• A “poker hand” consists of 5 unordered cards from a standard deck of 52. There are 52 5 = 2,598,9604 possible poker hands. Below, we calculate the probability of each of the standard kinds of poker hands. This hand consists of values 10,J,Q,K,A, all of the same suit. Since the values are fixed, we only need to choose the suit.
• A “poker hand” consists of 5 unordered cards from a standard deck of 52. There are 52 5 = 2,598,9604 possible poker hands. Below, we calculate the probability of each of the standard kinds of poker hands. This hand consists of values 10,J,Q,K,A, all of the same suit. Since the values are fixed, we only need to choose the suit.

In poker, the probability of many events can be determined by direct calculation. This article discusses computing probabilities for many commonly occurring events in the game of Texas hold 'em and provides some probabilities and odds for specific situations. In most cases, the probabilities and odds are approximations due to rounding. When calculating probabilities for a card game such as.

Thus, a probability of 1.0 (certainty) is 0:1 odds. 1:1 odds represents 0.50 probability ('even money'). Odds are useful because they reflect the fair payout on a bet: if you were to bet $1 on something with a 0.25 probability, the fair payoff would be $3.

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.

In the poker game of Texas hold 'em. There are 25 starting hands with a probability of winning at a 10-handed table of greater than 1/7. Limit hand rankings. Some notable theorists and players have created systems to rank the value of starting hands in limit Texas hold'em. These rankings do not apply to no limit play. Texas Hold'em Poker probabilities. When calculating probabilities for a card game such as Texas Hold'em, there are two basic approaches. The first approach is to determine the number of outcomes that satisfy the condition being evaluated and divide this by the total number of possible outcomes. Texas holdem reglas wikipedia Poker Room for additional details. Thanks for bringing this back. The payoff is 11 to 1. How Casinos Struggle with a Roulette Betting System The result of applying the theory mentioned in the previous section would be encouraging, as it claims that the probability of winning a bet on even chances is over 99. From Texas Hold'em For Dummies. By Mark Harlan. Texas Hold’em poker is everywhere these days — on TV, online, and in clubs and casinos. Before you sit down to a game of Texas Hold ’em, make sure you’re in good shape to be successful — take care of non-poker issues and check your physical, mental, and financial status. The format of Ultimate Texas Hold ‘Em is similar to other variants of poker available in most casinos and online poker sites. The player and the dealer will both get two cards. The former will then be allowed to look at their own cards and decide if they wish to check or raise four times the ante. The format of Ultimate Texas Hold ‘Em is similar to other variants of poker available in most casinos and online poker sites. The player and the dealer will both get two cards. The former will then be allowed to look at their own cards and decide if they wish to check or raise four times the ante.

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.

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.

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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:

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.
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Conclusion

Texas Holdem Poker Practice

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.

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.

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Figuring out poker odds and probabilities can seem like an overwhelming task. However, with a little research and memorization, this seemingly dense topic can be made much simpler. Once you understand basic situations, such as the likelihood of being dealt certain hands, hand versus hand scenarios, how your odds change with each new card, pot odds and implied odds, you will have the kind of primer you need to become a more effective player.

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How Come I Never Get Dealt Pocket Aces?

One of the most common complaints, especially among inexperienced players, is that they never get good cards. Sometimes that just comes down to a common misconception. Televised poker has the advantage of being edited down to where only the most interesting hands and the biggest pots are shown, so you are more likely to see players with pocket aces or ace-king suited than a 2-7 off-suit.

Televised poker highlights the best and most interesting hand. Your chances of being dealt pocket Aces in a game are not nearly as high.

In reality, however, your chances of being dealt any specific pair in a game of Texas Hold ‘Em are 220:1. And since most games you play at a casino will only deal about 30-45 hands an hour, even with professional dealers, you can’t expect a never-ending string of aces and kings.

Hand vs. Hand

This type of scenario comes into play most often in tournaments. And since there are so many basic scenarios that can be applied across a number of different types of match-ups, it’s mainly about memorizing those basic odds.

Poker Probability Texas Holdem Wikipedia

For instance, when you have a pocket pair and are facing only one opponent with a lower pocket pair, you are about a 4.5:1 favorite, meaning that for every eleven times the hand plays out you are likely to win nine. Another frequently seen scenario would be a pocket pair versus two over cards, e.g., pocket queens versus ace-king. In that case, which is often referred to as a race, the pocket pair is about a 1.2:1 favorite, essentially a coin flip.

Here are some odds for other basic hand-to-hand match-ups in Texas Hold ‘Em:

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• Pair vs. 2 Under Cards – 4.9:1
• Pair vs. 1 Over Card, 1 Under Card – 2.5:1
• 2 Over Cards vs. 2 Under Cards – 1.7:1

In-Hand Odds

This is where poker odds begin to get more complicated. No matter what kind of game you’re playing, as the hand progresses, more cards get dealt and bets get made, the odds change accordingly. In Hold ‘Em, just because you have the best hand on the flop doesn’t mean you’re the projected favorite to win once all the cards are dealt. Top pair on the flop may seem sweet, but if there are possible flush and straight draws, you may just be putting money into what will eventually become someone else’s pot.

One way to simplify the complexity of post-flop betting is to look at the odds certain types of draws face. That way you can decide whether it is worth it to draw against a better hand, or how much you need to bet to protect your hand. Like head-to-head odds, there are a few basic scenarios, and if you memorize the odds inherent in them, they can be applied as a rule of thumb. They will also help you better understand more difficult concepts like pot and implied odds.

Wikipedia Poker Probability Definition

For in-hand odds you need to know how many outs a certain hand has, e.g., in Hold ‘Em, a flush draw has nine outs – you have two of the suit in your hand and there are two on the board, meaning there are nine of that suit left that would give you a flush. Then it’s a matter of knowing what the chances are one of those outs will come on the turn and/or river.

Wikipedia Poker Probability Game

Here are some odds for common post-flop scenarios in Hold ‘Em:

Wikipedia Poker Probability Random

• Inside Straight Draw (4 outs) – 10.8:1 on the turn, 5.07:1 turn or river
• Open-Ended Straight Draw (8 outs) – 4.88:1 on turn, 2.18:1 on turn or river
• Flush Draw (9 outs) – 4.22:1 on turn, 1.86:1 on turn or river

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Pot and Implied Odds

Texas Holdem Probability Calculator

This final category of odds involves a little more subjectivity than the previous ones, and even invites a little bit of artistry on the player’s part. Pot odds are fairly simple. You look at the situation, i.e., you have a flush draw after the flop, and calculate how much money is in the pot and what it will cost you to call a bet. Then you figure out whether the potential payoff makes it worth calling to see another card.

In Hold ‘Em, with a flush draw on the flop you are likely to make your hand a little more than 1 out of every three times if you see the turn and river. But on the turn only the odds of hitting your flush are slightly worse than 4:1. So if there is already $120 in the pot and your opponent bets $40, you would be calling $40 to win $160 (the previous pot plus his bet), meaning you have 4:1 pot odds, since you would win four times the amount you are putting in.

In this particular scenario, the pot odds line up almost exactly: 4:1 on your money, approximately 4:1 that you hit on the next card. This is where experience and intuition would come into play. Analyze the situation, your opponent and your feeling about the likely direction of the hand to make a decision.

Implied odds are where you factor in the amount of money you think is likely to be added to the pot later in the hand by your opponent(s) after your bet or call. In that scenario, you may think your opponent will call even if you make your flush. That means, not only are you calling to win the $160, you think you’ll be able to get another x amount of dollars out of him on the turn or river.

Once you understand how to factor odds into your betting equation you will come to the table with far more tools and information at your disposal – and in poker, the more tricks and knowledge you have, the better your odds of winning.