Texas Holdem Flush Probability

• Texas Holdem Flush Probability Cheat
• Texas Holdem Flush Probability Calculator
• Texas Holdem Flush Probability Chart

To calculate the frequency of four of a kind, first note that there are 13 different ranks in which you can get four of a kind. For any given rank, the possible hands that give four of a kind in that rank all include the four cards of that rank as well as any three additional cards. There are C_48,3 = 17,296 different ways of choosing these three additional cards, so we have a total of 13 · 17,296 = 224,848 different four of a kind hands. This gives a frequency of (224,848/133,784,560) = 0.0017.

The worst case for a possible straight flush is holding something like A2s, AKs, A5s, etc., where there's only one possible way to flop the straight flush. In that case, the probability is one in 50C3, or 19600. The best case is 45s.TJs, which is 4 in 19600, because there are 4 ways to flop the straight flush.

To find the frequency of straight flushes, sort all straight flush hands by the high card of the highest straight flush in the hand. For ace high straight flushes in any of the four suits you need the A - K - Q - J - 10 of the given suit and then any 2 of the remaining 47 cards. This gives a total of C_47,2 = 1,081 distinct hands. For straight flushes that are not ace high the same argument holds except that one of the remaining 47 cards would give you higher straight flush if it were in your hand (for example, if you have 10 - 9 - 8 - 7 - 6 in hearts, if one of your two other cards was a jack of hearts you would have a jack high straight flush). Therefore, in these cases there are only C_46,2 = 1,035 distinct straight flush hands. So the total number of straight flush hands is (1,081 · 4) + (1,035 · 4 · 9) = 41,584 hands (the nine in the second parenthesis comes from the fact that there are nine different possible non-ace high cards for straights - a 2,3, or 4 high straight can not occur). The corresponding frequency is then (41,584/133,784,560) = 0.00031.

• Probabilities in Texas Hold'em Introduction An understanding of basic probabilities will give your poker game a stronger foundation, for all game types. This article discusses all the important, and interesting, probabilities that you should be aware of. Probabilities in poker Probability means the degree of certainty that a possible event will.
• The probability of the turn card being a heart is then 9/47 &asymp 0.19. The probability of the turn card notbeing a heart but getting a heart on the river is (38/47) (9/46) &asymp 0.16. Therefore the total probability of getting a flush is approximately 0.19 + 0.16 = 0.35.
• In Texas Hold'em a hand where aces, kings and queens pair up preflop is very rare. At a 9 player table this scenario unfolds roughly every 17,000 hands. The odds are 1:16,830 and the probability is 0.006%. Queens does happen every now and then, for example during this hand at the Bike.

To count the number of full house hands, we divide up the types of full houses by looking at the two cards that are not used as part of the final hand. These two cards can either be a pair (but of a different rank than the triple or the pair you are using for the full house, or else you would have four of a kind), one of the two cards could be of the same rank as your pair (giving you two triples and one card of some different rank), or the two cards could be of different ranks from each other, the triple, and the pair.

• We first consider the case of the unused cards being a pair. We can choose the rank for the triple in 13 ways. Once a rank is chosen we can pick the three cards for the triple in C_4,3 = 4 ways. We can then choose the two ranks for the two pairs in C_12,2 = 66 ways. For each pair, once we have chosen the rank we can choose the cards for the pair in C_4,2 = 6 ways. So we have a total of 13 · 4 · 66 · 6² = 123,552 full house hands of this type.
• Now we consider the case of two triples. We can choose the ranks for the triples in C_13,2 = 78 ways, and for each triple we can then choose the cards for the triple in C_4,3 = 4 ways. There are then 44 remaining cards from which to choose the last card of the hand, so we have a total of 78 · 4² · 44 = 54,912 hands of this type.
• Finally we consider the case of two cards of different rank from each other, the triple, and the pair. As above, the cards for the triple can be chosen in 13 · C_4,3 = 52 ways and the cards for the pair can then be chosen in 12 · C_4,2 = 72 ways. We can choose the two ranks for the remaining two cards in C_11,2 = 55 ways, and for each rank we can choose any of the four cards of that rank. This gives a total of 52 · 72 · 55 · 4² = 3,294,720 hands of this type.

Therefore, we have a total of 3,473,184 full house hands. This gives a frequency of (3,473,184/133,784,560) = 0.02696.

For additional calculations, as well as the frequencies for 5-card poker hands (which tend to be significantly easier to calculate), see for example Wikipedia.

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 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%.

Online gaming payment gateway. 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.

Texas Holdem Flush Probability Cheat

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. Used slot machine for sale. 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.

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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.

By Gerald Hanks

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.

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.

Texas Holdem Flush Probability Calculator

Texas Holdem Flush Probability Chart