities of each final hand of an arbitrary player, referred to as player two, given the poker value of the hand of the other player, referred to as player one. Combinations shown are out of a possible combin(52,5)×combin(47,2)×combin(45,2) = 2,781,381,002,400. The probability of being dealt a pair in Texas Hold’em is 5.88%, or odds of 1: 16. There are 13 pairs in Hold’em (22 – AA) and for each there are 6 ways to be dealt. There are 6 different ways to form a specific pair and there are 13 different pairs. Meaning there are unique hole card combinations that are a pair.

How to Count Cards: Counting Outs

Any good poker player needs to be able to count outs. Learning how to count outs will help you improve your game and give you excellent preliminary knowledge before you truly understand how to count cards in poker. So, what is an “out”? The term “out” in the context of poker refers to any card that will make your hand stronger or give you the potential of turning your hand into a winning one. To be able to identify cards that will do this to a hand, you need to have good knowledge of hand rankings. Thankfully, calculating outs is relatively simple:

Remember that counting cards is not an exact science. Unlike the example above, you will never know which cards your opponent is holding. As a result, you need to pay attention to how they play, when they produce their flop cards, how much they are betting while also considering the possible available combinations. Don’t forget that they could always be bluffing!

Counting Cards: Poker Pot Equity

You will be able to grasp pot equity when you get the hang of counting cards. It’s a natural extension of card counting and involves calculating the likelihood of your chances of forming a winning hand and thus taking the pot.

There’s a method for calculating pot equity, and it’s known as the “Rule of Two and Four”. It is only applied during the flop and river stages of a round, and that’s because they are the only two stages where more cards are revealed. Here the two simple rules within the Rule of Two and Four:

For example, if you had a draw with 12 possible outs on the flop, you would multiply this by four, giving you approximately a 48% chance of getting the right cards to complete your hand. Furthermore, if you are left with 12 outs on the turn, you would have a 24% chance of completing your hand. Calculating your pot equity can be extremely useful when it comes to determining your moves in a game, reducing the number of needless bets you have to make, and proving the importance of learning how to count cards in poker.

The underlying mathematics of this process is complex, but worth knowing if you want to calculate your equity on the fly. Say you have 10 outs on the turn with 46 cards left in the deck, your probability of hitting is 10/46. By imagining that there are 50 cards in the deck, the probability is 10/50, or 20/100, meaning that your chance of getting the pot equity is 20%.

However, the real probability of 10/46 is expressed as 21.7%, which would mean that the number of outs would have to be multiplied by 2.174 – an incredibly hard sum to do when your opponent just raised €50! Regardless of how you choose to use it, if you want to learn how to count cards, you need to know how to judge your pot equity.

Texas Hold’em Odds: Hole Cards

To give you a greater understanding of how difficult it can be to predict an opponent’s hand, as well as giving you a better insight into how to count cards effectively, it’s important to know the odds of receiving some of the best and worst hole cards. In addition, we’ll give you the probability of winning with these hands in a standard four-person game.

Now that you’ve learned the ins and outs of how to count cards, pot equity, and Texas Holdem odds, why not put your skills to the test of one of our online poker games?

Ever wondered where some of those odds in the odds charts came from? In this article, I will teach you how to work out the probability of being dealt different types of preflop hands in Texas Holdem.

It's all pretty simple and you don't need to be a mathematician to work out the probabilities. I'll keep the math part as straightforward as I can to help keep this an easy-going article for the both of us.

Probability calculations quick links.

A few probability basics.

When working out hand probabilities, the main probabilities we will work with are the number of cards in the deck and the number of cards we want to be dealt. So for example, if we were going to deal out 1 card:

The probability of dealing a 7 would be 1/52 - There is one 7 in a deck of 52 cards.
The probability of dealing any Ace would be 4/52 - There four Aces in a deck of 52 cards.
The probability of dealing any would be 13/52 - There are 13 s in a deck of 52 cards.

In fact, the probability of being dealt any random card (not just the 7) would be 1/52. This also applies to the probability being dealt any random value of card like Kings, tens, fours, whatever (4/52) and the probability of being dealt any random suit (13/52).

Each card is just as likely to be dealt as any other - no special priorities in this game!

The numbers change for future cards.

A quick example... let's say we want to work out the probability of being dealt a pair of sevens.

The probability of being dealt a 7 for the first card will be 4/52.
The probability of being dealt a 7 for the second card will be 3/51.

Notice how the probability changes for the second card? After we have been dealt the first card, there is now 1 less card in the deck making it 51 cards in total. Also, after already being dealt a 7, there are now only three 7s left in the deck.

Always try and take care with the numbers for future cards. The numbers will change slightly as you go along.

Working out probabilities.

Whenever the word 'and' is used, it will usually mean multiply.
Whenever the word 'or' is used, it will usually mean add.

This won't make much sense for now, but it will make a lot of sense a little further on in the article. Trust me.

Probability of being dealt two exact cards.

Multiply the two probabilities together.

So, we want to find the probability of being dealt the A and K. (See the 'and' there?)

Probability of being dealt A - 1/52.
Probability of being dealt K - 1/51.

Now let's just multiply these bad boys together.

P = (1/52) * (1/51)
P = 1/2652

So the probability of being dealt the A and then K is 1/2652. As you might be able to work out, this is the same probability for any two exact cards, as the likelihood of being dealt A K is the same as being dealt a hand like 7 3 in that order.

But wait, we do not care about the order of the cards we are dealt!

When we are dealt a hand in Texas Hold'em, we don't care whether we get the A first or the K first (which is what we just worked out), just as long as we get them in our hand it's all the same. There are two possible combinations of being dealt this hand (A K and K A), so we simply multiply the probability by 2 to get a more useful probability.

P = 1/2652 * 2
P = 1/1326

You might notice that because of this, we have also worked out that there are 1,326 possible combinations of starting hands in Texas Holdem. Cool huh?

Probability of being dealt a certain hand.

Two exact cards is all well and good, but what if we want to work out the chances of being dealt AK, regardless of specific suits and whatnot? Well, we just do the same again...

Multiply the two probabilities together.

So, we want to find the probability of being dealt any Ace andany King.

Probability of being dealt any Ace - 4/52.
Probability of being dealt any King - 4/51 (after we've been dealt our Ace, there are now 51 cards left).

P = (4/52) * (4/51)
P = 16/2652 = 1/166

However, again with the 2652 number we are working out the probability of being deal an Ace and then a King. If we want the probability of being dealt either in any order, there are two possible ways to make this AK combination so we multiply the probability by 2.

P = 16/2652 * 2
P = 32/2652
P = 1/83

The probability of being dealt any AK as opposed to an AK with exact suits is more probable as we would expect. A lot more probable in fact. Also, as you might guess, this probability of 1/83 will be the same for any two value of cards like; AQ, JT, 34, J2 and so on regardless of whether they are suited or not.

Probability of being dealt a range of hands.

Work out each individual hand probability and add them together.

What's the probability of being dealt AA or KK? (Spot the 'or' there? - Time to add.)

Probability of being dealt AA - 1/221 (4/52 * 3/51 = 1/221).
Probability of being dealt KK - 1/221 (4/52 * 3/51 = 1/221).

Probability Poker Hands Texas Hold Em

P = (1/221) + (1/221)
P = 2/221 = 1/110

Easy enough. If you want to add more possible hands in to the range, just work out their individual probability and add them in. So if we wanted to work out the odds of being dealt AA, KK or 7 3...

Probability of being dealt AA - 1/221 (4/52 * 3/51 = 1/221).
Probability of being dealt KK - 1/221 (4/52 * 3/51 = 1/221).
Probability of being dealt 7 3 - 1/1326 ([1/52 * 1/51] * 2 = 1/1326).

P = (1/221) + (1/221) + (1/1326)
P = 359/36465 = 1/102

This one definitely takes more skill with adding fractions because of the different denominators, but you get the idea. I'm just teaching hand probabilities here, so I'm not going to go in to adding fractions in this article for now! This fractions calculator is really handy for adding those trickier probabilities quickly though.

Poker Hand Probability Texas Holdem

Overview of working out hand probabilities.

Hopefully that's enough information and examples to allow you to go off and work out the probabilities of being dealt various hands and ranges of hands before the flop in Texas Holdem. The best way to learn how to work out probabilities is to actually try and work it out for yourself, otherwise the maths part will just go in one ear and out the other.